Exam IFM
The Agency Benefits of Leverage - Managerial Entrenchment
Because managers face little threat of being replaced, managers can run the firm to suit their interests. Leverage can provide incentives for managers to run a firm more efficiently and effectively due to: Increased ownership concentration. Reduced wasteful investment. Reduced managerial entrenchment and increased commitment.
Short position
Benefits from a decrease in the price of the asset.
Long position
Benefits from a increase in the price of the asset.
Butterfly Spread (Symmetric/Asymmetric)
Buy high- and low-strike options. Sell middle-strike option. Quantity sold = quantity bought.
Static/hedge-and-forget
Buy options and hold to expiration
Synthetic long forward is created by
Buying a stock and borrowing money (selling a bond) or buying a call and selling a put at the same strike
Costs of Asymmetric Information - Credibility Principle
Claims in one's self-interest are credible only if they are supported by actions that would be too costly to take if the claims were untrue. Managers consider how their actions will be perceived by investors in selecting financing methods for new investments: Issuing equity is typically viewed as a NEGATIVE signal as managers tend to issue equity when they believe that the firm's stock is overvalued. Issuing more debt is typically viewed as a POSITIVE signal as the company is taking on commitment to make timely interest and principal payments.
International Bonds
Classified into 4 broadly defined categories: Domestic bonds - issued by local, bought by foreign Foreign bonds - issued by foreign, bought by local Eurobonds - issued by local or foreign Global bonds
R&D-Intensive Firms
Firms with high research and development costs typically maintain low levels of debt.
Formula to guarantee a minimum selling price for a stock
Floor = +Stock + Put
Strike price effects - call
For K1 < K2 < K3: C(K1) >= C(K2) >= C(K3) C(K1) - C(K2) <= K2 - K1 European: C(K1) - C(K2) <= PV(K2 - K1) (C(K1) - C(K2))/(K2 - K1) >= (C(K2) - C(K3))/(K3 - K2)
Strike price effects - put
For K1 < K2 < K3: P(K1) <= P(K2) <= P(K3) P(K2) - P(K1) <= K2 - K1 European: P(K2) - P(K1) <= PV(K2 - K1) (P(K2) - P(K1)) / (K2 - K1) <= (P(K3) - P(K2)) / (K3 - K2)
Time until expiration
For T1 < T2: Camer(S, K, T1) <= Camer(S, K, T2) Pamer(S, K, T1) <= Pamer(S, K, T2) For a nondividend-paying stock: Ceur(S, K, T1) <= Ceur(S, K, T2) This is also generally true for European call options on dividend-paying stocks and European puts, with some exceptions.
Sizing Option
GROWTH options give the company an option to make additional investments when it is optimistic about the future. ABANDONMENT options give the company an option to abandon the project when it is pessimistic about the future.
Option Greeks
Delta: definition = dV/dS, long call = +, long put = - Gamma: definition = ddelta / dS = d^2V/dS^2, long call = +, long put = + Theta: definition = dV/dt, long call = -, long put = - Vega: definition = dV/dsigma, long call = +, long put = + Rho: definition = dV/dr, long call = +, long put = - Psi: definition = dV/ddelta, long call = -, long put = + Theta is usually negative. For short positions, just reverse the signs.
Other Anomalies for EMH - Super Bowl Effect
Historical data shows in the year after the Super Bowl, the stock market is more likely to do better if an NFC team won and worse if an AFC team won.
Round-trip transaction cost
Difference between what you pay and what you receive from a sale using the same set of bid/ask prices.
Costs of Asymmetric Information - Adverse Selection
A seller with private information is likely to sell you worse-than-average goods.
FFC - market portfolio
Accounts for EQUITY RISK. Take a long position in the market portfolio and finance itself with a short position in the risk-free asset.
FFC - small-minus-big portfolio
Accounts for differences in company size based on market capitalization. Buy small firms and finance itself by short-selling big firms.
FFC - high-minus-low portfolio
Accounts for differences in returns on value stocks and growth stocks. Buy high book-to-market stocks (i.e. value stocks) and finance itself by short selling low book-to-market stocks (i.e. growth stocks).
FFC - momentum
Accounts for the tendency of an asset return to be positively correlated with the asset return from the previous year. Buy the top 30% stocks and finance itself by short selling the bottom 30% stocks.
Profit
Accumulated value of cash flows at the risk-free rate.
Gap Option
K1 = Strike price K2 = Trigger price K1 determines the amount of the nonzero payoff. K2 determines whether the option will have a nonzero payoff. Payoff(gap call) = 0 when ST <= K2, ST - K1 when ST > K2 Payoff(gap put) = K1 - ST when ST < K, 0 when ST >= K GapCall = S(0) * e^-delta*T * N(-d1) - K1 * e^-rT * N(d2) GapPut = K1 * e^-rT * N(-d2) - S(0) * e^-delta*T * N(-d1) where d1 and d2 are based on K2 GapCall - GapPut = S(0)*e^-delta*t - K1*e^-rT
Bounds of option prices - European vs. American put
Ke^-rT >= P(eur) >= max(0, Ke^-rT - FP(S)) K >= P(amer) >= max(0, K - S)
Supporting Weak Form of EMH
Kendall found that prices followed a random walk model, i.e. past stock prices have no bearing on future prices. Brealey, Meyers, and Allen created a scatter plot for price changes of four stocks. They found no distinct pattern in the points, with the concentration of the points around the origin. No bias toward any quadrants, and autocorrelation coefficients were close to 0. Poterba and Summers found that the variance of multi-period change is approximately proportional to the number of periods.
Ratio Spread
Long and short an unequal number of calls/puts with different strike prices
Box Spread
Long call (put) bull spread + long put (call) bear spread
Collared Stock
Long collar + long stock
Straddle
Long put (K) + long call (K)
Strangle
Long put (K1) + long call (K2), K1 < K2
Collar
Long put (K1) + short call (K2), K1 < K2
Extreme Lookback Option definition
Lookback options with a fixed strike price
Standard Lookback Option definition
Lookback options with a floating strike price
Low Growth, Mature Firms
Mature, low-growth firms with stable cash flows and tangible assets benefit from high debt.
