Book 4

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양=주식 울타리=콜옵션 ...일 때, delta의 공식은? hedge 방법은?

1/Δ =콜(울타리)/주식(양) - 1/Δ=0.5 means that 0.5 call option is needed to hedge 1 share of stock. *뭘 long/short 할건진 자유 (양과 울타리가 서로 반대 포지션만 취하면 됨)

What is the realised return for a bond that is currently selling for $112 if it was purchased exactly one year ago for $105, paid a $2 coupon today, and paid a $2 coupon six months ago? Assume the coupon received six months ago was reinvested at an annual rate of 1%.

10.49%

2년만기 CB를 찢는다면?

1년만기 ZCB + 2년만기 ZCB = 2년만기 CB

Hazard rate을 활용해서 unconditional default probability를 계산하는 방법은? (공식)

1차년도 말의 부도율: 1-e^(-h1*t1) 2차년도 말의 부도율: e^(-h1*t1) - e^(-h2*t2) *h=average continuous hazard rates

Coupon effect란?

2 bonds with identical maturity but different coupon rates will have different interest rate sensitivity. - Smaller coupon rate = higher IR sensitivity

Most common key rates are? (4개) Where are they from?

2, 5, 10, 30년 rates, from US Treasury & related markets (most liquid bonds available; 위험 스프레드 없는 가장 기본 국채 등)

Sovereign credit risk) Foreign currency defaults 관련 제일 큰 사건은?

2012년 3월 그리스 사건 - $260 million 부도 & 투자자 손실 70% - 이 사건 외에도 2010-16년 사이에 8 차례 발생)

What is the estimate for the percentage price change in bond price from a 25 basis point increase in rates on a bond with a duration of 7 and a convexity of 243? A. 1.67% decrease. B. 1.67% increase. C. 1.75% decrease. D. 1.75% increase

With an initial PF value of 100.565 and face value of 100, the forward-bucket duration for a forward bucket '01 of 0.0095 is closest to: A. 0.9447. B. 0.9500. C. 94.47. D. 95.00.

A $1,000 par value US corporate bond pays a semiannual 10% coupon. Assume the last coupon was paid 90 days ago and there are 30 days in each month. Compute the accrued interest.

AI = $50*(90/180)=$25 (*Semiannual = 180 days)

Euler's theorem이란?

Allows a PF's homogeneous risk functions to be decomposed into their component contributions. - Determining the contribution of each underlying loan to the overall loan PF risk. (위험기여도 측정)

American option vs. European option - 둘 중 더 비싼 것은? 그 이유는?

American option이 더 비싸고, 그 이유는 조기행사권의 가치까지 포함하기 때문이다.

Carry Roll Down 시나리오의 가장 큰 특징은?

Assumes no change in IR term structure.

A PF consists of three options. Option 1 has a weighting of 20% and a delta of 0.75, Option 2 has a weighting of 35% and a delta of 0.45, and Option 3 has a weighting of 45% and a delta of 0.60. The PF delta is closest to: A. 0.27. B. 0.58. C. 0.60. D. 1.80.

A risk manager is looking to determine the contribution of each underlying loan to the overall loan PF risk. The risk manager should use: A. CreditMetrics. B. Euler's theorem. C. the Gaussian copula. D. the Vasicek model.

Suppose a 1-year European call option exists on XYZ stock. The current continuously compounded risk-free rate is 3% and XYZ does not pay a dividend. Assume an annual standard deviation of 8%. The risk-neutral probability of an up-move for the XYZ call option is: A. 0.31. B. 0.69. C. 0.92. D. 1.08.

The longer the time horizon, the higher the incidence of default for a given rating. This effect: A. is equal for low-rated and high rated bonds. B. is stronger for low-rated bonds than for high-rated bonds. C. is stronger for high-rated bonds than for low-rated bonds. D. has not been studied enough to be documented.

The parameters of a GARCH (1,1) model are ω=0.0003, α=0.04, and β=0.92. If daily volatility is estimated to be 1.5% and today's stock market return is 0.8%, what is the new estimate of the SD? A. 1.68% B. 1.55% C. 1.45% D. 2.74%

The present value of a 2-year, $1,000 bond with an annualised interest rate of 3% compounded monthly is closest to: A. $940.00. B. $941.84. C. $941.98. D. $942.60.

Which of the following statements regarding stress scenarios is incorrect? A contagion effect: A. results from a crisis event. B. increases diversification benefits. C. occurs with a rise in both volatility and correlation. D. causes a different return generating process in the underlying asset.

Duration vs. Convexity 각각 언제 더 useful?

[Duration] - Price change for relatively small & parallel changes in IR - Linear estimate (IR이 움직이는 방향이 +/- 상관 없이 price 변화폭 동일) [Convexity] - Price change for relatively big changes in IR - Measure of curvature between the changes in IR and bond price

With respect to the effect on the price of a bond, the effect of a bond upgrade will: A. be positive and stronger than the downward effect of a bond downgrade. B. be positive and weaker than the downward effect of a bond downgrade. C. have about the same negative effect, in absolute value terms, as a bond downgrade. D. be negative and about equal to that of a bond downgrade.

ρ(X+Y)≤ρ(X)+ρ(Y) is the mathematical equation for which property of a coherent risk measure? A. Monotonicity B. Subadditivity C. Positive homogeneity. D. Translation invariance.

Which of the following maturities is least likely associated with a key rate for US Treasuries? A. 2-year B. 10-year C. 15-year D. 30-year

예상주가 (expected value) 공식은?

E(St) = So*e^(μ*t) - μ = expected annual return (Uses the property of log-normal distribution.)

Calculate expected loss if a bank expects 1.8% default rate on its loans assuming the recovery rate in the event of default is 60%.

EL=1.8%*(1-0.60)=0.72%

Expected loss 공식은? Unexpected loss 공식은?

EL=EaD*PD*LGD UL=EaD*(WCDR-PD)*LGD - WCDR: worst case default rate; 99.9 percentile

Expected(mean) loss의 공식은?

EL=EaD*PD*LGD=EaD*PD*(1-RR)

KR01 은 무엇?

Effect of a dollar change of a 1bp shift around each KR.

(중요) Delta neutral hedging은 뭘 얼마나 사고 팔아야 하는가?

To completely hedge (delta neutral hedging) a short call position(대상; 주가 오르면 손해), an investor must purchase the # of shares of stock(수단; 주가 오르면 이득) equal to [delta x # of 대상 sold]

Risky 세상이나 risk-free 세상이나 차익거래 발생하면 안된다는 조건은 동일하다. True or false?

True

Option valuation) binomial model에서 Up-/down-factor들 공식은?

U=e^(σ*√t) D=e^(-σ*√t)=1/U (역수의 관계) *t=the length of the step in the binomial model (몇년 만기?) *σ=the annual volatility of 기초자산 return

What is the net realised return for a bond that is currently selling for $112 and paid a $2 coupon today if its purchase price of $105 was entirely financed at an annual rate of 0.6% exactly six months ago?

[(112+2-105)/105]-(0.6/2) = 8.27% *(0.6/2) = per period financing costs

Conditional distribution vs. unconditional distribution 비교하기!

[Unconditional distribution] - ND x - 시간의 개념 무시 (어떤 날이건 asset return의 mean & SD는 항상 동일) - 단, 비록 mean & SD가 같아도 각기 다른 시점의 샘플들을 추출하는 것은 fat tail을 유발 할 수 있음. [Conditional distribution] - ND 가정하지만, 샘플들의 mean & SD는 다름

Warrant 발행이 야기하는 기업의 손실은?

[Warrant 발행 개수 x Warrant 가치] *가치 구하는 공식은 캡처본 참고

회사채와 국채 중 더 유동성이 높은 것은?

국채

만약 cash price based on the bid & cash price based on the ask 두개가 주어졌다면 discount factor는 뭐임?

그 두 cash price들의 midpoint!

Vega란? e.g. 이게 만약 8이면 어떻다는거?

기초자산의 변동성 가정을 조금씩 바꾸면 옵션가격 얼마나 변동하는지!! e.g. Vega=8 : 변동성 1% 상승 -> 옵션가격 0.08 상승

Why is naked position risky?

기초자산이 없으면 사서 줘야 하니까, maximum potential gain은 capped at the level of premium. 반면 loss는 주가가 미친듯이 오를 수도 있기 때문에 unlimited.

PF delta 계산하는 방법은?

개별옵션의 Δ들 가중평균

Price-yield curve는 어떤 모양?

Bond PV와 YTM의 inverse & convex relationship

All rates can be determined as a function of a few _____.

Key rates.

SD of loss 공식은?

LGD*√{PD*(1-PD)}

Foreign currency debt와 local currency debt 중 신용등급이 높은 쪽은?

Local currency debt rating is often few notches higher.

Spread와 신용등급 중 신용 상태 판단에 더 용이한 것은? 이유는?

Spread (이유: more dynamic)

Key rates은 어디서 활용?

Used for measuring & hedging risk in bond PF.

BSM) European options with dividends - 배당=2%, continuous - 현재 주가=$50 - 콜옵션 행사가격=$45 - 잔여만기=3개월 - 무위험수익률=5% - Annualised SD of returns=12% Q. Dividend paying stock의 콜옵션 가격은?

주가: 그냥 So 말고 So*e^(-q*t)로 계산해야 함 - 50*e^(-0.02*0.25) = $49.75 위 주가 대입해서 d1, d2 구하고, 각각의 누적확률 N(d1), N(d2) 구해서 BSM에 대입하기~

IR나 credit spread 같은 risk factor가 더 모니터하기 어려운 이유는? 대안은?

주로 set of points(term structure)로 설명되기 때문. Popular한 만기의 YTM은 알 수 있지만, 독특하고 레어한 만기의 YTM은 not observable. 대안: "Linear interpolation" - Looking at the linear distances between 2 observable IRs in the term structure.

Put-call parity가 없다면 _____가 가능하다.

차익거래

Warrants란?

채권에 더해서 발행되는 신주인수권, 이거만 똑 떼서 발행 가능! (give the holder the right to purchase shares of a security at a stated price)

Duration = 0 의 의미는?

충격이 와도 PF 가치 불변이라는 의미

Bond valuation 하려고 계산기 사용할때 N이란?

쿠폰 총 몇번 주는지 (semiannual하게 10년이면 N=20)

DV01 가지고 hedging) [Hedge 대상] - DV01 = $340 - PF 총 가치 = 1 million [Hedge 수단] - DV01 = $285 - 대상 전체를 hedge 하려면 얼마나 필요한가?

1 million * (대상 DV01) = ? * (수단 DV01) ∴ ? = $1,192,982.46 만큼 홀드하면 대상 전체 hedge 가능!

Assume.a stock is currently priced at $25 with an expected annual return of 20%. Calculate the expected value of the stock in six months.

E(St) = $25 * e^(0.2*0.5) = $27.63

*문제 이상해 (1) 2년만기 CB (YTM=coupon rate=8%) (2) 1년만기 ZCB (YTM=4%, coupon rate=0%) (3) 2년만기 ZCB (YTM=8%, coupon rate=0%) 이렇게 있을 때, arbitrage 거래 하려면?

2년만기 CB 사서, strip 해서, 나눠진 두개의 ZCB를 팔기!

신용평가 agency의 key goal은?

Ratings stability (그래서 they only adjust ratings if there has been a long-term change in 의뢰회사's overall credit worthiness; 같은 맥락에서 through the cycle 방식을 선호)

Consider a PF that has the following asset returns: 5%, -4%, 9%, 6%. Calculate the return realised by this PF.

Realised PF return = [(1.05 * 0.96 * 1.09 * 1.06)^(1/4)] - 1 =3.9%

BSM model의 분포 가정은?

Stock prices are log-normally distributed. (lnSt ~ N(a,b) 해석: Since St is normally distributed, 95% of the values will fall within 1.96 SDs of the mean a.)

투기등급 회사들의 default probability가 시간이 지날수록 감소하는 이유는?

Surviving the first year of two represents an increased likelihood of an improvement in financial health. (1, 2차년도에도 살아남았으면 그 후에도 살 가능성이 크다는 것)

Forward bucket '01s가 활용되는 context는?

Swap & combination of bond/swap contexts

Historical data 이용해서 longer time periods로 scale 하는 방법은?

T 곱하기. (SD의 경우 Variance로 돌아가면 √T 를 곱한 셈이 됨!) - 기간이 짧다면 표준편차가 더 큰 영향 미침 - 기간이 길다면 평균의 영향이 더 큼

Homogenous expectation이란?

The means, SDs, and the correlations between investment returns are consistent from the perspective of all investors. (지배원리 성립, 지배원리가 뭔지는 캡처본 참고)

Due to systemic risk, defaults tend to _____.

cluster

Suppose that Stock XYZ is trading at $50, and there is a call option that trades on XYZ with an exercise price of $45, which expires in three months. The risk-free rate is 5%, and the SD of returns is 12% annualised. Determine the value of the call option's delta.

d1 = 1.99 (캡처본 참고) N(1.99) = delta = 0.9767 ∴ When the stock price changes by $1, the option price will change by 0.9767.

For a $100,000,000 PF, the expected 1-week PF return and SD are 0.00188 and 0.0125, respectively. Calculate the 1-week expected shortfall with a 95% confidence level.

This amount is larger than the VaR level calculated earlier of $1,874,500.

The mean-variance framework is unreliable when the assumption of _____ is not met.

normality

The parameters of a generalised autoregressive conditional heteroskedastic [GARCH (1,1)] model are ω=0.000003, α=0.04, and β=0.92. If daily volatility is estimated to be 1%, and today's stock market return is 2%, calculate the new estimates of volatility using the GARCH (1,1) model, and the implied long-run volatility level.

√(0.000075) = 0.866% = Long-run volatility

Assuming continuously compounded spot rates of 4.25 for 3 years and 4.40% for 3.5 years, calculate the forward rate for the period between Year 3 and Year 3.5.

(0.0440*3.5 - 0.0425*3.0)/(3.5-3.0) = 0.530, or 5.30%

What is the challenge in assessing credit risk? 해결방안은?

"Calculating required capital for banks include calculating the EaD (Loan의 EaD는 계산하기 쉽지만, 장외 파생상품의 EaD는 자주 변해서 계산이 어려움)." 해결방안: Basel rules set EaD for derivatives. <Current exposure + add-on amount> (1) Current exposure: amount that the bank could lose if its counterparty defaulted (2) Add-on amount: BUFFER if the exposure became worse by the time of the default (지금은 이 정도지만, 앞으로 얼마나 더 커질 수 있는지)

Historical-based approach란 어디서 사용되나? 그 안에서 크게 두가지로 나눈다면?

"Conditional distribution is estimated based on historical time series data." (1) Parametric approach - Requires specific assumptions regarding the asset return distribution (normal/log-normal, time-varying volatility) - 장점: data is more efficiently used. (샘플 가지고 모집단 추정하는거니까) (2) Non-parametric approach - Less restrictive (no assumptions of the asset returns distribution) -> 따라서 fat tails, skewness, other deviations 걱정할 필요 없음. - 예시: HS, MDE - 공통된 단점: large sample sizes required to precisely estimate.

Greek들을 활용한 hedging의 가장 큰 단점은? 보통 가장 많이 사용하는 지표들은?

"Expenses" associated with trying to maintain positions that are neutral to the Greeks. 따라서 일단 delta-neutral 만들고 나머지는 걍 monitor하는 걸로 만족..

Operational risk를 두개의 dimension으로 나누어 측정하는 방식은? 이때 사용하는 데이터 수집 방식은?

"Loss distribution approach" - Operational loss = Frequency x Severity - MC simulation 활용해서 데이터 수집 (1) Frequency (포아송 분포) - 공식은 캡처본 참고 - λ = expected(average) number of losses over a given time horizon (2) Severity (로그노말 분포: asymmetrical & fat-tailed) - 공식은 캡처본 참고 - e.g. 99th percentile = loss greater than 99% of the distribution's data - UL (at 특정 CL) = Loss at selected percentile - mean loss

Point-in-time 방식의 credit rating이 가지는 단점은?

"Procyclicality" (경기순응성) - 경기가 나쁠 때는 더 나쁘게, 좋을 때는 더 좋게 평가함. - e.g. 대출 증가, 기업 성과 굿, 경기 더 좋아짐, 대출 증가, ...

Stress testing validation 과정의 한계점은?

"The reality that stress tests represent rare events." *Rarity: correlation은 높고, RR은 낮은 상황!

Worst-case scenario analysis는 무엇이며, VaR이나 ES의 대안이 될 수 있나?

"Worst possible outcomes given an unfavourable event - an EL is then determined from it." Useful, but not an alternative of VaR or ES.

Calculating required capital must be done on a _____ basis. 이유는?

"counterparty basis" 이유: Unlike loans, derivatives are subject to netting agreements. Netting이 있으면 all transactions with a single counterparty는 무조건 single transaction. 따라서 거래가 몇개고 무엇인지 말고, 거래상대방이 누군지가 중요.

CreditMetrics model이란? Vasicek model과의 차이점은?

"금융기관에서 자체적으로 신용위험/자기자본 계산; 부도 위험 말고도 신용등급의 변화 가능성 역시 credit loss distribution으로 assess" - Borrowers are assigned both internal & external ratings. [Vasicek model과의 차이점] - Vasicek: 공식 사용 / credit risk = default risk - CreditMetrics: Gaussian Copula 활용한 MC simulation 사용 / credit risk = default risk + downgrade risk

Coherent risk measures란? 종류 4가지 대기

"범용성/일관성" 위한 위험척도 (1) Monotonicity - A PF with greater future returns will always have less risk. - R1 > R2, then ρ(R1) > ρ(R2) (2) Subadditivity (제일 중요) - The risk of a PF is at most equal to the risk of the assets within the RF. (분산효과) - ρ(R1+R2) ≤ ρ(R1) + ρ(R2) (3) Positive Homogeneity - 유동적 시장에 있다면 성립해야 하는 조건 - The size of a PF, β, will impact the size of its risk for all β>0. - ρ(βR) = βρ(R) (4) Translation invariance - PF risks relies on assets within the PF. - For all constants c(cash; 안전자산), ρ(c+R)=ρ(R)-c

GARCH model이란?

"장기평균이 있다고 가정하는 것" (EWMA는 이러지 않음) - 장점: it implicitly assume that variance tends to revert back to a 장기평균수준. - 특징: [w+α+β=1, α+β<1] for stability so that γ is not negative. *EWMA is nothing other than a special case of GARCH(1,1) with [w=0, α=(1-λ), β=λ].

Default probability를 계산할 때, unconditional percentage란? 특징은?

"출발선 대비" 몇% 부도 났는지. - 투자등급(AAA~BBB): 시간이 지날수록 부도율 증가 - 투기등급 (BB~CCC/C): 시간이 지날수록 부도율 증가하다가 감소

An invest has a $3 million investment with a duration of 6 and convexity of 25. The investor is looking to hedge the investment using two bonds: Bond 1 has a duration fo 7 and convexity 20, and Bond 2 has a duration of 5 and convexity of 19. Calculate the face amount of the bond required to hedge the investor's position.

$1 million long position in Bond 1 $5 million short position in Bond 2 *This ensures that the investment will be hedged against even larger parallel changes in rates.

If the observed loss for Bank Z is $5 million and it has $1 million in revenue, what will be the estimated loss size adjustment for Bank Y, which has revenue of $2 billion?

$5,864,175 Notice that this loss is much less than the proportional estimate of a $10 million loss given that Bank Y has twice the revenue.

(1) Bond 가격의 interest rate sensitivity 구하려면? (2) IR term structure 움직이는 형태는? ...single vs. multi-factor approach

(1) - Single factor approach: 1 risk factor = YTM - Multi-factor approach: multiple risk factors = 각 시점의 spot rates (S1, S2, ...) (2) - Single factor approach: the term structure shifts in a parallel fashion. (각기 다른 만기의 채권들이 있을 때, 1년 만기 채권 가격에 영향을 미치는 1년짜리 YTM의 변화는 다른 만기의 채권 가격에도 영향을 미침) - Multi-factor approach: the term structure shifts in a non-parallel fashion. (금리구조가 움직이는 이유는 각각의 금리들이 다 따로따로 움직이기 때문)

(중요) (1) Suppose that the stock of Vola, Inc., is trading at $50, and there is a call option on Vola available with an exercise price of $45 that expires in three months. The continuously compounded risk-free rate is 5%, and the annualised SD of returns is 12%. Using the BSM model, calculate the value of the call option. + Put option value 역시 계산하기) (2) 같은 조건, but the stock pays a continuous dividend yield of 2%.

(1) d1 = 1.99 (캡처본 참고하기) d2 = d1 - (0.12*√0.25) = 1.93 *Partial cumulative ND table에서 위 값들 찾기 N(d1) = 0.9767 N(d2) = 0.9732 C0 = [$50*0.9767]-[$45*{e^(-0.05*0.25)}*0.9732] = $5.59 + Put-Call Parity 사용하여 put option value 계산하기 P0 = $5.59 - $50.00 + [$45.00 * e^-(0.05*0.25)] = $0.03 (BSM put formula 사용해도 됨) ===== (2) The adjusted price of the stock: e^(-0.02*0.25)*$50/00 = $49.75 d1 = 1.91 (위 캡처본 "50" 자리에 "49.75" 대입) d2 = d1 - (0.12*√0.25) = 1.85 *Partial cumulative ND table에서 위 값들 찾기 N(d1) = 0.9719 N(d2) = 0.9678 C0 = [$49.75*0.9719]-[$45*{e^(-0.05*0.25)}*0.9678] = $5.34 + Put-Call Parity 사용하여 put option value 계산하기

[PF 조건] - 만기=3년 - 액면가=100 - 쿠폰=4.5, semiannual - IR term structure = flat, 3.5% What are: (1) the value of the bond before any shifts in the IR? (2) the value of the bond after shifting each bucket for 1bp? (1 year buckets) (3) the forward buckets 01'?

(1) (2.25/1.0175) + (2.25/1.0175^2) + (2.25/1.0175^3) + (2.25/1.0175^4) + (2.25/1.0175^5) + (2.25/1.0175^6) + (102.25/1.0175^6) = 102.8245 (2) - Shifting bucket 1: (2.25/1.01755) + (2.25/1.01755^2) + (2.25/1.01755^2*1.0175) + (2.25/1.01755^2*1.0175^2) + (2.25/1.01755^2*1.0175^3) + (102.25/1.01755^2*1.0175^4) = 102.8145 - Shifting bucket 2: (2.25/1.0175) + (2.25/1.0175^2) + (2.25/1.0175^2*1.01755) + (2.25/1.0175^2*1.01755^2) + (2.25/1.0175^3*1.01755^2) + (102.25/1.0175^4*1.01755^2) = 102.8148 - Shifting bucket 3: (2.25/1.0175) + (2.25/1.0175^2) + (2.25/1.0175^3) + (2.25/1.0175^4) + (2.25/1.0175^4*1.01755) + (102.25/1.0175^4*1.01755^2) = 102.8153 (3) - Forward Bucket '01 #1: 102.8245 - 102.8145 = 0.01 - Forward Bucket '01 #2: 102.8245 - 102.8148 = 0.0097 - Forward Bucket '01 #3: 102.8245 - 102.8153 = 0.0092 * 위에를 다 더하면 0.0289, which is the same as assuming all forward rates rise by 1bp. (2.25/1.01755 + ... + 2.25/1.01755^5 + 102.25/1.01755^6 = 102.7957, 102.8245-102.7957 = 0.0289)

BSM model 성질) (1) N(-d1) = ? (2) N(-∞) = ? (3) N(∞) = ?

(1) 1-N(d1) *좌우대칭인 성질 때문 (2) 0 (3) 1

(1) Duration으로 barbell PF 속 투자비중 계산하기 - 단기채: D=4.12 - 중기채: D=7.65 - 장기채: D=14.93 (2) 위에서 계산된 투자비중 활용해서 PF의 convexity 계산하기 - 단기채: C=21.9 - 중기채: C=59.8 - 장기채: C=310.5

(1) 4.12w1 + 14.93(1-w1) = 7.65 - w1 = 0.6735 - w2 = 0.3265 ∴ 단기채에 67.35% 투자, 장기채에 32.65% 투자 (2) 0.6735*21.9 + 0.3265*310.5 = 116.1 - Barbell PF's convexity (116.1) > Bullet PF's convexity (59.8) - Higher convexity is beneficial, because it improves the investor's position for parallel changes in IR ∴ Barbell PF가 bullet PF보다 유리

(1) SD of loss 공식은? (2) SD of the loss on a PF of loans 공식은? (3) If all loans have the same principal L and the SD of the loss from loan i is the same for all i, SD may simplify to? (4) 2번에 대한 답을 전체 PF size에 대한 %로 나타내면?

(1) = LGD*√{PD*(1-PD)} (2) ? (3) σ={√(PD-PD^2)} * [L(1-RR)] (4) 캡처본 참고

(1) A European put option has the following characteristics: S0=$50; X=$45; r=5%; T=1 year; and σ=25%. Which of the following is closest to the value of the put? A. $1.88. B. $3.28. C. $9.06. D. $10.39. (2) 그렇다면 value of the call? A. $1.88. B. $3.28. C. $9.06. D. $10.39.

(1) A (2) C [풀이] 다 같은 조건이지만, N(d1) = 0.7731 N(d2) = 0.6915 C = [50*0.7731] - [45*e^(-0.05)*0.6915] = 9.055

(중요) A delta-neutral position exhibits a gamma of -3,200. An existing option with a delta=0.50 exhibits a gamma of 1.5. (1) Which of the following will generate a gamma-neutral position for the existing PF? A. Buy 2,133 of the available options. B. Sell 2,133 of the available options. C. Buy 4,800 of the available options. D. Sell 4,800 of the available options. (2) Which of the following actions would have to be taken to restore a delta-neutral hedge to the gamma-neutral position? A. Buy 1,067 of the available options. B. Sell 1,067 of the available options. C. Buy 4,266 of the available options. D. Sell 4,266 of the available options.

(1) A (풀이: 3200/1.5=2,133 options) (2) B (풀이: Buying 2,133 options will increase the delta to 2,133*0.5=1,067. Therefore, one has to sell 1,067 shares to maintain a delta-neutral position.)

Monte-Carlo simulation의 장점 2가지는?

(1) Able to address multiple risk factors by assuming ND distribution & modeling the correlations among risk factors. (Risk factors 간의 ρ 알면 PF의 VaR 알 수 있기 때문; 캡처본 참고) (2) MC simulation은 risk factors 간의 ρ만 계산 가능하다면 ND 말고 다른 분포도 가정 가능! *핵심: 따라서 historical simulation보다 더 많은 시나리오를 생성하고 테스트 할 수 있음!

Day count conventions 3가지와 그 특징들 대기!

(1) Actual/actual : 가장 많이 사용, 미국채 (2) Actual/365 : Money market securities 사용, 캐나다/호주/NZ (3) 30/360 : 단기금리채시장, 미국 회사채/municipal bonds

Principal components analysis의 특징 3가지는?

(1) Analysing term structure movements in historical data. (2) Based on daily movements in rates across maturities. (3) Identifying uncorrelated factors.

Insurance의 문제점들 해결하려고 보험사가 내놓는 방안들은?

(1) Moral hazard - Deductibles - Policy limits - Coinsurance provisions (2) Adverse selection - Trying to understand each firm's internal controls -> auto-insurance처럼 premiums can be adjusted to adapt to different situations with different levels of risks. - 따라서 사전에 위험고지 하지 않은 손실에 대해서는 보상X

Operational risk의 규제자본이 얼마나 필요한지 determine하는 4가지 방식은?

(1) Basic indicator approach (기초접근법) - 15% of the bank's 3년 평균 annual gross income (interest earned + non-interest income - interest paid) (2) Standardised approach (기초접근법보다 더 세분화) - 회사 내 business unit 고려 (총 8개; 캡처본 참고) - 어떤 BU인지에 따라 가중평균 시 weight이 다름. (3) Advanced measurement approach (AMA) *(1)&(2)과 다르게 개별 은행의 독자적인 운영위험 역사나 내부적 관리능력 감안 - Banks want to take advantage of the possible lower capital requirements. - Internal criteria (qualitative, quantitative) 활용 - 단, 특정 조건들을 만족시키는 은행들만 이 방식 활용 가능! (UL 추정할 수 있어야 함) - Loss distribution approach: [Regulatory capital = loss at 99.9% CL - expected operational loss] (4) Standardised measurement approach (SMA) *AMA의 문제: large variability among banks in calculating risk capital -> 그래서 바젤이 16년도에 SMA발표. - Business indicator (BI): 은행의 영업사이즈 (이자손익 말고도 trading losses & operational expenses 고려한다는 점에서 다른 접근들과 차별화.) - Loss component (LC; operational loss exposure): 이거로 은행의 "internal loss multiplier (ILM)" 계산 - [Regulatory capital = BI*ILM] *BI>LC : ILM<1 (통제 잘했다는 의미) *BI<LC : ILM>1 *BI=LC : ILM=1

(1) Assume the stock price is currently $80, the stock price annual up-move factor is 1.15, and the risk-free rate is 3.9%. The value of a 2-year European call option with an exercise price of $62 using a two-step binomial model is closest to: A. $0.00. B. $18.00. C. $23.07. D. $24.92. (2) 같은 조건으로 put option이라면? A. $0.42. B. $16.89. C. $18.65. D. $21.05.

(1) C (캡처본 참조) (2) A [풀이] Put = call - stock + (exercise price * e^(-rT)) = $23.07 - $80 + [$62 * e^-(0.039*2)] = $0.42

Bond return decomposition하면 나오는 요소 3가지 말하기!

(1) Carry roll down (2) Rate change (3) Spread change

BSM의 가정 8가지는?

(1) Continuity - Trading is continuous. - BSM values options in continuous time. (2) No arbitrage - Put-call parity 성립 - P(시장)-P(이론) = C(시장)-C(이론) (3) Log normal distribution - 기초자산의 가격은 위 분포를 따름. - 장점: minimum of 0 (그냥 ND는 음수 가능, 비현실적) (4) Risk-free - The continuous risk-free rate is constant, known, & always available for borrowing/lending. (5) 기초자산의 volatility (캡처본 참고) - Volatility is constant and known. (비현실적) - Option values depend on volatility and IR. (6) Frictionless markets - No taxes, transaction costs, restrictions on short sales or the use of short sale proceeds. (7) Underlying assets have no CFs (dividends, coupons, etc.) (8) Only European options! - BSM에 American option은 적용 불가!

Stress testinig의 2가지 variables는?

(1) Core variables (2) Peripheral variables

Duration = f(1, 2, 3) 1, 2, 3 각각의 개념과 duration에 미치는 영향 대기!

(1) Coupon effect: inverse relationship (2) Maturity effect: 만기 길수록 duration 상승 (3) Initial yield level effect: 금리 낮을수록 duration 상승

Credit loss 해결방안 구체적으로 분류해서 말하기

(1) Credit reserve (대손충당금): EL 커버 (2) Capital reserve (자기자본; (1)번 이상으로 써야하면 사용): UL 커버 - Economic capital (은행 자기자본) - Regulatory capital (감독기관-바젤-에서 지시한 양)

"Large" operational risk 3가지는? 각각의 가장 큰 예시 및 특징들은?

(1) Cyber risks - 2011년 야후 사건: 3백만 유저의 개인정보 유출 - 2016년 방글라데시 중앙은행 사건 - 2017년 Equifax 사건: 143백만 유저의 개인정보 유출 (2) Compliance risk (unintentional; 규제에 걸린다는걸 몰랐음) - Can result from a small part of an organisation's global activities but can lead to hefty fines. - 솔루션: Designing adequate software & instituting internal training - HSBC: 벌금 물림 (lack of adequate anti-money laundering programmes) - BNP Paribas: 벌금 물림 (transacting with sanctioned countries) - Volkwagen: 벌금 물림 (cheating on emissions tests) (4) Rogue trader risk

VaR) SMA로 volatility 구하는 것의 단점 2가지는?

(1) Cyclical volatility + large n 이란 조건이 있으면 잘못 추정 가능. (따라서 장기평균 구할 때, 평탄화 시 유리한 모델) (2) Largely negative/positive returns from long ago could unduly impact the model. (시장의 극단적인 사건은 부적절한 영향 끼침; Echo effect, 큰 사건이 시간이 흘러도 동일한 비중으로 계속 영향 미치는 거)

Maturity/Coupon/Price 6 months/5.5%/101.3423 1 year/14.0%/102.1013 2 years/8.5%/99.8740 (1) Which of the following is the closest to the discount factor for the 6-month discount factor, d(1)? A. 0.8923. B. 0.9304. C. 0.9525. D. 0.9863 (2) Which of the following is the closest to the discount factor for the 1-year discount factor, d(2)? A. 0.8897. B. 0.9394. C. 0.9525. D. 0.9746.

(1) D (2) A

Suppose that a bank has a PF with 50,000 loans, and each loan is USD 1 million with a 1.1% PD in a year. The recovery rate is 40% and the correlation between loans is 0.2. Assume that L=1. (1) The SD of the loss from the loan PF is closest to: A. 0.01100. B. 0.01088. C. 0.04172. D. 0.06258 (2) The SD of the loss from the loan PF percentage of its size is closest to: A. 0.015288. B. 0.027988. C. 0.041622. D. 0.055975.

(1) D (2) B

An investor has a short position valued at $100 in a 10-year, 5% coupon bond with a 7% YTM. Assume discounting occurs on a semiannual basis. (1) Which of the following is closet to the dollar value of a basis point (DV01)? A. 0.033. B. 0.047. C. 0.056. D. 0.065. (2) Using a 20-year T-bond with a DV01 of 0.085 to hedge the interest rate risk in the 10-year bond with a DV01 of 0.065, which of the following actions should the investor take? A. Buy $76.50 of the hedging instrument. B. Sell $76.50 of the hedging instrument. C. Buy $130.75 of the hedging instrument. D. Sell $130.75 of the hedging instrument.

(1) D - 캡처본 참고 (2) A - Hedge ratio: 0.065/0.085=0.765 - Investor has a short position in the bond, meaning that the investor needs to buy $0.765 of par value of the hedging instrument for every $1 of par value for the 10-year bond.

DV01 vs. Duration 비교하기!

(1) DV01=$ vs. Duration=% (2) 각각 언제 convenient? - DV01: Trading or derivatives hedging (양쪽의 $ 금액이 다르기 때문에 %가 무의미) - Duration: Investing (high D would alert the investors) (3) PF 계산법 - DV01: 선형결합 (다 더하기) - Duration/convexity: 가중평균 (투자비중 반영)

VaR methods 총정리 하면? 그 중 parametric/non-parametric 한 것은? 그간의 차이점은?

(1) Delta-Normal: parametric (σ 알아내서 공식에 대입) - BSM의 implied volatility - Simple moving average (캡처본 참고) - EWMA - GARCH *Delta-Gamma 역시 존재. (2) Simulation 가족 (손실 data 모아서 CL 적용) -Historical simulation : non-parametric -Multivariate density estimation (MDE) : non-parametric -MC simulation : parametric *똑같이 parametric이지만 delta-normal과 MC simulation의 차이점: DN은 단순 곱의 계산, MC는 full evaluation 및 계속 price function으로 re-evaluate 가능.

Expected shortfall vs. VaR 3가지 부분에서 비교해보기

(1) ES는 VaR과 다르게 subadditivity를 만족시킴. (사실 ES는 모든 coherent risk measures 다 충족.) 따라서 when adjusting the holding period & the CL, an ES surface curve showing the interactions of both adjustments is convex. (2) ES는 loss magnitude를 제공하는데, VaR은 loss probability만 제공. (4) ES는 VaR보다 덜 restrictive한 가정들을 가짐. +캡처본 참고

Operational risk의 data limitation 2가지는?

(1) Estimation procedures - Operational risk의 historical loss data는 현재 부적절해서, 기업들은 database 열심히 쌓는 중! - Operational risk 특성상 "Objective data + subjective decision"의 조합이 필요함 - Frequency는 internal data만, severity는 internal+external 둘 다 사용하는게 좋음! (2) Potential bias - Vendor data is less reliable (inherent reporting bias; severity 측정에는 유리) - 보통 loss data는 내부통제가 약한 기업의 것이기 때문에 자연적으로 빈도가 높음. 그래서 frequency 관련해서는 우리 회사 데이터 사용하는게 유리!

Stress testing의 두가지 접근 방법과 각각의 장단점 대기!

(1) Examining historical crisis events -장점: no assumptions of 기초자산 return/normality needed -단점: limited to only events that have actually occurred (2) Analysing different predetermined stressed scenarios (정형화된 시나리오 미리 형성해두기) - 장점: not limited to the evaluation of risks that have actually occurred.