Defintion and Key Facts of Beta
Measures the sensitivity of the asset's return to the market return. Is defined as the expected percent change in an asset's return given a 1% chance in the market return. The beta for a stock, on average, is around 1. Cyclical industries (tech, luxury goods) tend to have higher betas. Non-cyclical industries (utility, pharmaceutical) tend to have lower betas.
EMH Underreaction/Overreaction Anomalies - New-Issue/IPO puzzle
Overreaction to new issues pushes up stock price initially.
Sharpe Ratio
Phi(option) = option's risk premium / option's volatility = (small gamma - r) / sigma(option) phi(stock) = stock's risk premium / stock's volatility = (alpha - r) / sigma(stock) phi(C) = phi(stock); phi(P) = -phi(stock)
Poisson with mean lambda
Pr[N = n] = (e^-lambda * lambda^n) / n!
Probability
Pr[ST < K] = N(-a2) Pr[ST > K] = N(+ a2) a2 = [ln(St/K) + (alpha - delta - 0.5*sigma^2)(T-t)] / [sigma*sqrt(T-t)]
Corporate Debt - Private Debt
Private debt is negotiated directly with a bank or a small group of investors. It is cheaper to issue due to the absence of the cost of registration.
Equity cost of capital / cost of equity / required return on equity
The rate of return that the equity holders require in order for them to contribute their capital to the firm.
Long call
Right (not obligation) to buy at the strike price Payoff = max(0, S - K) Profit = Payoff - AV(Prem) Max loss = AV(Prem) Max gain = infinity Strategy = insurance against high underlying price
Long put
Right (not obligation) to sell at the strike price Payoff = max(0, K - S) Profit = Payoff - AV(Prem) Max loss = AV(Prem) Max gain = K - AV(Prem) Strategy = insurance against low underlying price
Cost of Capital
The rate of return that the providers of capital require in order for them to contribute their capital to the firm.
Risk Premium
The risk premium of an asset is defined as the excess of the expected return of the asset over the risk-free return: risk premium(option) = small gamma - r risk premium(stock) = alpha - r small gamma - r = omega(alpha - r) sigma(option) = |omega| * sigma(stock)
At-the-money
The spot price is approximately equal to the strike price
True pricing
To calculate option price, discount the true expected option payoff at the expected rate of return on the option: V0 = e^-gamma*T * E[Payoff]
Reasons for using derivatives
To manage risk, to speculate, to reduce transaction cost, to minimize taxes/avoid regulatory issues
Reducing Agency Costs
To mitigate the agency costs of debt, firms and debt holders can: Issue short-term debt. Include debt covenants in bonds that place restrictions on the actions a firm can take.
Supporting Strong Form of EMH
Top performing fund managers in one year have only a 50% chance to beat their reference index the following year. The performance of actively managed mutual funds from 1971 to 2013 only beat the Wilshire 5000 index 40% of the time.
Futures compared to forwards
Traded on an exchange Standardized (size, expiration, underlying) More liquid Marked-to-market and often settled daily Minimal credit risk Price limit is applicable
GMAB with a Return of Premium Guarantee
Similar to GMDB with ROP guarantee, but the benefit is contingent on the policyholder surviving to the end of the guarantee period. The embedded option is a put option. Its value is: P(m) * Pr[Tx >= m]
Other Anomalies for EMH - Size Effect
Small-cap companies have outperformed large cap companies on a risk-adjusted basis.
Reasons Why the Market Portfolio Might Not be Efficient - Alternative Risk Preferences
Some investors focus on risk characteristics other than the volatility of their portfolio, and they may choose inefficient portfolios as a result.
Combining Risky Assets with Risk-free Assets
Suppose we invest a proportion of funds (x) in a risky portfolio and the remainder (1 - x) in a risk-free asset. Then we have: E[R(xP)] = x*E[R(P)] + (1 - x)*rf = rf + x(E[R(P)] - rf) sigma(xP) = x * sigma(P) Note: xP means x% is invested in P
Adding a New Investment
Suppose we have a portfolio, P, with an expected return of E[R(P)] and a volatility of sigma(P). Add the new investment to the portfolio if: Sharpe ratio(new) > rho(new, p) * Sharpe ratio(P) (E[R(new)] - rf) / sigma(new) > rho(new, P) * (E[R(P)] - rf) / sigma(P)
The Debt Cost of Capital
Two methods to estimate debt cost of capital: Adjustment from debt yield: r(d) = y - pL = yield to mat. - Pr(Default) * E[Loss Rate] CAPM using debt betas: rd = rf + beta(d) * (E[R(mkt)] - rf) Note that: It is difficult to get the beta estimates for individual debt securities because they are traded less frequently than stocks The average beta for debt tends to be low. However, the beta for debt does increase as the credit rating increases
Market Risk Premium
Two methods to estimate the market risk premium: The historical risk premium - uses the historical average excess return of the market over the risk-free interest rate. A fundamental approach - uses the constant expected growth model to estimate the market portfolio's expected return. P(0) = Div(1) / (E[R(mkt)] - g) => E[R(mkt)] = Div(1) / P(0) + g
Other Anomalies for EMH - Siamese Twins
Two stocks with claims to a common cash flow should be exposed to identical risks but perform differently.