External ratings 대신 internal ratings를 사용하는 3가지 이유는?

(1) External ratings may not be available. (2) Accounting standards requires banks to account for default probabilities when they value loans on their B/S. (등급이 존재하지 않는건 내부적으로 줘야 함.) (3) PD drive regulatory credit risk capital. (신용위험 노출 정도에 따라 규제 상환량이 상이.)

Principal components analysis) Factor들의 분산 관련 take-away 3가지는?

(1) Factor들의 분산 다 더하면 결과값 ΔY의 분산과 일치! (2) 각 factor의 분산이 결과값 ΔY의 분산의 몇%인지에 따라, 어떤 영향이 제일 중대한지 파악 가능 (해당 factor가 다양한 만기의 rates을 어떤 방향으로 움직이는지, +/- 일관적인지 고려하기) (3) 이때, 장단기금리가 다른 방향(+/-)으로 움직이면 Twist, 장단기금리는 같은 방향인데 중기금리만 다른 방향으로 움직이면 Butterfly.

Operational risk management의 종류 4가지는?

(1) Forward-looking approaches - 실수로부터 배우기 - 미래 potential risk에 대비하기 - Loss의 인과관계와 상관관계 분석하기 - RCSA - Key risk indicators (KRIs; 위험조기경고지표: employee turnover rate, number of failed transactions, etc) - Educating employees (3) Allocating operational risk capital - Less capital is allocated to BUs that are able to reduce the 빈도&강도 of risks. (이러면 return on invested risk capital 지표 개선) (4) The Power Law (캡처본 참고) - Operational risk는 tail에서 발생하니까 중요한 개념! - Top 5%의 x값들에 대부분 적용 가능 (5) Insurance - 문제 1: Moral hazard (사후적 개념) - 문제 2: Adverse selection (사전적 개념)

Senior management가 이사회에게 보고해야하는 3가지는?

(1) Governance (2) Validation (ST로 구한 값이 맞냐 X, 옳은 절차를 따랐냐 O) (3) Independent review of ST 여부

Operational risk management를 위한 key risk indicator가 되기 위한 조건은?

(1) Have a predictive relationship to losses (2) Be accessible and measurable in a timely fashion

(1) Delta (2) Gamma (3) Theta (4) Vega (5) Rho 각각 언제 제일 큰지 moneyness로 설명하기!

(1) ITM (2) ATM (3) ATM (4) ATM (5) ITM

CDS contract의 한계점은?

(1) Illiquid (2) Buyers are exposed to the default risk of the protection(contract) seller

Operational risk 7가지는?

(1) Internal fraud (허위보고, 횡령 등) (2) External fraud (제 3자가 저지른... robbery, hacking 등) (3) Employment practices & workplace safety (4) Clients, products, & business practices에게 의무를 다하지 못하는 것 - Mishandling confidential information - Breaches in fiduciary duty - Money laundering 등 (5) Damage of physical assets (테러, 지진 등) (6) Business disruption & system failures - Both internal & external - 공장가동 중지, 인터넷 문제 등 (7) Execution, delivery, & process management (실행 미진)

What are the 2 alternative methods to credit rating? 공통된 특징은?

(1) Kamakura (2) KMI *They use incorporate factors such as firm equity market value, equity volatility, debt in firm's capital structure, etc. - Estimates: "Point-in-time" - Goal: 변화하는 상황을 반영하는 것 (stability X)

Why are key rates appealing? (4 reasons)

(1) Key rates are affected by a combination of rates closest to them. (adjacent 한테서 영향 받음) (2) Key rates are mostly affected by the key rate. (3) Key rate effects are smooth. (Don't jump across maturity.) (4) A parallel shift across the yield curve results.

An American put option은 (1) likely to be early exercised for 큰 배당. The value of an American put option (2) with dividends.

(1) Less (2) Increases

PF insurance의 (1) 가장 대표적인 방법은? (2) 대안은?

(1) Long put: losses from the PF may be offset with gains from this position. (물론 적절한 풋옵션이 unavailable 할 수도 있음) (2) 대한: create a "synthetic put position with index futures contract." *Index futures contract란? - Selling index futures contracts - 얼만큼? proportion of the PF dictated by the Δ of the required 풋옵션. - Trading cost 낮고, 더 유동적이라는 장점 O.

Total country risk를 evaluate 하는 행위의 한계점 3가지는?

(1) Not every component is relevant to all investors. (2) No standardisation across the information providers. (비교하기 어려움) (3) Scores are better used as rankings than as true scores.

(1) An investor owns 60,000 shares of ABC stock that is currently selling for $50. A call option on ABC with a strike price of $50 is selling at $4 and has a delta of 0.60. Determine the number of call options necessary to create a delta-neutral hedge. (2) Calculate the effect on PF value of a $1 increase in the price of ABC stock. (3) The price of the underlying stock has moved to $51, and the delta of the call option with a strike price of $50 has increased from 0.60 to 0.62. How would the investor's PF of stock and options have to be adjusted to maintain the delta-neutral position? (4) Again, a call option on ABC with a strike price of $50 is selling at $4 and has a delta of 0.60. Determine the number of put options necessary to create a delta-neutral hedge.

(1) Number of options needed to delta hedge = 60,000/0.60 = 100,000 options = 1,000 call option contracts ∴ Because he has longed the stock, he needs to short the options. (2) Value of the call option would decrease by $0.60 (delta 때문) ∴ The net impact of the price change would be 0 (캡처본 참고) (3) Number of options needed to delta hedge = 60,000/0.62 = 96,744 options = 968 call option contracts ∴ 32 option contracts would need to be purchased (long) to maintain the delta-neutral position. *캡처본의 "decrease in option position"이 [96,774*(-0.62)] = -60,000 으로 바뀔 것 (다른 값들 대입해도 같은 결과) (4) (Put delta = call delta - 1 = 0.60 - 1 = -0.40) Number of options needed to delta hedge = -60,000/-0.40 = 150,000 options = 1,500 put option contracts ∴ Because he has longed the stock, he needs to long the options.

Convexity 차이 이용해서 차익거래하는 방법 설명하기 *Barbell/bullet의 context: (1) IR parallel shift (2) IR non-parallel shift (e.g. 단기/장기금리는 상승, 중기금리는 하락)

(1) Parallel shift니까 duration 걱정할 필요 없음 - Long Barbell (higher C, outperforms) - Short sell Bullet (2) Duration 걱정할 필요 있음!ㅋㅋㅋ - Barbell PF 채권가격 하락; 손실 - Bullet PF 채권가격 상승; 이득 ∴ 반대로 가야 함 (short sell Barbell, long Bullet)

3 ways of evaluating the total country risk?

(1) Political risk services (PRS) - Measures using 22 variables (2) Media outlets - Euromoney: 400명의 경제학자 동원 - The Economist: assesses currency risk/sovereign debt risk/banking risks to develop the country risk scores. (3) World Bank - Assess 6 areas: a. Corruption b. Gov. effectiveness c. Political stability d. Rule of law e. Accountability f. Regulatory quality

Carry Roll Down 시나리오의 3가지 타입은?

(1) Realised forward scenario (2) Unchanged term structure scenario (3) Unchanged yields scenario

What are the 5 challenges in quantifying credit risk?

(1) Regulators require that banks calculate "through-the-cyle" PD for the regulatory capital. - 이유: it removes the volatility of business cycles (while banks may prefer "point-in-time" PD for their internal purposes). (2) Economic downturn은 은행에게 "dual negative effect"를 미침. - Downturn 중에는 PD도 높아지는데 LGD 역시 높아짐. (3) Wrong-way risk - 장외파생의 기초자산은 내 exposure(가치)에도 영향을 주지만, 거래상대방의 부도율에도 영향. (4) 측정한 것과 실제 correlation은 다를 수 있음. (5) Credit risk 말고도 market/operation/liquidity/strategic risk 역시 규제자본에 영향 미침.

Stress testing을 시행하는 2가지 패턴은? (시점 관련)

(1) Routine basis (monthly, etc.) (2) Ad-hoc stress tests: Non-routine basis (그때그때, 선거나 정부 정책의 변화 반영)

Long stock position을 hedge하는 3가지 방법은? 필요한 수단의 개수를 계산하는 방법은?

(1) Short forward (2) Short call - # of calls to short = 1/Δc (3) Long put - # of puts to long = 1/(Δp) (Δp=Δc-1)

Sovereign default risk에 영향을 주는 요소 4가지는?

(1) Social security commitments (사회보장제도) (2) Tax base (3) Political risk (4) Implicit guarantees (e.g. Distressed EU countries tend to receive support from stronger EU countries.)

Bond initial price = 102.65 - Carry roll down: 0.85 - Rate changes: 0.30 - Spread changes: 0.08 Bond final price = 101.88 + Cash and carry coupon = 2 What is (1) the gain and (2) the gross return?

(1) The gain 101.88+2-102.65=1.23 (=carry roll down + rate changes + spread changes) (2) The gross return 1.23/102.65 = 1.198% *경우에 따라 financing & AI 역시 고려하기도 함.

4 sources of country risk?

(1) The level of a country's economic growth - 한 나라가 economic cycle에 어떻게 반응하느냐 역시 중요한 risk determinant. - 보통 developing country는 developed country보다 larger declines in real GDP growth rate during global downturn (이유: higher reliance on commodities). (2) Political risk - Democracy: Continuous & low risk - Dictatorship: Discontinuous & high risk - Smoothness of transferring political power (military coup OR election?) - Corruption - Physical violence (e.g. terror) - Nationalisation/expropriation risk (3) Legal risk - Protection of property rights - Efficiency of dispute resolution (timely settle) - 좋은 시스템은 기업들이 고소 당하게 놔둠. (4) Economic structure - Level of economic diversification (나라가 한 산업에 의존하면 위험) - Competitive advantage 여부 - Long-/short-term growth 여부

Non-parallel shift in the Yield Curve 종류 2가지 대기

(1) Twists (2) Butterfly

IR term structure 그래프 상 forward, spot, par rates 곡선 위치 순서는? (Upward/flat/downward 경우들 각각)

(1) Upward sloping: Forward > Spot > Par (2) Flat: Forward = Spot = Par (3) Downward sloping: Forward < Spot < Par

Warrants can be valued as a separate (1) on (2).

(1) call option (2) the firm's shares

The closer the American call option is to (1), the larger the (2), the more optimal (3) will become.

(1) expiration (2) dividend (3) early exercise (아메리칸 옵션의...)

CFs (1) put values & (2) call values.

(1) increase (2) decrease

When using a single discount rate, the (1) is added to the (2) to get the appropriate discount rate for all the (3).

(1) risk premium (2) risk-free rate (3) expected CFs

(1) KR01과 DV01의 관계는? (2) KR Duration과 그냥 duration의 관계는?

(1) Σ(KR01) = DV01 (2) Σ(KR Duration) = Duration

PF with 3 loans where: - Loan 1's SD: 1.2 - Loan 2's SD: 0.8 - Loan 3's SD: 0.8 *Correlations of the loans: (2)와 (3)은 0.6으로 correlated; 나머지는 다 무관. (1) PF's SD of total loss is? (2) What happens to (1) if the size of loan 1 increases by 1%? (3) 같은 방식으로 (2)와 (3) 역시 1%씩 올리고, 오일러 정리로 PF total loss의 SD를 계산한다면?

(1) √(1.2^2 + 0.8^2 + 0.8^2 + 2*0.6*0.8*0.8) = 1.87 (2) √(1.212^2 + 0.8^2 + 0.8^2 + 2*0.6*0.8*0.8)가 되어, (1)과의 차이를 구하면 0.007733. (3) 오일러 정리 활용: - Q1=0.007733/0.01=0.7733 - (2)와 (3) 역시 같은 방식으로 비중 1%씩 올려서 새로 계산한 뒤 차이 구해보면 Q2=0.5492, Q3=0.5492. ∴ 0.7733+0.5492+0.5492=1.87 (원래 PF's SD of total losses) - Decomposition of PF's SD of total losses into individual contributions of the 3 loans.

What are some consequences of sovereign debt? 과거와 현재 비교해보기

(1) 과거: military actions (2) 현재: different economic conditions - Loss of reputation - Reduced investment in stock/bond market - Economic downturn - Political instability

Strip의 장점 2가지는?

(1) 금융시장에 새로운 정보를 주기에 긍정적 (2) 만기가 긴 CB의 연수익률을 계산하기 용이

Scenario analysis는: (1) Bank's own previous experience & hypothetical scenarios 둘 다 사용 (2) Bank's own previous experience 만 사용 (3) Hypothetical scenarios 만 사용 (4) Bank's own previous experience & hypothetical scenarios 둘 다 사용하지 않고, 다른 지표 활용 (뭔지 대기)

(1) 둘 다 사용!

Puttable bond: - 채권 (1) 에게 유리 - 시장 금리 (2) 시 sell back at (3) 가능 - 채권 (4) 에게 손해 - 따라서 채권가 (5) - Puttable bond = Straight bond - (6)

(1) 매입자 (2) 상승 (3) Predetermined price (4) 발행자 (5) 높음 (6) Put value

LIBOR 특징 3가지 말하기

(1) 민간 sector 중 무위험 이자율 (2) 신용도 좋음 (3) 파생거래의 reference rate

Callable bond: - 채권 (1) 에게 유리 - 시장 금리 (2) 시 조기상환 가능 - 채권 (3) 에게 손해 - 따라서 채권가 (4) - Callable bond = Straight bond - (5) - 조기상환 시 (6)로 돌려줌 - 금리가 아무리 내려도 최대 채권가격은 (7)

(1) 발행자 (2) 하락 (3) 매입자 (4) 낮음 (5) Call value (6) 액면가 (기존 가격보다 높음) (7) Call price에 capped

YTM 정할 때: - Coupon 많이 주면: _____ spot rate 중요 - Coupon 적게 주면: _____ spot rate 중요 따라서 (1) IR term structure 우상향: _____ spot rate 중요 (2) IR term structure flat: _____ spot rate 중요 (3) IR term structure 우하향: _____ spot rate 중요

(1) 앞쪽 (2) 뒷쪽 (1) 뒷쪽 (앞쪽 spot은 낮으니까 채권가가 높고, 따라서 YTM 낮음) (2) Earlier spot = Final spot (3) 앞쪽 (앞쪽 spot은 높으니까 채권가가 낮고, 따라서 YTM 높음)

Historical simulation의 장단점은?

(1) 장점 - It may identify a crisis event that was previously overlooked for a specific asset class. - The focus is on identifying extreme changes in valuation. (2) 단점 - It is limited to actual historical data. - Separating the full sample of data into different market regimes reduces the amount of usable data.

Vasicek model과 관련된 공식들 써보기 (1) A company defaults if: (2) 99.9% CL에서 측정한 신용 PF 최대부도율: (3) UL under the IRB approach of Basel II:

(1) 캡처본 (2) 캡처본 (가정: 각 investment가 동일한 양인 homogeneous PF & 돈 빌려간 기업이 많음) (3) UL=(WCDR-average PD)*LGD*EaD - 이때, all loans have the same PD, correlation, and LGD. - For a PF of loans, the UL would be the sum of the individual loan ULs (ρ=1 가정에 단순합산 가능)

What is the gross realised return for a bond that is currently selling for $112 if it was purchased exactly six months ago for $105 and paid a $2 coupon today?

(112+2-105)/105=8.57%

(1) 운영위험은 최대한 일어나지 않도록 하는게 좋다. (2) 운영위험은 불가피하기 때문에 그 severity가 중요하다. 둘 중 옳은 statement은?

(2)

다음 중 풋/콜 옵션의 greek이 동일한 값인 경우는? 이유는? (1) Delta (2) Gamma (3) Vega

(2) Gamma, (3) Vega. 이유는 put-call parity 참고, 캡처본!

Stress testing은: (1) 1번만 진행해야 한다. (2) Series로 진행해야 한다.

(2) Series!

Stress testing은 다른 capital&adequacy measures의 _____. (1) Substitute (2) Supplement

(2) Supplement

Sovereign credit risk) Foreign currency default과 local currency default는 ____ 발생한다. 4가지 이유는? (1) 따로 (2) 동시에

(2) 동시에 ===== 이유 ===== 1. Gold Reserves: 1971년 이전에 어떤 나라들은 "gold standard"를 따라서, 돈을 찍어내려면 금이 필요; low flexibility of printing currency to repay debt. 2. Currency union (e.g. EU): 공통된 currency를 활용하면 당연히 돈 찍어내는게 제한적일 수 밖에 없음... 특히 EU 같은 경우는 중앙은행들이 아니라 ECB가 화폐 발행! 3. Currency debasement: 돈 찍어내는 행위 may debase a local currency & incur inflation; 비용 발생

Time horizon에 따른 increase in default rate은 _____이 더 dramatic. (1) 투자등급 (2) 투기등급

(2) 투기등급

Rho가 더 중요하게 작용하는 경우는? (1) 주식이 기초자산 (2) 채권이 기초자산

(2).

Delta-normal VaR 방식만으로 위험한 이유는?

(Normal 가정에 의존하기 때문에, 실제 VaR에 delta를 곱한만큼으로 overestimate, 이는 gamma로 커버쳐야 함) 단, Long call position: (1) Price estimation: underestimate (2) Loss(P/L) estimation: overestimate - 따라서 understates the prb. of high option values & overstate the prb. of low option values. *Short call position: 항상 위험 underestimate

Suppose we have a security paying $1,000 annually into perpetuity. The interest rate is 10%. Calculate the price of the perpetuity.

(계산기 불필요) PV=$1,000/0.10=$10,000

Clean price란?

(공시만 하는 가격) Clean price = quoted price = float price = dirty price - AI = annualised discount rate

Carry Roll Down 시나리오 중 unchanged yields scenario란?

(비현실적임!) - I/Y (=YTM) 가 불변이라는 가정 (2년짜리건 3년짜리건)

Dirty price란?

(실제로 지불) Dirty price = cash price = invoice price = full price = clean price + AI = FV - Q*(n/360)

Assume a stock has an initial price S0 = $25, an expected annual return of 12%, and an annual volatility of 20%. Calculate the mean and SD of the distribution of the stock price in three months.

(캡처본 참고) Since ln(St) is normally distributed, 95% of the values will fall within 1.96 SDs of the mean. Therefore, ln(St) will lie between 3.24 ± (1.65*0.1) 3.24 - (1.65*0.1) < ln(St) < 3.24 + (1.65*0.1) e^[3.24 - (1.65*0.1)] < St < e^[3.24 + (1.65*0.1)] 21.073 < St < 31.187

Effective duration이란? 그냥 Duration을 effective duration으로 바꾸려면?

*Convexity 때문에 +ΔP와 -ΔP의 크기가 다른 것을 반영 그냥 Duration을 effective duration으로 바꾸려면 YTM의 움직임에 따라 채권가격이 >평균< 얼마나 변동하는지 계산해야 함!

BSM) European options with dividends - 현재 주가=$100 - Volatility=20% - Rf=7% - 6개월 ATM 콜옵션=$7.43 (BSM 계산 결과) - Corresponding 풋옵션=$3.99 (BSM 계산 결과) - 배당: 2개월 후 $1, 5개월 후 $1 지급 Q. Dividend paying stock의 6개월 콜옵션 가격은?

*Dividend를 %(continuous)가 아니라 dollar amount(discrete)으로 줬다는게 이 문제의 핵심! - 각 배당 e^(-rf*t)로 현재가치 구하고, 그 현재가치들을 원래 주가에서 뺀게 adjusted stock price. - t는 12개월 기준으로 조정 (2/12=2개월 배당=0.1667, 5/12=5개월 배당=0.9713) (캡처본 참고) 구해진 So 가지고 d1, d2 계산, 누적확률분포표에서 N(d1), N(d2) 찾아서 BSM model의 Co, Po 계산하기~

Using KR01s for hedging) 아래 3개의 hedge 수단은 각각 얼마나 필요? KR/PF 변동분/Hedge 1/Hedge 2/Hedge 3 (가능한 양) KR01(1)/68/14/2/3 KR01(2)/106/3/12/4 KR01(3)/169/7/1/15

*Hedging involves setting KR01s = 0. (이렇게 되면 금리충격이 와도 PF 가치가 움직이지 않는다는 의미!) (1) 68 + 14(H1) + 2(H2) + 3(H3) = 0 (2) 106 + 3(H1) + 12(H2) + 4(H3) = 0 (3) 169 + 7(H1) + 1(H2) + 15(H3) = 0 ∴ 연립방정식 계산하면: take short positions of 2 H1s, 5 H2s, 10 H3s.

Sovereign credit risk 란? 종류는?

*Proxy for country risk = Risk that holders of government-issued debt fail to receive the full amount of promised interest & 원금 (1) Foreign currency defaults (외화로 국채 발행했으나 갚을 외화 부족): 이게 대부분 (2) Local currency defaults (국채)

Binomial model로 아메리칸 옵션 value 하는 방법은? - 현재 주가=$10 - πu=0.51, πd=0.49 - U=1.20, D=0.833 - Risk-free rate=2% - 만기=2년 - 행사가격=$12

*Two-step binomial model로만 아메리칸 옵션 value 가능 *결론의 $2.24라는 가격에는 내재가치 $2와 시간가치 $0.24가 포함되어 있음!

BSM) European options on foreign currencies - 주가 So 어떻게 조정?

*rfc = 외화이자

Carry Roll Down 시나리오 중 realised forward scenario란?

*고대로 parallel하게 옆으로 옮겨오는 것 잔존만기 2년일때 $100.785 였다면, 6개월 지나서 잔존만기 1.5년인 상황일때는: - Bond price = 1.0/1.005 + 1.0/(1.005*1.006) + 101/(1.005*1.006*1.007) = $101.188 (만기 더 짧아졌으니까 가격 더 오름) - 이때, 100.785*0.004 = $0.403, 이걸 더하면 새로운 채권가와 일치함.

Duration & convexity 둘다 이용해서 hedging 하는 방법은? [A=대상채권, B1=수단채권 1, B2=수단채권 2]

*공식은 캡처본 참고 - For a fully hedged investment, D=C=0. - 따라서: (1) -A*D(A) + -B1*D1 + -B2*D2 = 0 (2) A*C(A) + B1*C1 + B2*C2 = 0

Carry Roll Down 시나리오 중 unchanged term structure scenario란?

*대각선 아래로 옮겨오는 것 (최초 시작점 일치) - 사람들은 기간이 멀리 갈수록 risk premium을 요구하고, 이는 바뀌지 않는다는 가정.

Par rate 계산) swap rate & spot rate 사용의 차이는?

*둘다 캡처본의 공식을 사용하면 됨 (분자의 2는 semiannual basis이기 때문) (1) Swap rate 사용 - Par rate = Swap rate - 쿠폰을 스왑고정금리만큼 주면, 해당 만기의 채권을 par value로 발행 가능! (2) Spot rate 사용 - Par rate ≠ Spot rate - 쿠폰을 스팟금리만큼 줘도, 해당 만기의 채권을 par value로 발행하지 못한다!

(((((중요))))) - Investor owns 60,000 shares (현재 주가 $50) - Call option 가격 $4 (행사가격 $50) - 콜 델타 = 0.60 - 콜옵션 몇개 필요한가?

*문제에서는 [주식=양, 콜옵션=울타리]이기 때문에 원래 [S/C=1/Δ] 여서 1/Δ로 변환하는게 필요함. <(Δ or 1/Δ)*대상 개수=수단 개수>니까, (1/0.06)*60,000=100,000개의 콜옵션 필요함.

Discrete한 수익률과 continuous한 수익률의 차이는? - 공식 - 2 시점의 return을 정규분포로 보내면 주가가 어떻게 되는지 - 그래서 둘중에 뭐가 더 현실적인지?

*아무리 수익률이 음수여도, 주가는 절대 0 밑으로 내려갈 수 없기 때문에 continuous 수익률이 더 현실적이다!

Synthetic call replication이란? 공식은?

*지금 당장 콜옵션은 없지만, 그것을 들고있는 것과 같은 효과. Call Value (콜옵션 들고있으면 내가 가진 가치) = Δ*(주가 - PV(예상되는 최소 주가))

Volatility surface란? 그래프 그리기 +각 축의 x, y 변수 말고 또 영향 주는 변수는?

+ 잔여만기 역시 영향 주는 변수인데, (1) 짧은 잔여만기: smile 깊음 (2) 긴 잔여만기: smile 얕음

Risk-neutral valuation의 가정: _____ 시점에 _____ 가진 사람은 _____ 요구함.

0시점에 콜옵션 가진 사람은 무위험수익률 요구함. (위험프리미엄 요구하지 않음)

은행의 banksheet 그림 그려서 RWA와 regulatory capital 어떻게 결정되는지 정리하기!

- Asset 부분의 market/credit risk로 RWA가 결정 - Equity 부분에서 market/credit/operational risk 다 커버하는 자기자본 필요 ∴ Regulatory capital ≥ RWA*8%

Suppose an existing short option position is delta neutral but has a gamma of -6,000. Here, gamma is negative because we are short the positions. Also, assume that there exists a traded call option with a delta of 0.6 and a gamma of 1.25. Create a gamma-neutral position.

- Buy 4,800 options (6,000/1.25) - Now, the position is gamma-neutral, but the added options have changed the delta position of the PF from 0 to 2,880 (=4,800*0.6) - 2,880 shares of the underlying position will have to be sold to maintain not only a gamma-neutral position, but also the original delta-neutral position.

Economic/industrial cycle이 rating에 미치는 영향은? 이것에 신용평가기관들이 대응하는 방식의 장단점은?

- Since the rating should apply to a long horizon, rating agencies try to give a rating that incorporates the effect of an average cycle. - 장점: Ratings are stable over cycles - 단점: Economic condition이 평균에서 너무 멀면 over/under-estimate할 수 있음.

금리 충격의 2가지 종류와 각각이 등장하는 context는?

- ΔY = YTM (par rate) 충격; CB의 context - ΔS = Spot rate 충격; ZCB의 context

Ratings translation matrix란 무엇을 보여주나? 특징은?

- 앞으로 1년동안 부여받은 신용등급에서 다른 등급이 될 가능성을 illustrate. - 부여받은 신용등급을 유지할 가능성은 투자등급보다 투기등급 채권들이 더 큼. - Downgrade(upgrade) in one year has a higher likelihood of being followed by another downgrade(upgrade) in the next year.

BSM) European options on futures/forwards - 주가 So 어떻게 조정? - 이 context에서 등장하는 새로운 모형은?

- 주가 조정: 캡처본 참고 - 새 모형: Black's model

Historical simulation을 통한 VaR & ES 계산) 지난 300일 동안의 data가 모여져 있을 때: 1. 99% VaR은? 2. ES는?

1. 99% 신뢰수준이면 왼쪽 꼬리 0.01만큼에 해당하는 사건들을 봐야 하는 것: 300의 1%면 3! 따라서 300개의 사건들 중 third biggest loss가 바로 VaR. 2. 그렇다면 ES은 (first biggest loss + second biggest loss)/2. (산술평균 낸 것)

Default probability를 계산하는 3가지 방식은?

1. Cumulative percentage 2. Unconditional percentage 3. Conditional percentage

Arbitrage opportunity는 무엇에 따라 손익 발생하나? (2가지)

1. Level of price (방향 2. Volatility (변동성)

Strip의 두가지 종류는?

1. Principal: P-strips = TPs = Ps 2. Coupons: C-strips = TINTs = INTs

KMI credit rating의 2가지 특징은?

1. 머튼모형을 활용한다. 2. 기업이 가진 자본구조로 부도날 확율을 예측한다.

1. Probability of Default & Recovery Rate의 관계는? 2. Probability of Default & Loss given default의 관계는?

1. 역의 관계 2. 정의 관계 (애초에 LGD=1-RR이니까 위와 연관지어서 생각)

옵션 1계약이 커버하는 기초자산의 개수는?

100개

Sovereign credit risk) Local currency defaults 관련 제일 큰 두 사건은?

1990년 브라질 1998년 러시아

Operational risk란?

<All risk that is not credit or market risk> 바젤에서 내린 정의: "The risk of loss resulting from inadequate or failed internal processes, people & systems or from external events."

Expected shortfall이란?

= Conditional VaR = Expected tail loss (ETL) 이라고도 함. - 위험의 척도이나, ND와 무관하며 항상 subadditive하다. - EL given that the PF return already lies below the pre-specified worst case quantile return. (제일 손실 큰 애들 뒤에서 5명 나와, 너희들의 산술평균 구해) - 항상 VaR보다 큼.

Credit migration이란? 어떤 context에서 나오는 말?

= Possible deterioration (or improvement) in creditworthiness of borrower (Unexpected loss의 원인 중 하나!)

Required regulatory capital 계산 방식은?

=VaR - EL (over 1 year, at x% CL) *바젤은 CL 99.9%로 둠 (이 규제를 만족시킬 정도의 자기자본을 가지면 부도날 확률은 0.1% 라는 것)

A EUR 100,000 par value French corporate bond pays 3.5% coupon with a semiannual frequency. Assume the last coupon was paid 75 days ago and there are 30 days in each month. The accrued interest is closest to: A. 729 B. 1,094 C. 1,458 D. 2,917

All of the following represent rationales for developing internal rating systems except: A. they do not require testing like external ratings. B. situations where external ratings are not available. C. default probabilities driving regulatory credit risk capital. D. accounting requirements tied to recognising default probabilities on loan valuations.

The relationship between expected loss (EL), unexpected loss (UL), and actual loss can be best described as: A. actual loss = EL + UL B. actual loss = EL - UL C. actual loss = EL * UL D. actual loss = UL - EL

Which of the following is not one of the seven types of operational risk identified by the Basel Committee? A. Failed business strategies. B. Clients, products, and business practices. C. Employment practices and workplace safety. D. Execution, delivery, and process management.

A $1,000 par bond carries a 7.75% semiannual coupon rate. Prevailing market rates are 8.25%. What is the price of the bond? A. Less than $1,000. B. $1,000. C. Greater than $1,000. D. Not enough information to determine.

A (풀이: Because the coupon rate is less than the market interest rate, the bond is a discount bond and trades less than par.)

Which of the following reasons least likely explains local currency defaults? A. Countries may decide that the costs of higher inflation are less than the costs of default. B. Countries may decide that the costs of currency debasement are higher than the costs of default. C. Shared currencies like the euro make it difficult for countries to control their own monetary policy. D. Prior to 1971, the use of the gold standard prior made it more difficult for some countries to print money.

A (풀이: Countries may decide that the costs of higher inflation are higher than the costs of default.)

Fat-tailed asset return distributions are most likely the result of time-varying: A. volatility for the unconditional distribution. B. means for the unconditional distribution. C. volatility for the conditional distribution. D. means for the conditional distribution.

A (풀이: Examining a data sample at different points of time from the full sample could generate fat tails in the unconditional distribution, even if the conditional distributions are ND.)

Which of the following is not a reason that expected shortfall (ES) is a more appropriate risk measure than VaR? A. For normal distributions, only ES satisfies all the properties of coherent risk measurements. B. For non-elliptical distributions, the PF risk surface formed by holding period and confidence level is more convex for ES. C. ES gives an estimate of the magnitude of a loss. D. ES has less restrictive assumptions regarding risk/return decision rules than VaR.

A (풀이: VaR & ES both satisfy all the properties of coherent risk measures for ND, but only ES satisfies all the properties of coherent risk measures when the assumption of normality is not met.)

The role that rating agency evaluations of structured products played in the economic crisis of 2007-9 was largely a result of: A. underestimated correlations between defaults. B. the high degree of regulatory oversight in place. C. the low impact of potential profits to the agencies. D. the minimal impact of structured product performance modelling.

A (풀이: defaults often snowball and follow each other.)

Assuming a PF has the following asset returns: 6%, 2%, 8%, -3%, what is the realised PF return? A. 3.16%. B. 3.25%. C. 4.72%. D. 4.75%.

A Realised PF return = [(1.06*1.02*1.08*0.97)^(1/4)]-1 = 3.16%

A 1-year American put option with an exercise price of $50 will be worth either $8 at maturity with a probability of 0.45 or $0 with a probability of 0.55. The current stock price is $45. The risk-free rate is 3%. The optimal strategy is to: A. exercise the option because the payoff from exercise exceeds the PV of the expected future payoff. B. not exercise the option because the payoff from exercise is less than the discounted present value of the future payoff. C. exercise the option because it is currently in the money. D. not exercise the option because it is currently out of money.

A The payoff from exercising the option = exercise price - current stock price = 50 - 45 = $5 vs. The discounted value of the expected future payoff = (캡처본) ∴ It is optimal to exercise the option early because it is worth more exercised ($5.00) than if not exercised ($3.49).

A bond investor projects a decline in 2-year rates from 3% to 2.5% and a decline in 20-year rates from 6% to 4.5%. This situation is best described as: A. a bull flattener. B. a bear flattener. C. a bull steepener. D. a bear steepener.

A [풀이] A. A bull flattener - Decline in long-term rates > Decline in short-term rates B. A bear flattener - Rise in short-term rates > Rise in long-term rates C. A bull steepener - Decline in short-term rates > Decline in long-term rates D. A bear steepener - Rise in long-term rates > Rise in short-term rates

Which of the following key variable inputs is least likely to be incorporated into stress-test models? A. A 5% decrease in the stock market. B. A decline in GDP of 300 basis points. C. An increase in interest rates of 300 basis points. D. A 5% increase in the national unemployment rate.

A (풀이: A 5% decrease in the stock market could easily occur over the course of a few trading sessions.)

Compared to the value of a call option on a stock with no dividends, a call option on an identical stock expected to pay a dividend during the term of the option will have: A. a lower value in all cases. B. a higher value in all cases. C. a lower value only if it is an American-style option. D. a higher value only if it is an American-style option.

A (풀이: An expected dividend during the term of an option will decrease the value of a call option.)

In modelling risk frequency, it is common to: A. use of Poisson distribution. B. assume that risks are highly correlated. C. assume risk frequency and severity are the same. D. use straight-line projection from the most recent loss data.

A (풀이: It is common to use a Poisson distribution to model loss frequency; Poisson distribution has a single parameter than can be varied to accurately describe loss data.)

PF insurance payoffs would not involve which of the following? A. Selling call options in the proportion 1/delta. B. Buying put options one to one relative to the underlying. C. Buying and selling the underlying in the proportion of delta of a put. D. Buying and selling futures in the proportion of delta of a put.

A (풀이: PF insurance can be created by all of the statements except selling call options in the proportion 1/delta. This action generates a delta-neutral hedge, not PF insurance.)

A static hedging strategy will be least effective when the underlying stock price: A. increase from $4 to $6. B. increases from $20 to $21. C. decreases from $26 to $25. D. decreases from $35 to $34.

A (풀이: The largest dollar change & the largest percentage change)

A 1% increase in the risk-free rate (5% to 6%) will increase the value of the call option by approximately 0.01*10.81 = 0.1081.

Perpetuity란? 공식은?

Annuity PV 공식에서 만기를 ∞로 수렴시킨 것. [주기적 CF ÷ m 반영한 discount rate]

A key rate (KR01) for a two year spot rate will represent the increase in PF value from: A. a one basis point increase in the 2-year spot rate. B. a one basis point decrease in the 2-year spot rate. C. a one basis point increase in the 1- and 2-year spot rates. D. a one basis point decrease in the 1- and 2-year spot rates.

If the 3-years, continuously compounded spot rate is 6.25%, the discount factor will be closest to: A. 0.8125. B. 0.8290. C. 0.9394. D. 1.2062.

Which of the following measurement approaches for assessing operational risk would be most appropriate for small banks? A. The standardised approach. B. The basic indicator approach. C. The advanced measurement approach (AMA). D. Either the standardised approach or the AMA.

Which of the following statements most likely describes an advantage of using stressed risk metrics? A. The risk metric will be more realistic. B. The risk metric will be more conservative. C. The risk metric will mirror the PF returns. D. The risk metric will respond to current market conditions.

Which of the following statements is not a disadvantage of nonparametric methods compared to parametric methods? A. Data is used more efficiently with parametric methods than nonparametric methods. B. Identifying market regimes and conditional volatility increases the amount of usable data as well as the estimation error for historical simulations C. The full sample of data is not split into subgroups used to estimate conditional volatility. D. MDE requires a large amount of data that is directly related to the number of conditioning variables used in the model.

B 풀이: - The use of market regimes and identifying conditional means and volatility actually reduces - not increases - the amount of data from the full sample. - The full sample of data is split into subgroups used to estimate conditional volatility. - Therefore, the amount of data available for estimating future volatility is significantly reduced.