Options on currencies
Use the generalized BS formula in conjunction with the appropriate prepaid forward prices. Alternatively: S0 = x0, r = rd, delta = rf C = x0*e^-rf*T * N(d1) - Ke^-rd*T * N(d2) P = Ke^-rd*T * N(-d2) - x0*e^-rf*T * N(-d1) d1 = [ln(x0/K + (rd - rf + 0.5*sigma^2*T] / [sigma*sqrt(T)] d2 = [ln(x0/K + (rd - rf - 0.5*sigma^2*T] / [sigma*sqrt(T)] = d1 - sigma*sqrt(T)
Options on futures contract
Use the generalized BS formula in conjunction with the appropriate prepaid forward prices. The prepaid forward price for a futures contract is just the present value of the futures price: FP(F) = Fe^-rT Alternatively, S0 = F(0, TF), delta = r C = F(0, TF)*e^-rT * N(d1) - Ke^-rT * N(d2) P = Ke^-rT * N(-d2) - F(0, TF)e^-rT * N(-d1) d1 = [ln(F(0, TF)/K + 0.5*sigma^2*T] / [sigma*sqrt(T)] d2 = [ln(F(0, TF)/K - 0.5*sigma^2*T] / [sigma*sqrt(T)] = d1 - sigma*sqrt(T)
Cost of carry
r - delta
Capital Asset Pricing Model (CAPM)
r(i) = E[R(i)] = rf + beta(i) * E[R(mkt) - rf] E[R(mkt)] - rf is the market risk premium or the expected excess return of the market E[R(i)] - rf or beta(i) * E[R(mkt) - rf] is the risk premium for security i or the expected excess return of security i Since beta(i) only captures systematic risk, E[R(i)] under CAPM is not influenced by nonsystematic risk.
Constructing a binomial tree - general method
u = Su / S0 d = Sd / S0
Constructing a binomial tree - standard binomial tree
u = e^(r - delta)*h + sigma*sqrt(h) d = e^(r - delta) - sigma*sqrt(h) p-star = 1 / (1 + e^sigma*sqrt(h))
Knock-out Barrier Option
Goes out of existence if barrier is reached
Down vs. Up Barrier Option
If S(0) < barrier: up-and-in, up-and-out, up rebate If S(0) > barrier: down-and-in, down-and-out, down rebate
Levered vs. Unlevered
If a project is financed purely with equity, the equity is said to be unlevered; otherwise, it is levered.
Interpreting Beta
If beta = 1, then the asset has the same systematic risk as the market. The asset will tend to go up and down the same percentage as the market. If beta > 1, then the asset has more systematic risk than the market. The asset will tend to go up and down more than the market, on a percentage basis. If beta < 1, then the asset has less systematic risk than the market. The asset will tend to go up and down less than the market, on a percentage basis. If beta = 0, then the asset's return is uncorrelated with the market return.
The Effect of Correlation
If rho(i, j) = 1, no diversification. The portfolio's volatility is simply the weighted average volatility of the two risky assets. If rho(i, j) < 1, the portfolio's volatility is reduced due to diversification. It is less than the weighted average volatility of the two risky assets. If rho(i, j) = -1, a zero-risk portfolio can be constructed.
Important Logic for EMH
If something supports the stronger form of the EMH, then it also supports the weaker forms. If something violates the weaker form of EMH, then it also violates the stronger forms. However, the reverse is not necessarily true.
Margin call
If the margin balance falls below the maintenance margin, then the investor will get a request for an additional margin deposit. The investor has to add more funds to bring the margin balance back to initial margin.
Multi-factor Model
If the market portfolio is not efficient, a multi-factor model is an alternative. It considers more than one factor when estimating the expected return. An efficient portfolio can be constructed from other well-diversified portfolios. Also known as the Arbitrage Pricing Theory. Similar to CAPM, but assumptions are not as restrictive.
Naked writing
If the option writer does not have an offsetting position in the underlying asset, then the option position is said to be naked.
Covered writing
If the option writer has an offsetting position in the underlying asset, then the option position is said to be covered.
Breakeven
If the price of the underlying stock changes by one standard deviation over a short period of time, then a delta-hedged portfolio does not produce profits or losses. Assuming the BS framework, given the current stock price, S, the two stock prices after a period of h for which the market-maker would break even are: S +- S*sigma*sqrt(h)
Interest Rate Conversion
(1 + i)^t = (1 + i^m / m)^mt = e^rt
Annualized forward premium rate
(1/T)*ln(F(0,T)/S(0))
Write a covered call
-Cap = +Stock - Call
Write a covered put
-Floor = -Stock - Put
4 Common Types of Corporate Debt
1. Notes (unsecured) 2. Debentures (unsecured) 3. Mortgage bonds (secured) 4. Asset-backed bonds (secured)
Two Types of Municipal Bonds
1. Revenue Bonds 2. General Obligation Bonds
2 Types of Private Debt
1. Term loan 2. Private placement
Component 1 of the Present Value of Financial Distress Costs
1. The costs of financial distress and bankruptcy, in the order they occur. Direct costs - fees to outside professionals like legal and accounting experts, consultants, appraisers, auctioneers, and investment bankers. These are higher for firms with more complicated business operations. These are also typically higher, in percentage terms, for smaller firms. Indirect costs - loss of customers, loss of suppliers, loss of employees, loss of receivables, fire sale of assets, inefficient liquidation, cost to creditors. These are difficult to measure and are often much larger than direct costs. They may even occur prior to bankruptcy if the potential perceived threat of future bankruptcy is high. Companies with marketable tangible assets (e.g. airlines, steel manufacturers) have lower costs of financial distress than companies without these assets (e.g. information technology companies, companies in the service industry) because tangible assets can be sold relatively easily.
4 Types of Treasury Securities
1. Treasury bills 2. Treasury notes 3. Treasury bonds 4. Treasury inflation-protected securities (TIPS)
Standard Steps to Launching a Typical IPO
1. Underwriter typically manage an IPO and they are important because they market the IPO, assist in required filings, and ensure the stock's liquidity after the IPO. 2. Companies must file a registration statement, which contains two main parts: preliminary prospectus/red herring, and final prospectus. 3. A fair valuation of the company is performed by the underwriter through road show and book building. 4. The company will pay the IPO underwriters an underwriting spread. After the IPO, underwriters can protect themselves more against losses by using the over-allotment allocation or greenshoe provision.
Component 2 of the Present Value of Financial Distress Costs
2. The probability of financial distress and bankruptcy occurring. Companies with a higher debt-to-equity ratio have a higher probability of bankruptcy. This probability increases when the volatility of a firm's cash flows and asset values increases. Firms with STEADY cash flows (e.g. utility companies) can use high levels of debt and still have low probability of default. Firms with VOLATILE cash flows (e.g. semiconductor firms) must have low levels of debt in order to have a low probability of default.
Supporting Semi-Strong Form of EMH
3 months prior to a takeover announcement, the stock price gradually increased. At the time of announcement, stock price instantaneously jumped. After the announcement, the abnormal returns dropped to zero.
Component 3 of the Present Value of Financial Distress Costs
3. The appropriate discount rate for the distress costs. The higher the firm's beta: The more likely it will be in distress. The more negative the beta of the distress costs. The lower the discount rate of the distress costs. The higher the PV of distress costs.
Catastrophe bond
A bond issued to investors where repayments and principal payments are contingent on there not being a catastrophe which causes large losses for the insurer. Thus, investors who buy these bonds face the risk of not receiving coupon payments or repayment of their principal. In general, cat bondholders typically receive higher interest rates for taking on this risk.
The Agency Costs of Leverage - Excessive Risk-taking and Asset Substitution
A company replacing its low-risk assets with high-risk investments. Shareholders may benefit from high-risk projects, even those with negative NPV.
Compound Call
A compound call allows the owner to buy another option at the strike price
Compound Put
A compound put allows the owner to sell another option at the strike price
Overnight Profit
A delta-hedged portfolio has 3 components: Buy/sell options buy/sell stocks borrow/lend money (sell/buy bond) Overnight profit is the sum of: profit on options bought/sold profit on stocks bought/sold profit on bond Alternatively, overnight profit is the sum of: gain on options, ignoring interest gain on stocks, ignoring interest interest on borrowed/lent money For a market-maker who writes an option and delta-hedges the position, the market-maker's profit from time t to time t + 1 is: deltat[(S(t+h) - St) - (V(t + h) - Vt) - (e^rh - 1)(delta*S - V)] = -0.5*E^2*gamma - h*theta - (e^rh - 1)(delta*S - V) where E = S(t + h) - St If h is small, then e^rh - 1 = rh
Enterprise Value
A firm's enterprise value is the risk of the firm's underlying business operations that is separate from its cash holdings. It is the combined market value of the firm's equity and debt, less any excess cash: V = E + D - C To determine the enterprise value, we use the firm's net debt: Net debt = debt - excess cash and short-term investments The beta of the firm's underlying business enterprise is: beta(U) = w(E) * beta(E) + w(D) * beta(D) + w(C) * beta(C) where w(E) = E / (E + D - C), w(D) = D / (E + D - C), w(C) = -C / (E + D - C)
Funding Round
A funding round occurs when a private company raises money. An initial funding round might start with a "seed round", and then in later funding rounds the securities are named "Series A", "Series B", etc.
GMDB with a Return of Premium Guarantee
A guarantee which returns the greater of the account value and the original amount invested: max(ST, K) = ST + max(K - ST, 0) The embedded option is a put option. Its value is: E[P(Tx)] = the integral from 0 to infinty of P(t)*fTx(t) dt
Capital Allocation Line
A line representing possible combinations from combining a risky portfolio and a risk-free asset. E[R(xP)] = rf + {(E[R(P)] - rf) / sigma(P)} * sigma(xP) At the intercept, the portfolio only consists of the risk-free asset. At point P, the portfolio only consists of risky assets. The line extending to the right of sigma(P) represents portfolios that invest more than 100% in the risky portfolio P. This is done by using leverage (borrow money to invest). A portfolio that consists of a short position in the risk-free asset is known as a levered portfolio.
Efficient Frontier
A portfolio is efficient if the portfolio offers the highest level of expected return for a given level of volatility. The portfolios that have the greatest expected return for each level of volatility make up the efficient frontier.
Expected Returns and the Efficient Portfolio
A portfolio is efficient iff the expected return of every available asset equals its required return. Thus, we have: E[R(i)] = r(i) = rf + beta(i, eff) * (E[R(eff)] - rf)
Arbitrage
A transaction which generates a positive cash flow either today or in the future by simultaneous buying and selling of related assets, with no net investment or risk. Arbitrage strategy is buy low, sell high.
Required Return on New Investment
Adding the new investment will increase the Sharpe ratio of portfolio P if its expected return exceeds its required return, defined as: r(new) = rf + beta(new, P) * (E[R(P)] - rf) In general, the beta of an asset i with respect to a portfolio P is: beta(i, P) = Cov[R(i), R(p)] / sigma(P)^2 = rho(i, P) * sigma(i) / sigma(P)
Haircut
Additional collateral placed with lender by short-seller. It belongs to the short-seller.
Takeover Offer
After the initial jump in the stock price at the time of the announcement, target shocks do not appear to generate abnormal subsequent returns on average. Stocks that are ultimately acquired tend to appreciate and have positive alphas, while stocks that are not acquired tend to digress and have negative alphas.
Assumptions of Mean-Variance Analysis
All investors are risk-averse. The expected returns, variances, and covariances of all assets are known. To determine optimal portfolios, investors need only know the expected returns, variances, and covariances of returns. There are no transaction costs or taxes.
Firm Commitment
All shares guaranteed to be sold at the offer price. Most common.
Alternatives to Bankruptcy - Prepackaged Bankruptcy
Also known as "prepack". The company will first create a reorganization plan with the agreement of its primary creditors, then file Chapter 11 reorganization to implement the plan.
Systematic Risk
Also known as common, market, or non-diversifiable risk Fluctuations in a stock's return that are due to market-wide news
Payoff
Amount that one party would have if completely cashed out.
Asset Backed Securities
An asset-backed security is a security whose cash flows are backed by the cash flows of its underlying securities. Banks also issue ABS using consumer loans, such as credit card receivables and automobile loans. A private ABS can be backed by another ABS. This new ABS is known as collateralized debt obligation.