Which of the following statements regarding challenges in calculating credit risk capital for derivatives is least accurate? A. It is easier to calculate EAD for loans than derivatives. B. Netting arrangements simplify calculating credit risk capital for derivatives. C. Basel rules set EAD for derivatives as current exposure plus an add-on amount. D. For derivatives with netting arrangements, calculating required capital must be done on a counterparty basis.

B (풀이: Netting arrangements make calculating risk capital for derivatives more challenging because all derivatives with a single counterparty are considered a single derivative if the counterparty defaulted, and therefore calculating required capital must be done on a counterparty basis rather than on a transaction basis.)

GARCH (1,1) models can only be used to estimate volatility in the case where: A. α+β>0 B. α+β<1 C. α>β D. α<β

B (풀이: Otherwise the model is unstable.)

XYZ Bank is trying to forecast the expected loss on a loan to a mid-size corporate borrower. It determines that there will be a 75% loss if the borrower does not perform the financial obligation. This risk measures is: A. the PD. B. the loss rate. C. the unexpected loss. D. the EAD.

B (풀이: loss rate = loss given default)

The annual standard deviation for Baker stock is 11%. The continuously compounded risk-free rate is 3.5% per year, and Baker pays dividends at a yield of 2%. The risk-neutral probability of a downward move πd is closest to: A. 0.366. B. 0.459. C. 0.541. D. 0.634.

B U = e^(0.11*√1) = 1.116278. D = 1/U = 0.895834 *이거 가지고 πu와 πd 계산! (캡처본)

There are 3 million outstanding shares of ABC stock currently selling at $42 each. ABC is considering issuing 1 million warrants with a strike price of $45 exercisable in one year. If the current value of a 1-year European call option is $2.12, the expected stock price after announcing the warrant (assuming no perceived benefit to issuance) will be closest to: A. $40.41. B. $41.47. C. $42.53. D. $43.59.

B [풀이] The value of each warrant = [3,000,000/(3,000,00+1,000,000)]*$2.12 = $1.59 The total warrant cost = 1,000,000*$1.59 = $1.59 million The initial stock price = [1,000,000/(3,000,00+1,000,000)]*$2.12 = $0.53 ∴ Stock price = $42.00 - $0.53 = $41.47

Which of the following statements about documentation of stress tests is most appropriate? A. Institutions are not concerned if their vendors document stress-testing activities. B. Institutions should incentivise documenting stress tests to increase efficiency. C. Documentation is not useful for stress-test developers, but it is important to senior management. D. Documentation should not include a description of the types of stress tests and methodologies, but it should include a description of the key assumptions and limitations.

B [풀이] - A가 아닌 이유: Institutions should ensure ~ - C가 아닌 이유: Documentation is useful for both ~ - D가 아닌 이유: Documentation should include both ~

Which of the following statements regarding partial '01s is most accurate? A. They reflect a 100 basis point change in rates. B. They are derived from highly liquid instruments. C. They cannot be used to hedge risk in swap portfolios. D. They differ significantly from key rate exposures in terms of functionality.

B [풀이] - A가 아닌 이유: They reflect a 1 basis point change in rates. - C가 아닌 이유: They are often used to hedge risk in swap portfolios. - D가 아닌 이유: They are similar to key rate exposures in terms of functionality.

An investor is analysing a 1-year European call option with an exercise price of $18. The stock value in the up state is $30, while the value in the down state is $10. The delta for this option is closest to: A. 0.40. B. 0.60. C. 0.67. D. 0.90.

B [풀이] If the stock price goes up to $30, the call option with an exercise price of $18 will be worth $12. If the stock price goes down to $0, the call option will be worth $0. ∴ delta = (12-0)/(30-10) = 0.60

Which of the following is not an assumption underlying the BSM options pricing model? A. The underlying asset does not generate cash flows. B. Continuously compounded returns are log-normally distributed. C. The option can only be exercised at maturity. D. The risk-free rate is constant.

B (풀이: "Asset price (not returns) follows a lognormal distribution.")

Using a binomial model, the price of a call option is equal to $3.46. For the same option, the BSM model produces a price of $3.38. If the binomial model are shortened, the expectation is that: A. $3.38 will get closer to $3.46. B. $3.46 will get closer to $3.38. C. the price for both models will be approximately $3.42. D. there will be no change in the gap between prices.

B (풀이: As time intervals are shortened, the price produced by the binomial model will converge toward the BSM model price.)

Stop-loss strategies with call options require purchasing the underlying asset for: A. a naked call position when the asset falls below the option's strike. B. a naked call position when the asset rises above the option's strike. C. a covered call position when the asset falls below the option's strike. D. a covered call position when the asset rises above the option's strike.

B (풀이: Stop-loss strategies with call options are designed to limit the losses associated with short option positions.)

Titan Bank enters into a 3-year, pay floating, received fixed interest rate swap. The fixed rate is set at 3.50% and the floating rate is tied to LIBOR+1%. In which of the following situations will Titan have to pay the counterparty? A. LIBOR falls below 2.50%. B. LIBOR rises above 2.50%. C. The OIS rate falls below the fixed rate of 3.50%. D. Fixed rates on newer swaps rise to 4.00%.

B (풀이: The OIS rate and the fixed rates on newer swaps are not relevant.)

The value of a 3-year bond is 103.960. If forward rates are increased by one basis point, the value falls to 103.925. Which combination of three forward-bucket '01s is feasible for this bond? A. 0.010, 0.009, 0.008. B. 0.014, 0.012, 0.009. C. 0.017, 0.017, 0.017. D. 0.020, 0.011, 0.006.

B (풀이: The difference in price = 103.960-103.925 = 0.035. The combination of three forward-bucket '01s must equal 0.035.)

A bond rating is most likely to fall during a contractionary business cycle using which of the following rating methodologies? A. Forecast. B. Point-in-time. C. Historical driven. D. Through-the-cycle.

B (풀이: The point-in-time approach is likely to result in a bond rating decline during a contractionary cycle, while the through-the-cycle approach is used to achieve ratings consistency by not having rating changes as the economy progresses through business cycles.) *다른 두개(forecast & historical driven)는 not actual methodologies.

If the risk-free rate is 3% and the time to maturity is nine months, the delta of a forward position is closest to: A. 0.98. B. 1.00. C. 1.02. D. 2.25.

B (풀이: The relationship between a forward and the underlying asset is one-to-one, making the delta=1.)

An investor takes a short position in a call option on ABC stock with a current price of $15 and a strike price of $18. If the investor does not own the underlying stock, the biggest risk to the investor is: A. a loss on the premium paid. B. the stock price rising above $18. C. the stock price falling below $15. D. a decline in the overall stock market.

B (풀이: the higher the price goes above the strike price, the more likely it is that the call will be exercised and the investor will then have to go out into the market and buy the stock at now higher prices to cover the call.)

Which of the following statements is most accurate in regard to a ratings matrix? A. A bond has a greater likelihood of following a ratings upgrade with a downgrade. B. A bond with a AAA rating in one year is likely to keep that rating in the next year. C. A bond with a CCC rating in one year is likely to keep that rating in the next year. D. A bond has a greater likelihood of following a ratings downgrade with an upgrade.

B (풀이: the ratings matrix phenomenon has shown that a ratings upgrade(downgrade) is likely to be followed by another upgrade(downgrade).)

Ravi Chowdhury, a PF manager for a hospital foundation, is considering the inclusion of sovereign bonds in the fixed income portion of the foundation's PF. Chowdhury, much to the surprise of his colleagues, plans to purchase the bonds of a country that has long been under authoritarian rule. He cites "lower political risk" when asked about his investment decision. Which of the following statements best supports his assertion of lower risk? A. Authoritarian regimes are more likely to control corruption in government agencies. B. Government policies that may affect debt repayment are often more stable under an authoritarian regime. C. Relative to a democracy, risks are greater on a day-to-day basis but the effects are less detrimental overall. D. In most authoritarian countries, property rights are protected and property disputes are settled quickly.

B (풀이: 반면 democratic country는 선거에 따라 gov. policies 급변 가능/ C가 아닌 이유: Risks in a democracy are continuous, but usually low. In contrast, risks is a dictatorship are discontinuous; policy changes are less frequent, but more severe.)

The current yield curve shows a 1-year spot rate of 2.5%, a 2-year spot rate of 3.25%, and a 10-year spot rate of 4.75%. Which of the following par and forward rate combinations for the 10-year bond is most likely to be found on this curve? A. Forward rate of 4.25%; par rate of 5.00% B. Forward rate of 5.25%; par rate of 4.50% C. Forward rate of 4.75%; par rate of 4.75% D. Forward rate of 4.50%; par rate of 4.50%

B (풀이: 유일하게 forward rate > par rate)

Implied volatility 사용하는 방법은?

BSM 공식에서 σ 빼고 다 관측 가능하니까 σ 빼고 다른 변수들 다 대입 후, market price of option에 같다고 두고 σ에 대해 구하기 (구해진 σ는 VaR 계산에도 대입해서 사용 가능!)

BSM 공식에서 long call option의 delta 유도하고, 범위와 그 이유 얘기하기! +만약 배당이 있는 콜옵션이라면?

BSM 콜가격에 미분해서 N(d1)=Δ 얻기. 범위: 0과 1 사이, 이유는 캡처본 참고 ===== 배당이 있다면 BSM 공식의 So가 So*e^(-q*t)로 변함! 반영한 미분 결과는 N(d1)=e^(-q*t)*Δ

Basel I과 Basel II의 규제자본 관련 조항은 어떻게 달라졌나?

Basel II에서는 규제자본 계산방식에 credit risk 말고도 operation & market risk 역시 포함하도록 바뀜

An investor has a $1.7 million investment with a duration of 6.5. The investor is looking to hedge the investment using a bond with a duration of 7.9. Calculate the face amount of the bond required to hedge the investor's position, and demonstrate why this will result in a fully hedged position.

Bond value = -$1.7 million * (6.5/7.9) = -$1.4 million The investor should therefore short $1.4 million bonds.

(답 이상함) Stock ABC trades for $60 and has 1-year call and put options written on it with an exercise price of $60. The annual SD estimate is 10%, and the continuously compounded risk-free rate is 5%. The value of both the call and put using the BSM option pricing model are closest to which of the following? (Call; Price) A. $6.21; $1.16 B. $4.09; $3.28 C. $4.09; $1.16 D. $6.21; $3.28

Assume an investor is very risk-averse and is creating a PF based on the mean-variance model and the risk-free asset. The investor will most likely choose an investment on: A. the left-hand side of the efficient frontier. B. the right-hand side of the efficient frontier. C. the line segment connecting the risk-free rate to the market PF. D. the line segment extending to the right of the market PF.

Each of the following statements accurately reflects why stress testing is an appropriate risk management tool except: A. normal market conditions can present a false sense of security. B. the extreme scenarios that are modelled are unlikely but still possible. C. Extreme events tend o have a high probability of occurrence with a moderate impact. D. an institution must have sufficient liquid assets and capital to survive an extreme event.

In constructing the operational risk capital requirement for a bank under the advanced measurement approach (AMA), risks are aggregated for: A. commercial and retail banking. B. investment banking and asset management. C. each of the seven risk types and eight business lines that are relevant. D. only those business lines that generate at least 20% of the gross revenue of the bank.

The λ of an EWMA model is estimated to be 0.9. Daily SD is estimated to be 1.5%, and today's stock market return is 0.8%. What is the new estimate of the SD? A. 1.68% B. 1.55% C. 1.45% D. 2.74%

Under Basel II regulations, banks using the advanced measurement approach must calculate the operational risk capital charge at: A. a 99 percentile confidence level and a 1-year time horizon. B. a 99 percentile confidence level and a 5-year time horizon. C. a 99.9 percentile confidence level and a 1-year time horizon. D. a 99.9 percentile confidence level and a 5-year time horizon.

Which of the following statements best reflects the responsibilities of an internal audit? A. An internal audit should not assess the staff involved in stress-testing activities. B. An internal audit must independently assess each stress test used. C. An internal audit should review the manner in which stress-testing efficiencies are identified and tracked. D. The internal audit function need to be impartial but does not need to be independent.

Which of the following statements regarding linear and nonlinear derivatives is true? A. The delta of a linear derivative is equal to one. B. A forward contract is an example of a nonlinear derivative. C. A linear derivative's delta must be constant for all levels of value for the underlying factor. D. The value of the call option changes at a constant rate with the change in the value of the underlying stock.

C (풀이: 반대로 delta of a nonlinear derivative changes for different levels of the underlying factor. A가 아닌 이유: The delta does not necessarily equal to one. D가 아닌 이유: The value of the call option does not change at a constant rate with the change in the value of the underlying stock.)

(중요) Assume you are given the following bonds and forward rates: Maturity/YTM/Coupon/Price 1 year/4.5%/0%/95.694 2 years/7%/0%/87.344 3 years/9%/0%/77.218 *1-year forward rate 1 year from today = 9.56% *1-year forward rate 2 years from today = 10.77% *2-year forward rate 1 year from today = 11.32% Which of the following statements about the forward rates, based on the bond prices, is true? A. The 1-year forward rate 1 year from today is too low. B. The 2-year forward rate 1 year from today is too high. C. The 1-year forward rate 2 years from today is too low. D. The forward rates and bond prices provide no opportunities for arbitrage.

C Given the bond spot rates on the ZCB, the appropriate forward rates should be as the attached. Therefore, the 1-year forward rate 2 years from today is too low.

(아리까리) The problem of fat tails when measuring volatility is least likely: A. in an unstable distribution. B. in a conditional distribution C. in a regime-switching model. D. in an unconditional distribution.

C 풀이: A regime-switching model... - Captures the conditional normality and may resolve the fat-tailed problem and other deviations from normality. - Allows for conditional means and volatility. ∴ The conditional distribution can be normally distributed even if the unconditional distribution is not.

Which of the following bonds is likely to have the highest yield? A. A bond rated AA by S&P. B. A bond rated BB by S&P. C. A bond rated B by Moody's. D. A bond rated Baa3 by Moody's.

C (풀이: The bond with the highest yield will be the one with the lowest rating.)

A security sells for $40. A 3-month call with a strike of $42 has a premium of $2.49. The risk-free rate is 3%. What is the value of the put according to put-call parity? A. $1.89. B. $3.45. C. $4.18. D. $6.03.

C (풀이: p = c + Xe^(-rT) - S = 2.49 + 42*e^(-0.03*0.25) - 40 = $4.18

The 3-year continuously compounded rate is 2.25%, and the 4-year continuously compounded rate is 2.375%. An investor will borrow for 3 years, invest for four, and make a profit if the forward rate for the fourth year is: A. 2.75% B. 2.25% C. less than 2.75% D. greater than 2.75%

C F=(0.02375*4 - 0.0225*3)/(4-3)=0.0275, 2.75% (Year 3에서 1년동안 적용되는 금리 구하는 것!) If the anticipated forward rate for the fourth year is less than 2.75%, the cost of borrowing in the fourth year would make the overall cost of borrowing lower than the 2.375% earned on the 4-year investment.

Which of the following statements about the Greeks is true? A. Rho for fixed-income options is small. B. Call option deltas range from -1 to +1. C. A vega of 10 suggests that for a 1% increase in volatility, the option price will increase by 0.10. D. Theta is the most negative for out-of-money options.

C [풀이] - A가 아닌 이유: Rho for equity options is small. - B가 아닌 이유: Call option deltas range from 0 to +1. - D가 아닌 이유: Theta is the most negative for at-the-money options.

Which of the following statements about governance structure is accurate? A. Senior management has ultimate oversight responsibility and accountability for an entire institution. B. The BoD has responsibility for implementing authorised stress-testing activities. C. The BoD can change an institution's capital levels and exposures following a review of stress-test results. D. Senior management should use scenario analysis, not stress testing, to evaluate an institution's risk decisions.

C [풀이] - A가 아닌 이유: The BoD has ultimate oversight responsibility and accountability for an entire institution. - B가 아닌 이유: Senior management has responsibility for implementing authorised stress-testing activities. - C가 맞는 이유: 실제로 이사회는 stress testing의 결과에 따라 액션을 취할 수 있어! - D가 아닌 이유: Senior management should use stress testing to evaluate an institution's risk decisions.

Assume the following KR01s along with 2 hedging instruments. Key Rates / PF / Hedge 1 / Hedge 2 KR01(1) / 41 / 3 / 2 KR01(2) / 48 / -4 / 1 To properly hedge this PF, which of the following positions should an investor take? A. A short position of 1 and long position of 3 for X1 and X2 B. A long position of 6 and short position of 4 for X1 and X2. C. A long position of 5 and short position of 28 for X1 and X2. D. A long position of 46 and long position of 45 for X1 and X2.

C 연립방정식 풀면 X1=5, X2=-28.

If the hazard rate is constant at 4% per year, the probability of default for a bond by the end of the fifth year is closest to: A. 1.87%. B. 3.92%. C. 18.13%. D. 20.00%

C (풀이: 1-e^(-0.04*5)=0.1813)

Assume a current stock price of $35 with a continuously compounded dividend yield of 2.5%. There is a 6-month call option on the stock with an exercise price of $33. What is the adjusted stock price to use for the BSM model? A. $30.12. B. $32.59. C. $34.57. D. $35.44.

C (풀이: Adjusted stock price = $35 * e^(-0.025*0.5) = $34.57

A shortcoming of the risk and control self-assessment (RCSA) program is that it does not consider: A. the expert opinion of managers. B. the identification of expected losses. C. the independent verification of risk identification and measurement. D. the ongoing assessment of the effectiveness of risk management activities.

C (풀이: An RCSA provides no independent verification of risk measurement and identification.)

The duration of a PF can be computed as the sum of the value-weighted durations of the bonds in the PF. Which of the following is the most limiting assumption of this approach? A. All weights must be different. B. The PF must be equally weighted. C. The rate changes are assumed to be parallel. D. All the bonds in the PF must be in the same risk class or along the same yield curve.

C (풀이: Duration measures assume a parallel shift in the yield curve. Duration is not a good measurer of nonparallel shift.)

An option with a strike price of $12 and a current stock price of $12 that has one week until expiration is likely to have a gamma to an option seller that is: A. positive & large. B. positive & small. C. negative & large. D. negative & small.

C (풀이: Gamma is the most positive for ATM options near expiration for an option seller.) *High, positive theta = Large, negative gamma

Which of the following day count conventions would most likely be used in pricing an Australian money market security? A. 30/360 B. Actual/360 C. Actual/365 D. Actual/actual

C (풀이: The actual/365 day count convention is typically used for money market securities in Canada, NZ, and Australia.)