Efficient Market
An efficient market is a market in which security prices adjust rapidly to reflect any new information, i.e. security prices reflect all past and present information.
Initial Public Offering (IPO)
An initial public offering (IPO) is the first time a company sells its stock to the public. Advantages of IPO: greater liquidity and better access to capital Disadvantages of IPO: dispersed equity holdings and compliance is costly and time-consuming
Option pricing: replicating portfolio
An option can be replicated by buying small delta shares of the underlying stock and lending B at the risk-free rate. small delta = e^-delta*h((Vu - vd) / (S(u - d)) B = e^-rh(u*Vd - d*Vu) / (u - d) V = small delta*S + B Call: (small delta = +, B = -) Put: (small delta = -, B = +) To replicate a call, buy shares and borrow money. To replicate a put, sell shares and lend money.
Shout Option definition
An option that gives the owner the right to lock in a minimum payoff exactly once during the life of the option, at a time that the owner chooses. When the owner exercises the right to lock in a minimum payoff, the owner is said to shout to the writer.
Lookback Option definition
An option whose payoff at expiration depends on the maximum or minimum of the stock price over the life of the option.
Rainbow Option definition
An option whose payoff depends on two or more risky assets.
No-arbitrage condition
Arbitrage is possible if the following inequality is not satisfied: 0 < p-star < 1 <=> d < e^(r - delta)*h < u
Asian Option
Arithmetic average = A(S) = sum(St) / N Geometric average = G(S) = (product of St) ^1/N G(S) <= A(S) Payoff(call): average price = max[0, Sbar - K], average strike = max[0, S - Sbar] Payoff(put) = average price = [0, K - Sbar], average strike = [0, Sbar - S] The value of an average price Asian option is less than or equal to the value of an otherwise equivalent ordinary option. As N iincreases: value of average price decreases, value of average strike option increases
Bid-ask spread
Ask price - bid price
Risk-neutral pricing
Assume alpha = gamma = r. To calculate option price, discount the risk-neutral expected option payoff at the risk-free rate: V0 = e^-rT*E[Payoff]
Capital Market Line
Assume investors have homogenous expectations: All investors have the same efficient frontier of risky portfolios, and thus the optimal risky portfolio and capital allocation line. Every investor will use the same optimal risky portfolio: the market portfolio. When the market portfolio is used as the risky portfolio, the resulting capital allocation line is the capital market line. The equation for the capital market line is: E[R(xM)] = rf + {(E[R(M) - rf) / sigma(M)} * sigma(xM) Only efficient portfolios plot on the capital market line. Individual securities plot below this line.
Generalized Black-Scholes formula
Assume the current time is time 0 and the options expire at time T. C = FP(S)*N(d1) - FP(K)*N(d2) P = FP(K)*N(-d2) - FP(S)*N(-d1) d1 = [ln(FP(S) / FP(K)) + 0.5*sigma^2*T] / [sigma*sqrt(T)] d2 = [ln(FP(S) / FP(K)) - 0.5*sigma^2*T] / [sigma*sqrt(T)] = d1 - sigma*sqrt(T) sigma = sqrt(Var[ln(St)] / t) = sqrt(Var[ln(F(S)] / t) = sqrt(Var[ln(FP(S)] / t) The generalized BS formula can be applied to various assets, including stocks, futures contracts, and currencies. If the stock pays discrete dividends, then the volatility of the prepaid forward price should be used as the volatility parameter. For a stock that pays continuous dividends, the generalized BS formula can be written as: C = S0*e^-delta*T * N(d1) - K*e^-rT * N(d2) P = K*e^-rT * N(-d2) - S0*e^-delta*T * N(-d1) d1 = [ln(S0 / K) + (r - delta + 0.5*sigma^2)*T] / [sigma*sqrt(T)] d2 = [ln(S0 / K) + (r - delta - 0.5*sigma^2)*T] / [sigma*sqrt(T)] = d1 - sigma*sqrt(T)
Lognormal model for stock prices
Assume the current time is time T: For T > t, ln[ST / St] ~ N[m, v^2] m = (alpha - delta - .5*sigma^2)(T - t) v^2 = sigma^2(T - t) ST/St ~ LogN(m, V^2) For T > t, ln[ST] ~ N[m, v^2] m = lnSt + (alpha - delta - .5*sigma^2)(T - t) ST ~ LogN(m, v^2) E[ST] = E[ST | St] = St*e^(r - delta)(T - t) Var[ST] = Var[ST | St] = (E[St])^2 * (e^v^2 - 1) ST = St*e^(alpha - delta - .5*sigma^2)(T - t) + sigma*sqrt(T - t)*Z, Z ~ N(0, 1) Cov(St, ST) = E[ST / St] * Var[ST | S0] To find the pth percentile of ST: 1. Determine the corresponding pth percentile of the standard normal random variable Z. 2. Substitute the resulting value of Z into the expression for ST. Median = 50th percentile = St*e^(alpha - delta - .5*sigma^2)(T - t) = E[ST] * e^-0.5*sigma^2(T - t)
Cost of Capital and Beta for Levered and Unlevered Projects
Assuming a project is financed entirely with equity, we can estimate the project's cost of capital and beta based on the asset or unlevered cost of capital and the beta of comparable firms. Beta: All-equity: beta(U) = beta(E) Levered: beta(U) = w(E) * beta(E) + w(D) * beta(D) Cost of capital: All-equity: r(U) = r(E) Levered: r(U) = w(E) * beta(E) + w(D) * beta(D)
Trade-Off Theory
Balance the value-enhancing effects of debt on a firm's capital structure with the value-reducing effects. VL = VU + PV(Interest tax shield) - PV(financial distress costs) - PV(agency costs of debt) + PV(agency benefits of debt) The optimal level of debt, D, occurs at the point where the firm's value is maximized. It balances the benefits and costs of leverage.
Calculating Beta
Beta(i) = beta(i, mkt) = Cov[R(i), R(mkt)] / sigma(mkt)^2 = rho(i, mkt) * sigma(i) / sigma(mkt) Beta(P) = the sum from i = 1 to n of x(i) * beta(i) Beta can be estimated using linear regression: R(i) - rf = alpha(i) + beta(i) * (R(mkt) - rf) + epsilon(i)
Process of short-selling
Borrow an asset from a lender. Immediately sell the borrowed asset and receive the proceeds (usually kept by a lender or a designated third party). Buy the asset at a later date in the open market to repay the lender (close/cover the short position).
Other Anomalies for EMH - Bubbles
Bubbles also violate market efficiency. It happens when the market value of the asset significantly deviates from its intrinsic value.
Exchange Option
C(A, B) = FP(A) * N(d1) - FP(B)*N(d2) P(A, B) = FP(B)*N(-d2) - FP(A)*N(-d1) d1 = [ln(FP(A) / FP(B)) + 0.5*sigma^2*(T - t)] / [sigma*sqrt(T - t)] d2 = d1 - sigma*sqrt(T - t) sigma = sqrt[Var(ln(A/B)) / t] = sqrt[sigma^2(A) + sigma^2(B) - 2Cov(A, B)] = sqrt[sigma^2(A) + sigma^2(B) - 2*rho*sigma(A)*sigma(B)]
PCP for exchange options
C(A,B) - P(A,B) = FP(A) - FP(B) C(A,B) = P(B,A) where C(A,B) = receive A, give up B P(A,B) = give up A, receive B
PCP for bonds
C(B,K) - P(B,K) = FP(B) - Ke^-r(T-t) where FP(B) = B(t) - PV(coupons) B(t) = bond price at time t
PCP for futures
C(F,K) - P(F,K) = Fe^-r(T-t) - Ke^-r(T-t)
PCP for stocks
C(S,K) - P(S,K) = FP(S) - Ke^-r(T-t)
PCP for currency options
C(f,K) - P(f,K) = x0*e^-rf*T - Ke^-rd*T where S(0) = x0, r = rd, delta = rf, and x0 is in d/f
Capital Market Line (CML) vs. Security Market Line (SML)
CML: the x-axis is based on TOTAL risk (i.e. volatility) and it only holds for EFFICIENT portfolios (because all combinations of the risk-free asset and the market portfolio are efficient portfolios) SML: the x-axis is based on SYSTEMATIC risk (i.e. beta) and it holds for ANY security or combination of securities (because the CAPM can be used to calculate the expected return for any security)
Breakeven Analysis
Calculate the value of each parameter so that the project has an NPV of zero. The internal rate of return (IRR) is the rate at which the NPV is zero.
Bear Spread
Call Bear: Short call (K1) + long call (K2), K1 < K2 Put Bear: Short put (K1) + long put (K2), K1 < K2
Bull Spread
Call Bull: Long call (K1) + short call (K2), K1 < K2 Put Bull: Long put (K1) + short put (K2), K1 < K2
Put-call parity for compound options
CallonCall - PutonCall = C(eur) - x*^-rt1 CallonPut - PutonPut = P(eur) - x*e^-rt1
American-style options
Can be exercised at any time during the life of the option
Bermudan-style options
Can be exercised during bounded periods (i.e., specified periods during the life of the option)
European-style options
Can only be exercised at expiration
Sensitivity Analysis
Change the input variables ONE AT A TIME to see how sensitive NPV is to each variable. Using this analysis, we can identify the most significant variables by their effect on the NPV. The range is the difference between the best-case NPV and the worst-case NPV.
Formula to guarantee a maximum purchase price for a stock
Cap = Call - Stock
Scenario Analysis
Change SEVERAL input variables at a time, then calculate the NPV for each scenario. The greater the dispersion in NPV across the given scenarios, the higher the risk of the project. The underlying variables are interconnected.
The Black-Scholes formula assumptions
Continuously compounded returns on the stock are normally distributed and independent over time. There are no sudden jumps in the stock price. Volatility is known and constant. Future dividends are known The risk-free rate is known and constant (i.e, the yield curve is flat) There are no taxes or transaction costs. Short-selling is allowed at no cost. Investors can borrow and lend at the risk-free rate.
Who bears the financial distress costs?
Debt holders recognize that when the firm defaults, they will not be able to obtain the full value of the assets. As a result, they will pay less (or demand higher yields) for the debt initially. It is the equity holders who most directly bear the financial distress costs.
Decision Tree
Decision tree is a graphical approach that illustrates alternative decisions and potential outcomes in an uncertain economy. 2 kinds of nodes in the decision tree: The square node is the decision node where you have control over the decision. The circular node is the information node where you have no control over the outcome.
Reasons Why the Market Portfolio Might Not be Efficient - Proxy Error
Due to the lack of competitive price data, the market proxy cannot include most of the tradable assets in the economy.
Risk and Return of a Single Asset
E[R] = sum from 1 to n of p(i)*R(i) Var[R] = E[(Ri - E[R])^2] = sum from 1 to n of p(i) * (Ri - E[R])^2 = E[R^2] - (E[R])^2 SD[R] = sqrt(Var[R])
Conditional and partial expectations
E[ST | ST < K] = [PE(ST | ST < K)] / Pr(ST < K) = (St*e^(alpha - delta)(T - t) * N(-a1)) / N(-a2) E[ST | ST > K] = [PE(ST | ST > K)] / Pr(ST > K) = (St*e^(alpha - delta)(T - t) * N(+a1)) / N(+a2) a1 = ln(St / K) + (alpha - delta + 0.5*sigma^2)(T -t) / sigma*sqrt(T -t)
Expected option payoffs
E[call payoff] = St*e^(alpha - delta)(T - t) * N(a1) - KN(a2) E[put payoff] = K*N(-a2) - St*e^(alpha - delta)(T - t) * N(-a1)
Payoff on a risk-free ZCB
Equals ZCB's maturity value
Selling a risk-free ZCB
Equals borrowing at the risk-free rate
Buying a risk-free ZCB
Equals lending at the risk-free rate
Profit on a risk-free ZCB
Equals zero
Estimating the Debt Overhang
Equity holders will benefit from a new investment requiring investment I only if: NPV / I > (beta(D) * D) / (beta(E) * E)
Secondary Offerings
Existing shares sold by current shareholders.