Assume the 1-year spot rate is 4%, the 1-year forward rate starting in 1 year is 5%, and the 1-year forward rate starting in 2 years is 6%. Under the realised forward scenario, the realised 1-year rate in 1 year would be: A. 4%. B. 4.5% C. 5%. D. 5.5%.

C (풀이: Under the realised forward scenario, as forward rates are realised, they will be equal to the expected future spot rates. As a result, the realised 1-year rate in 1 year would be 5%.)

BSM 주요 formula 4개 대기!

C0, P0, d1, d2 계산할 수 있어야 함!

What is the best known one factor model?

CAPM

How does CDSs provide important information about country/sovereign credit risk?

CDS의 credit protection은 프리미엄을 지불하면 보험처럼 제공 받을 수 있는데, 이 프리미엄이 얼마인지에 따라 그 나라의 위험을 판단할 수 있음.

(아리까리) Calculate VaR of the fifth percentile using historical simulation and the data provided below: Six lowest returns/Historical simulation weight/Cumulative weight -4.70%/0.01/0.0100 -4.10%/0.01/0.0200 -3.70%/0.01/0.0300 -3.60%/0.01/0.0400 -3.40%/0.01/0.0500 -3.20%/0.01/0.0600

Calculating VaR of 5% requires identifying the 5th percentile. 그러나 observations must be thought of as a random event with a probability mass centred where the observation occurs, with 50% of its weight to the left and 50% of its weight to the right. Therefore, the "5th percentile" is somewhere "between the 5th and 6th" observation. (그냥 5번째 값은 represents the 4.5th percentile.) ∴ [(-3.40% + -3.20%)/2] = -3.30%

Call option과 warrant의 차이점은?

Call option - 이미 발행된 (outstanding) 주식 사들일 권리 vs. Warrant - "새로" 발행해달라고 말할 권리 - 기업의 본질은 그대로 (no benefits)

Call & Put의 가장 중요한 차이점은 무엇?

Call: Δ=0과 1 사이 Put: Δ=-1과 0 사이 (마찬가지로 BSM 공식 미분해서 Δ 구하면 Δ=N(d1)-1)

For PFs with many non-normal variables, the _____ lets the PF to be approximately normal. 이렇게 하면 VaR이나 ES 계산할 때 어떻게 되나?

Central Limit Theorem, mean(뮤) 빼고 계산해도 됨.

Knock-on effect란?

Consequences of other firm's reactions to adverse effects.

Call option 가치를 결정하는 변수들은? 그 중 유일하게 관측 불가능한 변수는?

Ct = f(St, σ, T-t, X, r) 이 중 σ(voltility)는 unobservable, 따라서 항상 그냥 추정치를 사용함~

Sovereign default/country risk 측정하는 방법은? Corporate ratings과 비교하면?

Corporate ratings: debt-to-equity ratio 사용 Sovereign default/country risk: debt-to-GDP ratio 사용, rating transition 역시 중요하게 측정!

Suppose an analyst is looking to estimate the updated correlation between two asset returns. The analyst observes on day n-1 that return X is 2% and Y is 4%, and the correlation between X and Y is 0.3. The volatility of return X and Y is 1% and 2% respectively. The analyst estimates a value for λ of 0.92. When the volatilities of X and Y on day n are now updated to 1.11% and 2.23%, calculate the new coefficient of correlation.

Corr x,y = 0.0001192/(0.0111*0.0223) = 0.48

EWMA를 활용한 correlation estimates) - Return Xn-1 = 2% - Return Yn-1 = 4% - Correlation(xy) = 0.3 - Volatility Xn-1 = 1% - Volatility Yn-1 = 2% - λ = 0.92 Q. Cov(n)과 Cov(n-1) 구하기!

Cov(n-1) = Cov(xy) = Corr(xy)*σx*σy 따라서 = 0.3*0.01*0.02=0.00006 Cov(n) = 0.92*0.00006 + 0.08*0.02*0.04 = 0.0001192

A call option and an MBS are good examples of: A. a linear and nonlinear derivative, respectively. B. a nonlinear and linear derivative, respectively. C. linear derivatives. D. nonlinear derivatives.

Key rate duration is most accurately described as: A. the dollar change in PF value associated with a 1 basis point change in yield. B. the dollar change in PF value associated with a 100 basis point change in yield. C. the percentage change in PF value associated with a 1 basis point change in yield. D. the percentage change in PF value associated with a 100 basis point change in yield.

Reinvestment risk would not occur if: A. interest rates shifted over the time period the bond is held. B. the bonds were callable. C. bonds are issued at par. D. only ZCBs are purchased.

The mean variance framework is inappropriate for measuring risk when the underlying return distribution: A. is normal. B. is elliptical. C. has a kurtosis equal to three. D. is positively skewed.

XYZ stock has a current price of $30 and an expected value in nine months of $34. The expected annual return is closest to: A. 11.76%. B. 12.52%. C. 13.33%. D. 16.69%.

The parameters of a GARCH (1,1) model are ω=0.0003, α=0.04, and β=0.92. These figures imply a long-run daily SD of: A. 1.68% B. 1.55% C. 1.45% D. 2.74%

D 0.00003/(1-0.04-0.92)=0.00075, √0.00075=0.0274=2.74%

An annuity pays $10 every year for 100 years and currently costs $100. The YTM is closest to: *계산기 이용!!!!!!!!!!!!! A. 5%. B. 7%. C. 9%. D. 10%.

D N=100 PMT=10 PV=-100 CPT ∴ I/Y=10%

Which of the following statements accurately reflects a Basel Committee stress-testing principle? A. Stress-testing models should be reviewed at least twice per year. B. Stress-test results should not be communicated beyond senior management and the board. C. The risk captured in a stress-testing framework should be comprehensive, ranging from mild to extreme. D. Stress-testing framework objectives should be aligned with the overall risk management framework.

D [풀이] - A가 아닌 이유: 굳이 1년에 2번씩 할 필요는 없음 - C가 아닌 이유: stress tests는 mild risk 잡을 필요 없고 extreme만 잡음

Which of the following statements about expected loss (EL) and unexpected loss (UL) is true? A. Expected loss always exceeds unexpected loss. B. Unexpected loss always exceeds expected loss. C. Expected loss requires quantifying the actual loss. D. Expected loss is directly related to the exposure amount.

D [풀이] - EL increases with increases in the exposure amount. - UL typically exceeds EL, but they are both 0 when PD=0. - UL requires quantifying actual loss.

A BBB rated bond has a 1.78% chance of defaulting within five years. If the expected loss on bond principal is 0.445%, the recovery rate will be closest to: A. 5%. B. 15%. C. 25%. D. 75%.

D (풀이: 1-(0.00445/0.0178)=0.75)

An investor has purchased an equity security which is part of the S&P 500 index. Relative to the normal distribution, she can reasonably expect the security's returns to: A. be less peaked. B. exhibit zero skewness. C. have no excess kurtosis. D. be more extreme in both directions.

D (풀이: Equity security = risky asset, don't tend to follow the ND but rather has fatter tails.)

Which of the following reasons best explains why institutions use reverse stress tests? A. To identify liquidity risk. B. To identify risk concentrations. C. To assess where multiple risks occur simultaneously. D. To test events that threaten the viability of the institution.

D (풀이: Use reverse stress-tests to assess the events that are outside of normal business expectations and could threaten the institution's viability.)

A $1,000 par bond carries a coupon rate of 10%, pays coupons semiannually, and has 13 years remaining to maturity. Market rates are currently 9.25%. The price of the bond is closest to: *계산기 이용!!!!!!!!!!!!! A. $586.60 B. $1,036.03. C. $1,055.41. D. $1,056.05.

D N=26 PMT=50 I/Y=4.625 FV=1,000 CPT ∴ PV=$1,056.05.

An investment pays $50 annually into perpetuity and yields 6%. Which of the following is closest to the price? A. $120. B. $300. C. $530. D. $830.

D PV=C/I=50/0.06=$833.33

Assume the SD for factor scores are 10.25, 7.16, 4.12, and 3.08. How much of an impact do the first two factors together have relative to the total variance? A. 41.65% B. 57.48% C. 70.74% D. 85.52%

D Total variance: (10.25)^2 + (7.16)^2 + (4.12)^2 + (3.08)^2 = 182.79. First two variances: (10.25)^2 + (7.16)^2 = 156.33 ∴ 156.33/182.79 = 85.52%

Relative to other measures of risk, stress testing is more likely to: A. use relatively short time horizons B. capture both positive and negative events. C. capture a large number of extreme scenarios. D. be forward-looking without providing probabilities for loss distributions.

D [풀이] Stress tests are: - Forward-looking - Do not provide probabilities for loss distributions. - Time horizons are typically long. - Only negative events are captured. - The number of extreme scenarios tends to be relatively small.

Which of the following statements regarding foreign currency defaults is most accurate? A. In recent years, defaults have often been followed by military actions. B. Greater central bank independence means less difficulty for a country to print money. C. Prior to the 20th century, no country had ever defaulted on funds borrowed in a foreign currency. D. Countries are more likely to default on funds borrowed from foreign banks than on sovereign bond issues.

D (A가 아닌 이유: 최근엔 military actions 없었음 / B가 아닌 이유: more difficulty / C가 아닌 이유: 여러 나라가 경험함)

신용등급이 _____ 때, 시장에 더 큰 shock이 온다. 이유는?

Downgrade 되었을 때. 이유: 굿뉴스(upgrade)는 보통 기업이 먼저 기사를 내기 때문에 시장이 잘 흡수하는데, 밷뉴스(downgrade)는 그렇지 않음.

In an attempt to understand country risk, an analyst at Global Funds examines multiple sources of information to determine the truest measure of risk. She considers sovereign risk ratings, default risk spreads, and composite measures of risk. Which of the following sources relies on surveys of several hundred economists to measures sovereign risk? A. Political Risk Services. B. The Economists. C. World Bank. D. Euromoney.

D (풀이: Euromoney surveys 400 economists who assess country risk factors and rank countries from 0 to 100, with higher numbers indicating lower risk.)

Which of the following choices will effectively hedge a short call option position that exhibits a delta of 0.5? A. Sell two shares of the underlying for each option sold. B. Buy two shares of the underlying for each option sold. C. Sell the number of shares of the underlying equal to half the options sold. D. Buy the number of shares of the underlying equal to half the options sold.

D (풀이: For every 2 options sold, buy a share of the underlying security.)

The models developed by organisations such as KMV are: A. driven by a goal of ratings stability. B. based on a through-cycle approach. C. non-responsive to equity and debt volatility. D. highly responsive to changing circumstances.

D (풀이: KMV models are highly responsive to changing circumstances (특히 equity and debt volatility) because they are more point-in-time while external rating agencies which tend to strive for ratings stability and use a through-the-cycle methodology.)

Which of the following statements is most accurate regarding implied volatility in the BSM model? A. Volatility is constant across strike prices. B. Volatility is most accurately applied using historical data. C. The process for estimating volatility involves two steps at most. D. Volatility is often derived using the BSM market price and the other inputs.

D (풀이: Volatility is not directly observable, and so to estimate it, the price of the option using the BSM model and the other observable inputs are put into the model to derive volatility.) - A가 아닌 이유: Volatility is not constant across strike prices. - B가 아닌 이유: Historical data는 Volatility 추정에 유용하나, 현재 혹은 미래의 volatility를 추정하진 못한다! - C가 아닌 이유: 2개 이상의 단계를 거친다!

Which of the following actions is least likely a component of the validation and independent review of stress tests? A. Using expert-based judgement. B. Testing data during non-stress periods. C. Communicating stress-test results to all stress-test users. D. Reviewing the qualitative but not the judgmental aspects of stress tests.

D (풀이: 둘다 리뷰해야 함)

액면가 = $100,000 쿠폰 = 6%, semiannual 만기 = 5년 현재 채권가 = $100,750.00 <이때, parallel shift of IR 10bp> - 그 결과 채권가 = $101,181.44 - DV01은? (계산기 사용!!)

DV01 = -(101,181.44-100,750.00)/10 = $43.14 - 이자율이 1bp 변하면 채권가격은 $43.14 변화한다. [계산기 사용] N=10 PMT=3,000 PV=-100,750 FV=100,000 CPT I/Y=2.9 (semiannual basis, annual basis=5.825) - 이때, 문제처럼 10bp 조정하면 5.725, semiannual basis=2.863. - 저 위의 계산기 process에 I/Y=2.863으로 넣으면 PV=-101,181.44 도출 가능!

(중요) Suppose that a bond with a face value of $100,000 and a coupon of 6% (compounded semiannually) matures in five years. The bond is currently priced at $100,750.00. Also, suppose there is a parallel shift in the interest rate term structure by 10 basis points, and that the bond's price increases to $101,181.44. Compute the DV01. (+채권가격의 변동 직접 유도해보기) *계산기 이용!!!!!!!!!!!!!

DV01 = -(101,181.44-100,750.00)/10 = $43.14 ∴ 1bp에 채권가격 $43.14 만큼 변화한다. *채권가격 변동 유도 N=10 PMT=3,000 PV=-100,750 FV=100,000 CPT I/Y=2.9 <Annual로 변환하면 5.825, 문제처럼 10bp 조정하면 5.725, 다시 semiannual로 변환하면 I/Y=5.725/2=2.863> N=10 PMT=3,000 FV=100,000 I/Y=2.863 CPT PV=101,181.44

신용등급 관련) Watchlist란?

Designed to reflect a short term (3달 이하) anticipated change, which can be positive or negative.

Bond valuation이 coupon date들 중간중간에 일어난다면 계산에 줘야 하는 변화는?

Dirty price를 계산해야 함

Discount rate과 discount factor는 어케 다른겨?

Discount rate = implicit interest = e.g. 2% Discount factor = e.g. 0.98

Historical volatility 사용하는 방법은?

Discrete return = Continuous return 으로 두고, e^r = 1+R 풀면 연속수익률 r=ln(1+R), r의 SD가 곧 historical volatility.

"Dollar" duration과 그냥 duration의 차이는?

Dollar duration = 채권가격의 dollar 변화 = DV01 = 캡처본 참고 Duration = 채권가격의 % 변화

Callable/Puttable bond의 가격 구하는 방법은?

Effective duration 사용 (1) Straight bond의 가격 구하기 (2) Call/put value 구하기 ( 채권의 P+, P- 알 수 있음) *Callable/Puttable bond = straight bond - call/put value

An investor's $1 million PF has a DV01 of $340. This PF can be hedged with a 3% coupon, 5-year bond with a DV01 of $285. Calculate the face amount of the bond required to hedge the investor's initial position.

FV of hedging instrument = 1 million * (340/285) = $1,192,982.46 Therefore, the investor's $1 million PF with a DV01 of $340 can be hedged with a $1,192,982 FV 3% coupon, 5-year bond with a DV01 of $285.

Factor scores란?

Factors가 PF를 얼마나 변동시키는지 숫자로 나타낸 변수값 - Variable values relating to a specific data point covering daily changes, with SDs aligned with the relative importance of each factor. (캡처본 참고)

Stochastic volatility란?

Fat tail = 분포의 volatility와 mean이 시간이 흐름에 따라 변해서 생기는 것. 이때, volatility가 unpredictable하게 변하면 확률적으로 변하는 stochastic volatility!

For a 1% increase in the volatility of the option (12% to 13%), the value of the option will increase by approximately 0.01*1.375 = 0.01375.

Forward value 계산하는 방법은?

Forward value = 현재주가 - PV(약속가격)

Operational risk의 loss severity should be adjusted for _____.

Inflation

Gamma measures the rate of change in the option's delta, so for a $1 change in the price of the stock, the delta will change by 0.0183.

One factor model이란? Gaussian Copula model과의 차이점은?

Gaussian Copula model은 만약 large number of distribution이 있다면 correlation 계산이 쉽지 않은데 (parameter 역시 많이 필요해서), 이 문제를 해결해주는게 One factor model! - 종속변수: 기업의 성장률, etc. - 독립변수: 동일한 market factor가 있고, specific factor는 기업별로 차별화할 수 있음

Monte-Carlo approach란? 실행 단계는?

Generating scenarios using random samples & simulates thousands of valuation outcomes for 기초자산들, from diverse risk factors. (1) Using current values of risk factors, value the PF today. (2) Apply sampling techniques from the multivariate normal prb. distribution for changes in x. (3) Using the sampled values of Δx, determine the values of the risk factors at the end of the period. (4) Revalue the PF using the updated risk factor values. (5) Subtract the revalued PF value from the current value to determine the amount of loss. (6) Repeat above steps to create a loss distribution. - 이 모든 과정이 다 끝나면 그 분포를 토대로 VaR, ES 계산 가능.

신용등급을 매길 때 grade, notch, outlook의 차이점은?

Grade: AAA-AA-A-... Notch: +, _, - 같이 알파벳 뒤에 붙는거 Outlook: Positive, neutral, negative (전망)

Rho는 언제 제일 큰가?

ITM. 이유: 이자율이 오르면? (1) Larger increases for ITM call prices than OTM. (2) Larger decreases for ITM put prices than OTM.

롱 콜옵션의 payoff 그래프와 연관지어 delta를 표기하자면? (delta는 언제 제일 큰가?)

ITM일 때 제일 큼.

Binomial option pricing model의 시점들을 증가시키면 생기는 일은?

If we continue to shrink the length of intervals in the model until "arbitrarily small", we approach continuous time where we use the BSM model.

PR risk와 number of asset의 관계는 (1) Perfect vs. (2) Imperfect

Imperfect. 아무리 자산 개수 늘려도 (아무리 diversify 해도) systemic risk는 존재함!

Value of each warrant 공식은? 공식으로부터 얻을 수 있는 한가지 implication은?

Implication: company's stock price will decline by [M/(M*N) * value of regular call option].

Law of one price란?

Instruments with identical risk & CF should sell for the same price.

Investor A expects an upward-sloping term structure to flatten in the coming months, with long-term rates falling and short-term rates rising. Investor B expects the same term structure to go in the opposite direction. Describe the appropriate strategies for each investor.

Investor A: long long-term bonds & short short-term bonds. - Long-term rates falling = long-term bonds' 가격 상승 - Short-term rates rising = short-term bonds' 가격 하락 Investor B: short long-term bonds & long short-term bonds. - Long-term rates rising = long-term bonds' 가격 하락 - Short-term rates falling = short-term bonds' 가격 상승 *Bond rate과 price를 구분하여 생각하는게 중요한 문제

EWMA model의 장점은?