Forward premium
F(0,T) / S(0)
Relationship between F(S) and FP(S)
F(S) = accumulated value of FP(S) = FP(S)*e^r(T-t)
Option on futures contracts
F(t, Tf) = St*e^(r - delta)(Tf - t) T = expiration date of the option Tf = expiration date of the futures contract T <= Tf St = F(t, Tf), delta = r uF = e^sigma*sqrt(h) dF = e^-sigma*sqrt(h) p-star = (1 - dF) / (uF - dF) small delta = (Vu - Vd) / F(uF - dF) B = e^-rh[p-star*Vu + (1 - p-star)*Vd]
Bounds of option prices - European vs. American call
FP(S) >= C(eur) >= max(0, FP(S) - Ke^-rT) S >= C(amer) >= max(0, S - K)
Forward Start Option
For a call option expiring at time T whose strike is set on future date t to be XSt: C(St, XSt, T - t) = St*e^delta(T - t) * N(d1) - XSt*e^-r(T - t) * N(d2) = St[e^-delta(T - t) * N(d1) - Xe^-r(T - t) * N(d2)] d1 = [ln(St / XSt) + (r - delta + 0.5*sigma^2)(T - t)] / [sigma*sqrt(T - t)] = [ln(1 / x) + (r - delta + 0.5*sigma^2)(T - t)] / [sigma*sqrt(T - t)] d2 = d1 - sigma*sqrt(T - t) The time-0 value of the forward start option is: V(0) = FP(S) * [e^-delta(T - t) * N(d1) - Xe^-r(T - t) * N(d2)] Same analysis applies to a put option.
Other Anomalies for EMH - Political Cycle Effect
For a given political administration, its first year and last year yield higher returns than the years in between.
Chooser Option
For an option that allows the owner to choose at time t whether the option will become a European call or put with strike K and expiring at time T: Vt = max[C(St, K, T - t), P(St, K, T - t)] = e^-delta*(T - t) * max[0, Ke^-(r-delta)(T - t) - St] + C(S(0), K, T)
Mortgage Loan as Put
For an uninsured position, the loss to the mortgage lender is max(B + C - R, 0), where B is the outstanding loan balance as default C is the lender's total settlement cost R is the amount recovered on the sale of property This is a put payoff with K = B + C and S = R
Probability
For n periods, let k be the number of "up" jumps needed to reach an ending node. Then, the risk-neutral probability of reaching that node is given by: (n choose k)*p-star^k(1 - p-star)^n -k, k = 0,1,...,n
Dynamic Hedging
Frequently buy/sell assets and/or derivatives with the goal of matching changes in the value of guarantee
Uniform on [a,b]
Fx(x) = (x - a) / (b - a) E[X] = (a + b) / 2 Var[X] = (b - a)^2 / 12
Exponential with mean theta
Fx(x) = 1 - exp(-x / theta)
Monte Carlo Simulation
General steps: 1. Build the model of interest, which is a function of several input variables. Assume a specific probability distribution for each input variable. 2. Simulate random draws from the assumed distribution for each input variable. 3. Given the inputs from step 2, determine the value of the quantity of interest. 4. Repeat steps 2 and 3 many times. 5. Using the simulated values of the quantity of interest, calculate the mean, variance, and other measures. Inversion method: Set Fx(x) = u.
Estimating return and volatility
Given S0, S1,...Sn, where the observations are at intervals of length h, we can estimate the lognormal parameters as follows: 1. Calculate the continuously compounded returns: ri = ln(S(i) / S(i - 1), i = 1,2,...,n 2. Calculate the sample mean of the returns: r bar = sum from i = 1 to n of ri divided by n 3. Estimate the standard deviation of returns by taking the square root of the sample variance of the returns: sigma hat h = sqrt(sum(ri - r hat)^2) / (n - 1) 4. Annualize the estimate of the standard deviation: Var[ln(S t + h) / St] = sigma hat^2*h => sigma hat h / sqrt(h) 5. Annualize the estimate of the return: E[ ln((St+h)/(St)) = (alpha hat - delta - .5sigma hat^2)*h => alpha hat = r bar/h + delta + .5*sigma hat^2
Timing Option (Call Option)
Gives a company the option to delay making an investment with the hope of having better information in the future. Can be valued using the Black-Scholes formula: S = current market value of asset K = initial investment required T = final decision date rf = risk-free rate sigma = volatility of asset value Div = free cash flow (FCF) lost from delay Discount FCF at the cost of capital. Discount K at the risk-free rate.
Shortcut method for graphing payoff of all calls
Go left-to-right on payoff diagram, and evaluate slope of the payoff diagram at each strike price. Going left-to-right means that a positive slope is one that increases left-to-right, and a negative slope is one that decreases left-to-right.
Shortcut method for graphing payoff of all puts
Go right-to-left on payoff diagram, and evaluate slope of the payoff diagram at each strike price. Going right-to-left means that a positive slope is one that increases right-to-left, and a negative slope is one that decreases right-to-left.
Knock-In Barrier Option
Goes into existence if barrier is reached
Other Types of Debt
Government entities issue sovereign debt and municipal bonds to finance their activities. Sovereign debt is issued by the national government. In the US, sovereign debt is issued as bonds called "treasury securities". Municipal bonds are issued by the state and local governments.