It requires few data points. All we need: (1) Current estimate of the variance (2) Most recent squared return on day n-1

Bond ratings are considered _____ term ratings. Why? (1) Long (2) Short

Long 이유: Bonds offer periodic interest payments & principal payback at maturity to their debtholders.

Bond spread는 무엇과 무엇의 차이? Spread 없애려면 어떻게 해야하나?

Market price와 computed price의 차이 - Spread 없애려면 S&F들에 "Z-spread" 더해서 조정해야 함

Assume a stock has an expected annual return of 12% and an annual volatility of 20%. Calculate the mean and SD of the probability distribution for the continuously compounded average rate of return over a four-year period.

Mean = 0.12 - (0.2^2/2) = 0.10 SD = 0.2/(√4) = 0.10

Duration이란?

Measure of bond price's volatility

Sovereign credit spread란?

Measure of sovereign default risk - Reflects the sovereign bond yield compared to a riskless investment (동일 통화, 만기) - e.g. 두개의 나라(미국 vs. 신흥국; 둘 다 달러화 활용)가 동일 통화로 채권을 발행하면, 그 채권들 간의 가격 spread을 통해 어느 나라가 신용위험이 더 큰지 파악 가능!

Forward bucket '01s는 무엇을 위해 존재하는가?

Measuring risk based on changes in the shape of the yield curves

What are the respective key guiding lines for Moody's and S&P? Key guiding lines란 무엇을 의미하는가?

Moody's: Baa3 S&P: BBB- *Key guiding lines: 저기까지는 투자등급, 그 아래로는 투기등급.

Autocracies are ____ likely to default than democracies.

Bond valuation 하려고 계산기 사용할때 눌러야 하는 6가지 버튼들은?

N PMT FV I/Y CPT PV

Suppose a fixed-income instrument offers annual payments in the amount of $100 for 10 years. The YTM for this instrument is 10%. Compute the price (PV) of this security. *계산기 이용!!!!!!!!!!!!!

N=10 PMT=100 I/Y=10 CPT ∴ PV=$614.46

Suppose a fixed-income instrument offers annual payments in the amount of $100 for 10 years. The current value for this instrument is $700. Compute the YTM on this security. *계산기 이용!!!!!!!!!!!!!

N=10 PMT=100 PV=-700 CPT ∴ I/Y=7.07%

Suppose now that the security in the previous example plays the $100 semiannually for five years. Compute the periodic yield and the YTM on this security. *계산기 이용!!!!!!!!!!!!!

N=10 (notice how this is 동일 with 만기 10년 annual payment) PMT=100 PV=-700 CPT ∴ I/Y=7.07% *단, 이건 semiannual payment. To compute the annual YTM, multiply the periodic yield by the number of periods per year, 2. This produces a YTM of 14.14%

Naked position이란? Covered position이란?

Naked position: one party selling a call option without owning 기초자산. Covered position: one party selling a call option while owning 기초자산

Risk는 무조건 줄이는게 좋다. True or false?

No! 특정 리스크를 줄이는 cost가 그 이점을 outweigh 한다면 그대로 두는게 좋음.

Is delta constant?

No. It will change over time. (+ Gamma measures the speed of change)

주가에 배당이 미치는 영향은? 이 영향을 완화시키려면 어떻게 해야하나?

On an ex-dividend date(배당락일; 이날 주식 사면 다음 배당 받을 권리 없음), the stock price will naturally decline. 따라서 best approach is to remove stock price changes on ex-dividend dates from datasets.

투자등급 회사들의 default probability가 시간이 지날수록 증가하는 이유느?

Over time, there is greater chance that the financial health of the issuing entity declines. (시간이 지날 수록 초창기의 좋은 financial health가 유지될거라는 보장이 적어짐)

Duration이 가진 위험은?

Overestimation of risk - Long position이면 채권가격 하락시 손실이지만, 직선만큼 떨어지진 않음

Assume we have a non-dividend-paying stock with a current price of $100 and volatility of 20%. If the risk-free rate is 7%, the price of a 6-month at-the-money call option, according to the BSM model, will be $7.43. The corresponding put option price will be $3.99. Now, assume that the same stock instead pays a $1 dividend in two months and a $1 dividend in five months. Compute the value of a 6-month call option on the dividend-paying stock.

PV of the first dividend: 1*e^(-0.07*0.1667) = 0.9884 PV of the second dividend: 1*e^(-0.07*0.4167) = 0.9713 Stock price = 100 - 0.9884 - 0.9713 = $98.04 이 주가 정보 가지고 캡처본 참고! *Since the dividend reduces the value of the stock, the call value decreased, and the put value increased compared to the non-dividend-paying stock.

Calculating partial KR01s) KR01(1), KR01(3), KR01(1)은? Bond maturity: 1년/5년/10년/15년 Decrease in PF value (as spot rate increases): 48.75/302.65/595.10/856.45 Bond maturity: 1년/5년/10년/15년 1년 spot rate: 1/0.50/0/0 3년 spot rate: 0/0.50/0.40/0.15 10년 spot rate: 0/0/0.60/0.85

Partial KR01s) - KR01(1) = 48.75*1 + 302.65*0.50 = 200.08 - KR01(3) = 302.65*0.50 + 595.10*0.40 + 856.45*0.15 = 517.83 - KR01(10) = 595.10*0.60 + 856.45*0.85 = 1,085.04

Binomial model로 아메리칸 옵션 value 할때, 조기행사 조건은?

Payoff from 1기간 (intrinsic value) > option's value (= PV of expected payoff from 2기간)

Bond's net realised return 공식

Per-period financing cost 빼주기

Bond valuation 하려고 계산기 사용할때 I/Y는 무엇인가?

Periodic YTM (10%인데 annual이면 I/Y=10, semiannual이면 I/Y=5%)

액면가=$1,000 쿠폰=10%, semiannual 마지막 이자지급일 = 90일 전 위와 같은 상황일 때의 accrued interest (경과이자) 계산하기!

Periodic coupon = 1,000*(10%/2) = $50 AI = 50*(90/180) = $25 *분모는 semi-annual이니까 180!

Bond valuation 하려고 계산기 사용할때 PMT란?

Periodic 쿠폰 (% 말고 실제 dollar 금액)

신용등급 평가 시, 가능한 outlook의 종류는?

Positive, negative, stable (unlikely to change), developing (변할 수 있지만, 방향은 미정)

Total regulatory capital이 너무 많으면 안좋은 이유는?

ROE를 낮추기 때문

Monte-Carlo simulation의 단점 1가지는?

Processes are slow & computationally intensive. 따라서 이 time-consuming 하다는 특징 떄문에 large PF 관련 계산할 때 MC simulation 많이 돌림.

Put-call parity의 기반이 되는 옵션들은? 그래서 Put-call parity 공식은?

Protective put option (나중에 팔기 위한 기초자산 소지) & fiduciary call option (나중에 권리 행사할 때 필요한 금액 소지) *이 두 옵션은 동일한 CF & risk, 그래서 PV 역시 동일함! 공식: Co + Xe^-rf = Po + St (캡처본 참고)

The current price of Downhill Ski Equipment, Inc., is $20, the risk-free rate if 4% per year, and the price of a 1-year call option with a strike price of $20 is $1.76. Compute the value of a 1-year European put option on Downhill Ski Equipment with a strike price of $20. (*Use Put-Call Parity)

Put = $1.76 - $20 + $20*e^(-0.04*(1)) = $0.98

Operational risk management에서 등장하는 RCSA란?

Risk & control self-assessment = surveying the managers responsible for each BU. *단, do not expect the managers to disclose risks that are out of control.

Option) Risk-neutral valuation이란? 이게 필요한 이유는?

Risky (불확실한) 상황임에도 불구하고 risk-free rate으로 option의 expected payoff를 discount하는 것. 왜 하나? The Law of one price dictates that if 2 investments (캡처본 참고) provide the same CFs at the same times, they both should sell for exactly the same price.

Stress testing을 실행하는 주체는?

Senior management

Senior management는 _____가 발견됨에 따라 _____를 업데이트 해야 함!

Senior management는 new risk가 발견됨에 따라 stress testing model을 업데이트 해야 함.

KR Duration이란?

Sensitivity of a PF value to a 100bp Δ in yield. Σ{ΔP(pf)/P(pf)}/ΔYn (where Yn = KRs)

Convexity란?

Sensitivity of duration to changes in IR *Duration: sensitivity of price to changes in IR

Money market instruments' ratings are considered _____ term ratings. Why? (1) Long (2) Short

Short 이유: MMFs last 1 year or less & simply return 1 final payment

Stop-loss strategy란?

Short call 포지션의 손실 막는 전략. <Purchasing the 기초자산 for a naked call position when the asset rises above 행사가격> *Asset sold as soon as it goes below 행사가격. *Works well when the option is initially ITM. 정리: Naked when OTM, covered when ITM. 한계: (1) 주가는 행사가격 전후로 자주 왔다갔다 하니, 비용이 큼 (2) 주가 예상해서 결정하는게 아니라 동향을 봐서 사후거래 하는거여서 만기의 가격이 uncertain.

BSM) European options with dividends - 주가 So 어떻게 조정?

So를 So*e^(-q*t)로 바꾸기! (q=continuously compounded rate of the dividend payment)

금리들 중 진정한 화폐의 시간가치는? Par rate이 될 수 없는 이유는?

Spot rate. Par rate은 쿠폰 노이즈 때문에 불가!

Assuming K=100 (an estimation window using the most recent 100 asset returns), estimate a conditional mean assuming the market is known to decline 15%.

The daily conditional mean asset return, μt, is estimated to be -15 bps/day: μt = -1500 bps/100 days = -15 bps/day

Spot rate provides the same information as _____.

The discount factor d(t).

Theta per trading day: -2.49/252 = -0.00988

Theta는 언제 제일 큰가? 그림으로 표현하면? 예외적인 상황은?

Theta는 시간이 지나면서 옵션가치가 떨어지는 속도 - ATM+짧은 잔존만기에 제일 큼 (하락속도 가속화; 음수의 절대값 증가) *유러피안 옵션이면서 ITM이면, positive theta도 가능함. (아메리칸은 절대 불가) - 잔여만기가 긴거보다 짧은게 더 가치있다는건데, 특히나 deep ITM일수록 시간 더 흐르면 어케 될지 모르고 걍 빨리 돈 챙기고 싶으니 만기 얼마 안남은게 좋음.

Gamma는 언제 제일 큰가? 그림으로 표현하면? 예외적인 상황은?

Theta처럼 ATM+짧은 잔존만기에 제일 큼. *단, deep ITM이거나 OTM이면 changes in stock price have little effect on gamma.

Determine how a risk manager could estimate the VaR of an equity index futures contract. Assume a one-point increase in the index increases the value of a long position in the contract by $500.

This relationship is shown mathematically as: Ft=$500St, where Ft is the futures contract and St is the underlying index. The VaR of the futures contract is calculated as the amount of the index point movement in the underlying index, St, times the multiple, $500 as follows: VaR(Ft)=$500VaR(St)

신용등급 평가 시 사용되는: (1) Through the cycle (2) Point-in-time ratings 두 방식 비교해보기

Through the cycle 방식이 더 많이 사용됨!

Delta hedging의 목적은?

To make a riskless position

Backward induction이란?

Two-step binomial model에서 만기 시점 수치들로 시작하는 것

The current price of Downhill Ski Equipment, Inc., is $20. The annual SD is 14%. The continuously compounded risk-free rate is 4% per year. Assume Downhill pays no dividends. Compute the value of a 1-year European call option with a strike price of $20 using a one-period binomial model. (+ two-period binomial model이라면?) (+ 반대로 같은 성질의 put option의 가치는?)

Up-move factor: U=e^(0.14*√1)=1.15 Down-move factor: D=1/1.15=0.87 The risk-neutral probabilities of an up-move and down-move are: πu = 0.61 πd = 1-πu = 0.39 Expected value of the option in 1 year: ($3*0.61)+($0*0.39)=$1.83. ∴ PV of the option's expected value = 1.76 ===== + Two-period이라면: Expected value of the option in 2 year: (0.61*0.61*6.45)+(0.61*0.39*0)+(0.39*0.61*0)+(0.39*0.39*0)=$2.40 ∴ PV of the option's expected value = 2.40/e^(0.04*(2)) = $2.21 ===== +The value of the put option: (풋-콜 패리티 사용) Put = $1.76 - $20 + $20*e^(-0.04*(2)) = $0.67

GARCH model의 장점은?

Useful when modelling volatility clustering

Bond valuation using par rates) 공식은?

V (% of par) = 1 + {(c-p)/2} + Σ현가계수

기초자산과 파생상품의 VaR 관계는?

VaR of 파생상품 = Δ*VaR of 기초자산

For a $100,000,000 PF, the expected 1-week PF return and SD are 0.00188 and 0.0125, respectively. Calculate the 1-week VaR with a 95% confidence level.

VaR=[0.00188-1.65(0.0125)]*100,000,000 =-$1,874,500 The manager can be 95% confident that the maximum 1-week loss will not exceed $1,874,500.

만약 IR이 _____ 하다고 예상하면 barbell PF 선택할 것! 왜 그런지 이유 2가지도 대기

Volatile (1) Convexity 클수록 (더 심하게 휠수록) 직선(D)보다 outperform. (2) IR가 잔잔하면 그냥 bullet PF가 더 수익률(I/Y=YTM) 높음. (캡처본 참고)

주가 희석화 (dilution) 이란?

Warrant 발행할 때, 유통주식수량 증가해서 주가 하락하는 것.

VaR란? 공식은?

Worst possible loss under normal conditions with a certain CL over a specified period (1-10 days). *풀어 쓰자면, 10일간 손실이 발생한다면 최대 손실액은 95%의 확률로 $-. >>>>>캡처본에서 뮤+ 틀림!!! 뮤- 여야 함<<<<<

Suppose there is a 15-year option-free noncallable bond with an annual coupon of 7% trading at par. If interest rates rise by 50 basis points (0.50%), the estimated price of the bond is 95.586. If interest rates fall by 50 basis points, the estimated price of the bond is 104.701. Calculate the convexity of this bond.

[104.701+95.586-2*(100)]/[(100)*(0.005)^2] = 114.8

주식을 공매도하여 가진 PF를 콜옵션으로 hedge할 때, net asset position은? (binomial model 활용) - 현재주가 = $100 - 최대 주가 = $200 - 최소 주가 = $50 - 콜옵션 행사가격 = $125 - Δ = 0.5

[Δ = 0.5 이기 때문에 콜옵션(양) 2개 long, 주식(울타리) 1개 short-sell] - 주가가 200으로 오르면 (1) Long한 콜옵션은 총 [200-125=75] 이익, 2개 long 했으니까 총 150만큼 이익. (2) Short-sell한 주식은 100 받고 팔았는데 200 주고 갚아야 하니 100 손해, 따라서 총 이익은 [150-100=50]. - 주가가 50으로 내리면 (1) Long한 콜옵션은 총 [50-125=음수], 행사하지 않아 무가치. (2) Short-sell한 주식은 100 받고 팔았는데 50만 줘도 사서 갚을 수 있으니 50만큼 이익. ∴ Net asset position: $50 (no matter which way the stock price moves, the hedged PF will be worth $50.) ∴ PV of the strategy: 50*e^(-0.08*1) = $46.16

Stressed risk metrics란? 장단점은?

[장점] - Conservative(보수적 접근) relative to the typical scenarios incorporated in VaR or ES. - Forward-looking - Focuses on longer periods than VaR or ES. [단점] - Do not necessarily respond to current market conditions. (현재 시장상황은 고려X) - Loss distribution/정확한 possible outcome 제공X - Backtesting(맞는지 나중에 점검해보는) data X

옵션가격에 영향 미치는 function과 연관지어 알아야 하는 Greek 종류 5개 대기!

c=f(S, σ, T, X, r) S: Delta, Gamma (기초자산 가격이 옵션가격에 미치는 영향) σ: Vega (Volatility가 옵션가격에 미치는 영향) T: Theta (잔존만기가 옵션가격에 미치는 영향) X: 행사가격은 고정, 옵션가격과 관계 없음 r: Rho (이자율이 옵션가격에 미치는 영향)

Vega는 언제 제일 큰가? 이것의 implication은? 작을때는 언제?

delta, gamma와 같이 이거도 ATM일 때 제일 큼. This means that options are most sensitive to changes in volatility when ATM. *Deep ITM/OTM: vega = 거의 0

Gamma란? delta와의 관계는? Gamma=0.8이라는 것을 풀어서 설명하면?

delta에서 미분 한번 더 해서, curvature를 표현한 것. 따라서 delta hedging할 때 error 줄이기 위해 gamma 사용하기도 함. - 감마가 0.8이면 주가가 $1 변할 때 Δ가 $0.8 변한다는 것.

Arbitrage opportunities exist when C-Strips' price and P-Strips' price _____.

diverges. *Although transaction costs may negate the profit.

Black's model 이란?

e^(-r*t)를 밖으로 뺐다는게 핵심!

Stress testing은 VaR&ES에 비해 ____ time horizon.

longer

Binomial model로 옵션가격 valuation하는 순서 연상하기

~~~ 연상하세요 대충 머... up/down factor 1시점의 기초자산가격에 곱해서 2시점의 branches 만들고, 행사가격과 비교해서 차익 구하고, 거기에 π들로 가중평균에서 expected payoff 계산 후 무위험이자율로 할인해서 현재가치로 가져오기!

양=콜옵션 울타리=주식 ...일 때, delta의 공식은? hedge 방법은?

Δ = 콜(양)/주식(울타리) - Δ=0.5 means that 0.5 stock is needed to hedge 1 옵션 계약. *뭘 long/short 할건진 자유 (양과 울타리가 서로 반대 포지션만 취하면 됨)

Δ 의 공식은 _____이며 이것은 _____의 개수이다. 1/Δ 의 공식은 _____이며 이것은 _____의 개수이다.

Δ 의 공식은 (Cu - Cd)/(Su - Sd)이며 이것은 perfect hedging을 위해 필요한 주식울타리의 개수이다. 1/Δ 의 공식은 (Su - Sd)/(Cu - Cd)이며 이것은 perfect hedging을 위해 필요한 콜옵션울타리의 개수이다.

Duration만 이용해서 hedging 하는 방법은? [A=대상채권, B=수단채권]

ΔA = -D(A) * A * ΔY ΔB = -D(B) * B * ΔY - Duration: parallel shift에만 적용되니 ΔY는 A & B에서 동일 - For a fully hedged investment, ΔA = ΔB. - 따라서 B = -{A*D(A)}/D(B)

Using the duration/convexity approach, estimate the effect of a 150 basis point increase and decrease on a 15-year, 7%, option-free bond currently trading at par. The bond has a duration of 9.115 and a convexity of 114.8. (+Duration만 사용했다면 일어났을 일은?)

ΔP = -9.115*100*0.015+(1/2)*114.8*100*(0.15^2)=-12.381 Using duration alone would have implied a price change of -9.115*100*0.015=-13.675. (A bond price decline of $13.675) *Duration would have overestimated the decline.

Duration으로 채권가 변동 구하기) ΔP = ___ * ___ * ___

ΔP = -D * P * ΔY

Suppose a 15-year bond with an annual coupon of 7% is currently trading at $98,550. The bond has a duration of 3.28. Compute and interpret the bond's duration for a 25 basis point decrease in all rates.

ΔP = -D*P*Δy = -3.28 * $98,550 * -0.0025 = $808.11 Therefore, for a given 25 basis point decrease in rates, the bond's price is expected to increase by $808.11.

Binomial model) πu와 πd는 어디에 쓰이나? 공식은?

πu = 기초자산 가격 오를 확률 = 공식은 캡처본 참고 πd = 기초자산 가격 내릴 확률 = 1-πu

Suppose that a bank has a PF with 10,000 loans, and each loan is EUR 1 million and has a 0.5% PD in a year. Also, assume that the recovery rate is 30% and the correlation between losses is 0.2. Calculate the SD of the loss from the loan PF and the SD of the loss as a percentage of its size. Assume in this example that L=1.

σ = 0.04937 α = 0.02208

Why is short (공매도) position risky?

가격 오르면 손해인데, 그 손실액이 점점 증가하기 때문.

One factor model에서 결합부도율 구하는 방법은?

각 회사의 one factor model 공식에서 market factor의 계수들끼리 곱한 것!

IR term structure가 flat할 때, discount rate을 (1) par rate으로 둔 경우와 (2) spot & forward으로 둔 경우로 나눠서 [forward=spot=par] 유도해보기

결국 y(par)=Sn=nFn, if y=Sn

Arbitrage 거래 할 때 short은 어떤 방식으로? 무엇을 위해?

공매도! 자기 돈 들이지 않기 위함.

각 부서에게 할당되는 regulatory capital은 좋은건가 나쁜건가?

나쁜 거... 일종의 penalty 개념!

Bankruptcy-free PF란? - 현재주가 = $100 - 최대 주가 = $200 - 최소 주가 = $50 - 콜옵션 행사가격 = $125 - Δ = 0.5 ...일 때의 hedging 방식과 연관지어 생각해보기!

남의 돈도 조금 빌려서 옵션을 사고싶을 때, 아무리 주가가 떨어져도 절대 부도 나지 않을만큼을 차입하는 것. (1기간 후 갚아야 함) - Binomial model에서 알 수 있는 down factor를 곱한 주가가 곧 예상 가능한 최소 주가인데, 그만큼의 PV를 차입하는 것! - 따라서 long 옵션에 들어가는 내 돈은 [주가 - PV(최소주가)] ===== - 주식 PF니까, Δ와 콜옵션 가치에 따라 이 PF를 얼마나 들고있어야 perfect hedging 가능한건지 알 수 있음. Δ = 0.5니까 주식(울타리) 0.5개 long 해야 콜옵션(양) 1개 short 한거 hedge 가능하다는 의미! - 이때, PF는 주가 오르면 [200-50(차입금)=150] 이익, 내리면 [50-50(차입금)=0 이익! - 콜옵션은 주가 오르면 [200-125(행사가격)=75] 이익, 내리면 [50-125(행사가격)=음수] 굳이 행사하지 않아 가치 없음. - 따라서 이 주식 PF의 정확히 절반 (오르면 150/2=75 이익, 내리면 0/0=0 이익) 가지면 콜옵션의 가치와 일치!

Default probability를 계산할 때, cumulative percentage란?

누적부도율; 시간 지날수록 무조건 부도율 증가

IR term structure가 downward sloping인 경우 말로 풀어서 설명하기 (언제 주로 그러는지, 등)

단기금리>장기금리 (단기자금 구하기 어려움) - 주로 금융위기 시 발생 - 실제로 많이 벌어지는 일은 아님!

Stress testing을 하는 사람과 리뷰하는 사람은 _____ 함. (1) 같아야 (2) 달라야

달라야지!!!!!!

External ratings는 주로 _____ 방법을 사용하며, internal ratings는 주로 _____ 방법을 사용한다. (1) Through the cycle / Point-in-time (2) Point-in-time / Through the cycle

답: (1)

The decay factor in an exponentially weighted moving average model is estimated to be 0.94 for daily data. Daily volatility is estimated to be 1%, and today's stock market return is 2%. Calculate the new estimate of volatility using the EWMA model.

답: 1.086%

(1)는 (2)를 체인으로 가진 것과 동일하다. 1. (1) Forward (2) Futures 2. (1) Futures (2) Forward +따라서 시간가치는 어느 쪽이 더 큰가?

답: 1번 +시간가치는 forward가 더 큼.

Default probability를 계산할 때, conditional percentage란? 특징은?

대신 "비교대상(올해 초, 이번 분기 초, 등) 대비" 몇% 부도 났는지. (최초 출발선이 중요한게 아님!) - 투자등급(AAA~BBB): 시간이 지날수록 부도율 증가 - 투기등급 (BB~CCC/C): 시간이 지날수록 부도율 감소 (앞쪽에 높은 부도율이 몰려있기 때문.)

Talk about the policies/procedures/documentation of stress testing.

대충 혀.. 중요X

What is the major implication of fatter tails?

더 위험한 자산들에서 자주 보이며 (현실적), 극단적인 수치가 더 자주 가능하다는 것. 캡처본 참고하기.

Bond price change 와 DV01의 크기는 +X bp 보다 -X bp의 경우 더: (1) 크다. (2) 작다.

더 크다!

Gaussian Copula model이란?

두개의 확률변수가 각자 가진 분포(ND 아닐 수도 있음; 물론 표준 ND면 correlation 결합이 쉬움)를 유지하면서 결합되어 새로운 joint 확률분포를 생성할 경우 두개의 독립된 확률분포를 결합하는 "함수" - e.g. Mapping non-normal distributions into the standard ND. *결합하는 이유: PF처럼 합쳐진 위험을 알고 싶어서!

The _____ the convexity, the _____ the volatility.

둘 다 "higher"

Delta neutral & gamma neutral hedging의 차이는? 둘 간의 관계는?

둘 다 hedging strategy로 가지고 있는게 제일 확실하고 좋음. (Gamma & delta 둘 다 0으로 만들어야 기초자산으로부터 완전히 자유로움) Delta neutral은 주가의 비교적 작은 변화를 hedge할 때 유리하고, 큰 변화는 gamma neutral 활용하는게 좋음.

Call option valuation 시, 만기 때와 만기 이전의 계산방식 비교하기!

만기 이전: Co = [So*N(d1)] - [X*e^(-rf*t)*N(d2)] *BSM 공식 사용 만기 때: Ct = Max(St-X,0) *일반적인 payoff

Longer time horizon을 계산하기 위해서는 σ(volatility)에 √T를 곱하면 되는데, 어떨 때는 이게 정확한 값을 도출하지 못한다. 어떤 상황에서 그런가?

만약 underlying risk factor가 mean-reverting이라면 σ√T를 사용하는건 risk를 "overestimate." *σ√T는 변동성/분산 불변(σt=σ)을 가정하는데, 현실은 그렇지 않음.

감마는 _____ 휠수록 좋다. (1) 적게 (2) 많이

많이 휠수록 좋음! (이유: 높은 수익률 가능)

OIS (overnight index swap) 특징 얘기하기!

매월 고정금리 지급, daily 콜금리 (한달분량) 기하평균 낸 만큼 수취

무디스랑 S&P/Fitch 각각의 rating 순서 나열하기

무디스: Aaa-Aa-A-Baa-Ba-B-Caa-Ca-C-D (Aa1, Aa2, ... 등으로 추가 확장) S&P/Fitch AAA-AA-A-BBB-BB-B -CCC-CC-C-D (AA+, AA, AA-, ... 등으로 추가 확장)

Factor loadings란?

민감도...그냥...이런거... (캡처본 참고)

d(0.5)라는 것은 이 discount factor의 context가 며칠이라는 것?

반년이라는 것.

배당 있을 때는 언제 콜옵션 행사하는게 유리한가? 풋옵션은?

배당 직전 행사할 경우 payoff = St-1 - X 배당락이 난 다음에 행사할 경우 payoff = [St - D] - X*e^(-r*(T-t)) (배당락 가격) 따라서 St-1 - X > [St - D] - X*e^(-r*(T-t)) or D > X(1-e^(-r*(T-t)), 즉 [확실한 CF > 행사 시 포기하는 행사금액의 이자]일 때 옵션 행사하는게 유리하다. *풋옵션은 반대.

Forward rate을 계산하는데 있어 변동금리채(FRN)과 고정금리채의 차이는?

변동금리채: spot rate들을 공식에 대입해서 forward rate 구함 vs. 고정금리채: forward rate까지 이미 다 정해져 있음!

Regime switching의 개념이란?

보통 volatility 높으면 그 주변 시점 역시 높고, 낮으면 그 주변 시점 역시 낮음. 그러나 volatility는 갑자기 빠르게 변할 수도 있는데, 보통 "시장의 국면이 바뀔 때" 그럼!!! = Regime switching ===== 원인: unexpected central bank/government announcements, etc. 결과: market volatility spikes immediately, then sharply declines once markets absorb the news.

Risk premium은 _____에 대해서만 발생

분산투자해도 소용없는 시장risk에 대해서만 발생

Bond's gross realised return with reinvested coupons 공식

분자에 reinvested 금액을 더해준다는 것이 차이점

Barbell PF와 bullet PF 비교하기! 둘 중 선택하는 기준은?

선택은 투자자의 view on IR에 달림! (IR이 얼마나 변할지)

Delta-normal approach는 _____자산(e.g. 주식)에만 적용 가능. 이유는?

선형 정규분포 가정 하는데, 옵션 같은 비선형 자산은 수익률이 정규분포 따르지 않기 때문.

Hazard rate이란? 이걸 알아야 하는 이유는?

순간부도율 = Default intensity = Unconditional default probability를 계산하기 위한 알아야 함.

Vasicek model이란? 용도와 목적은?

신용위험을 internal-ratings-based 방식으로 계산하는 모델 - 이자율의 확률가정모형 (stochastic process) - 용도: assessing counterparty risk by using both EL&UL - 목적: determining regulatory capital (규제자본)

PF insurance란?

아래 두가지를 동시에 들고 있는 것: (1) An underlying instrument (2) Either cash(무위험자산) or a derivative that generates a floor value for the PF in the event that market values decline. (=아무리 기초자산의 시장가격이 내려도 적어도 "이 정도"는 남게 해줄 수 있는,,)

Hedge Ratio 공식은?

양 DV01/울타리 DV01

중앙은행이 independent 할 수록 정부로 하여금 화폐 발행이 ____. (1) 쉬움 (2) 어려움

어려움.

배당이 없을 때, European/American option은 차이 있나 없나? 이유는?

없다. (캡처본 참고) 배당 없을 경우: 중간에 설사 ITM 이어도 내재가치만 얻고 시간가치 손해 보기 때문에 굳이 행사하지 않음. 차라리 옵션 팔면 전체 가치 (시간+내재) 얻을 수 있기 때문에 나음. vs. 배당 있을 경우: [배당>행가가격의 이자소득]이라면, 배당 노리고 행사가격 지불하는게 나음.

사람들의 옵션거래를 하는 이유는?

옵션은 그냥 기초자산 들고있는거보다 레버리지가 크기 때문!

What are "stressed" VaR or ES?

원래 VaR or ES는 보통 comprehensive data 사용하는데, "stressed" VaR or ES는 오직 specifically stressed period의 데이터만 활용. (Conditional on recurrence of a specific stressed period) - If we had a repeat of a specific stressful period, would we be able to survive that period?

Annuity란? PV와 FV 각각의 공식은?

원리금 균등

Put-call parity는 _____ 옵션에만 적용 가능하다.

유러피안

Operational risk의 loss를 측정할 때 "scale adjustment"는 무엇인가?

은행의 상대적 revenue 크기에 따라 estimated loss를 조정하는 것 (베타 = 통상적으로 0.23)

Multivariate Density Estimation이란?

이거도 HS, MC simulation 처럼 data set 만들어서 VaR & ES 측정하는 것. (1) An analysis is first done to see which past periods correspond to the current period. (2) "Weight" is assigned to the historical data depending on "how similar it is to the current data." *Very flexible in introducing dependence on economic variables.

Modifying the binomial model) Valuing a stock that pays a continuous dividend yield - 공식은? - 그 이유는?

이유: 배당수익률 q를 미리 받으면 가격상승으로 보게 될 이득이 그만큼 (rf-q)로 줄어들기 때문.

Modifying the binomial model) Valuing options on futures - 공식은? - 그 이유는?

이유: 현물자산(stock, IR, bond, commodities, etc.)은 가지고 가는데 비용이 발생해서 rf라는 수익률이 필요하지만, 파생상품(futures, swap, option, forward, etc)은 costless 하기 때문에 수익률이 0이라 e^0=1.

Hazard rate을 활용해서 1차년도 기준 2차년도의 conditional default probability를 계산하는 방법은? (공식)

일단 unconditional default probability를 알아야 함: - 1차년도 말의 부도율: 1-e^(-h1*t1) - 2차년도 말의 부도율: e^(-h1*t1) - e^(-h2*t2) 1차년도 기준 2차년도의 conditional default probability: - {e^(-h1*t1) - e^(-h2*t2)}/e^(-h1*t1)

Non-parallel shift in the Yield Curve) Butterfly 그래프 그리고 특징 얘기하기

장단기금리보다 중기금리의 변화폭이 더 큰 shift (Changes in degree of curvature = 볼록성) - Positive butterfly: less curved - Negative butterfly: more curved

Non-parallel shift in the Yield Curve) Twist 그래프 그리고 특징 얘기하기

장단기금리차의 변화로 인한 shift (1) 만약 new curve가 steepened (장단기금리차 상승) - Bear steepener: 단기금리가 많이 감소해서 - Bull steepener: 장기금리가 많이 증가해서 (2) 만약 new curve가 flattened (장단기금리차 하락) - Bear flattener: 단기금리가 많이 증가해서 - Bull flattener: 장기금리가 많이 감소해서

Scenario analysis의 장단점은?

장점: 아직 일어나지 않은 사건들에 대해서 management가 immunise 할 수 있게 함. 단점: 시나리오 및 contingency plan 생성하는 시간

Duration은 항상 실제 price change를 convexity보다 (적게/크게) 계산한다.

적게! 따라서 항상 convexity로 positively(+) adjust해야 함.

Bootstrapping method 순서 연상해보기! 무엇부터 시작? (만기가 다른 5개의 채권의 coupon, maturity, bond price가 주어짐)

첫번째 채권(가장 짧은 만기)의 (주어진 채권가격/원리금)으로 d(1) 구하고, 두번째 채권의 [채권가격 = 이자*d(1) + 원리금*d(2)] 에서 d(2) 구할 때 대입하여 사용하기. + 세번째 채권: 채권가격 = 이자*d(1) + 이자*d(2) + 원리금*d(3) ... *항상 이자는 주어진 채권의 특정 이자에서 m 반영한 값

- 현재 주가 = $20 - Annual SD = 14% - Risk-free rate = 4% (연간, 연속) - No dividends - 1년 만기 유러피안 콜옵션 (행사가 = $20) 위 조건들로 binomial model option valuation 해보기!

캡처본 참고~

Historical data 이용해서 shorter time periods로 scale 하는 방법은? (e.g. 1년 -> 1일 수익률)

캡처본의 T 자리에 1/250 넣기! (1년=휴일,주말 제외 250일)

투자등급과 투기등급 중 economic/industrial cycle에 더 영향을 많이 받는 쪽은?

투기!!! - 투기등급 bond: 부도율 is correlated with cycles - 투자등급 bond: 부도율 is fairly stable.

Two-step binomial model 만들어서 call option valuation 하기! - 현재 주가=$20 (무배당) - Annual SD=14% - Risk-free rate=4% (연간) - 만기=2년, 유러피안 옵션 - 행사가격=$20 - πu=0.61, πd=0.39 가정 + 같은 조건의 풋옵션은?

풋옵션은 put-call parity 활용! ∴ Put = 2.21 - 20 + 20*e^(-0.04*2) = $0.67

Forward buckets란? 특징은?

하나의 버킷 안의 움직임은 다 동일함!

EWMA(exponentially weighted moving average) model 이란?

핵심: "Weights are assumed to decline exponentially back." *λ: decay factor (충격소멸계수), 0과 1 사이, 문제에서 주어짐. *Adaptive volatility estimation 이라고도 함.

배당 없을 때 아메리칸 옵션 (행사한다/안한다.)

행사 안한다. 행사하면 옵션에서 주식으로 바뀌는건데 그러지 말고 걍 옵션 자체를 팔아서 돈을 챙겨라.

Futures의 현재가치는? 이거로 futures delta 구하기 +배당이 있다면?

현재가치: Vt(T)=Ft(T) - Ft-1(T) = St*e^(r*(T-t)) - Ft-1(T) *둘째 항은 어제의 값이며 상수 Forward delta = e^(r*(T-t)) ≠ 1 ===== 배당이 있다면 St*e^(r*(T-t)) 대신 St*e^((r-q)*(T-t)), delta 역시 e^((r-q)(T-t))로 바뀜.

Forward의 현재가치는? 이거로 forward delta 구하기 +배당이 있다면?

현재가치: Vt(T)=St-Fo(T)*e^(-r*(T-t)) Forward delta = 1 ===== 배당이 있다면 St 대신 St*e^(-q*(T-t)), delta 역시 e^(-q*(T-t))로 바뀜.

1년 만기 채권에 투자 vs. 5년 만기 채권에 투자, 1년 후 매매 둘 중 더 높은 수익률을 얻을 수 있는 전략은?

후자 5년 만기 채권 매입 1년 후 4년 만기인 채권을 매매하면, 차익 얻을 수 있음 (5년 만기채 가격 < 4년 만기채 가격)

Book 4

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