Portfolio Greek and Elasticity
Greek for a portfolio = sum of the Greeks Elasticity for a portfolio = weighted average of the elasticities omega = [delta * S] / V = sum(w * omega) small gamma - r = omega(alpha - r)
Guaranteed minimum death benefit (GMDB)
Guarantees a minimum amount will be paid to a beneficiary when the policyholder dies
Guaranteed minimum accumulation benefit (GMAB)
Guarantees a minimum value for the underlying account after some period of time, even if the account is less
Guaranteed minimum withdrawal benefit
Guarantees that upon the policyholder reaching a certain age, a minimum withdrawal amount over a specific period will be provided
Guaranteed minimum income benefit
Guarantees the purchase price of a traditional annuity at a future time
Required Return on a Leveraged Project
If the project is financed with both equity and debt, then use the weighted-average cost of capital (WACC): r(WACC) = w(E) * r(E) + w(D) * r(D) * (1 - tau(C)) = r(U) - w(D) * r(D) * tau(C) where w(D) * r(D) * (1 - tau(C)) is the effective after-tax cost of debt. Note that: WACC is based on the firm's after-tax cost of debt while the unlevered cost of capital is based on the firm's pretax cost of debt. Unlevered cost of capital is also called the asset cost of capital or the pretax WACC. When we say "WACC" with no qualification, we mean "after-tax WACC".
Underdiversification
Individual investors fail to diversify their portfolios adequately. They invest in stocks of companies that are in the same industry or are geographically close. Explanations: Investors suffer from familiarity bias, favoring investments in companies they are familiar with. Investors have relative wealth concerns, caring most about how their portfolio performs relative to their peers.
Investor Attention, Mood, and Experience
Individual investors tend to be influenced by attention-grabbing news or events. They buy stocks that have recently been in the news. Sunshine has a positive effect on mood and stock returns tend to be higher on a sunny day at the stock exchange. Major sports events have impacts on mood. A loss in the World Cup reduces the next day's stock returns in the losing country. Investors appear to put inordinate weight on their experience compared to empirical evidence. People who grew up during a time of high stock returns are more likely to invest in stocks.
Excessive Trading and Overconfidence
Individual investors tend to trade very actively. Explanations: Overconfidence bias - they often overestimate their knowledge or expertise. Men tend to more overconfident than women. Trading activity increases with the number of speeding tickets an individual receives - sensation seeking.
Herd Behavior
Investors actively try to follow each other's behavior. Explanations: Investors believe others have superior information, resulting in INFORMATION CASCADE EFFECT. Investors follow others to avoid the risk of underperforming compared to their peers (RELATIVE WEALTH CONCERNS) Investment managers may risk damaging their reputations if their actions are far different from their peers. If they feel they are going to fail, then they would rather fail with most of their peers than fail while most succeed.
Reasons Why the Market Portfolio Might Not be Efficient - Non-Tradable Wealth
Investors are exposed to significant risks outside their portfolio. They may choose to invest less in their respective sectors to offset the inherent exposures from their human capital.
Assumptions of CAPM
Investors can buy and sell all securities at competitive market prices. There are no transaction costs or taxes. Investors can lend and borrow at the risk-free rate. Investors hold only efficient portfolios of traded securities. Investors have homogenous expectations regarding the volatilities, correlations, and expected returns of securities. The consequence of these assumptions is that the MARKET portfolio is the EFFICIENT portfolio.
Reasons Why the Market Portfolio Might Not be Efficient - Behavioral Biases
Investors may be subject to systematic behavioral biases and therefore hold inefficient portfolios.
Holding on to Losers and the Disposition Effect
Investors tend to hold on to investments that have lost value and sell investments that have increased in value. People tend to prefer avoiding losses more than achieving gains. They refuse to "admit a mistake" by taking the loss. Investors are more willing to take on risk in the face of possible losses. The disposition effect has negative tax consequences.
EMH Underreaction/Overreaction Anomalies - Earnings Announcement puzzle
Investors underreacted to the earnings announcement.
Early exercise of American options - American put
It is rational to early exercise if PV(interest on strike) > PV(divs) + implicit call It may be rational to early exercise if PV(interest on strike) > PV(divs)
Barrier option formulas and relationships
Knock-in + knock-out = ordinary option barrier option <= ordinary option Relationships: If S(0) <= barrier <= strike: up-and-in call = ordinary call, up-and-out call = 0 If S(0) >= barrier >= strike: down-and-in put = ordinary put, down-and-out put = 0
Other Anomalies for EMH - Neglected Firm Effect
Lesser-known firms yield abnormally high returns.
Costs of Asymmetric Information - Pecking Order Hypothesis
Managers prefer to make financing choices that send positive rather than negative signals to outside investors. The pecking order (from most favored to least favored financing option: internally generated equity (i.e. retained earnings) > debt > external equity (i.e. newly issued shares).
The Agency Benefits of Leverage - Empire Building
Managers tend to take on investments that increase the size, rather than the profitability, of the firm.
Maintenance margin
Minimum margin balance that the investor is required to maintain in the margin account at all times
Factors Affecting the Timing of Investment
NPV of the investment: Without the timing option, invest today if NPV of investing today is positive. With the timing option, invest today only if NPV of investing today exceeds the value of the option of waiting, assuming the NPV is positive. Volatility: When huge uncertainty exists regarding the future value of the investment (i.e. high volatility), the option to wait is more valuable. Dividends: It is better for an investor to wait unless the cost of waiting is greater than the value of waiting.
Primary Offerings
New shares sold to raise new capital.
Early exercise of American options - American call
Nondividend-paying stock: early exercise is never optimal, C(amer) = C(eur) Dividend-paying stock: it is rational to early exercise if PV(divs) > PV(interest on strike) + implicit put, it may be rational to early exercise if PV(divs) > PV(interest on strike)
Features of futures contracts
Notional value = # of contracts * muliplier (S&P = 250) *futures price Bal(t) = Bal(t-1) * e^rh + Gain(t) where Gain(t) = # contracts * multiplier * price change(t) (for long position) Gain(t) = - # contracts * multiplier * price change(t) (for short position) Price change(t) = future price(t) - future price(t-1)
Long forward
Obligation to buy at the forward price Payoff = S - F Profit = Payoff Max loss = F Max gain = infinity Strategy = guarantee/lock in purchase price of underlying
Short put
Obligation to buy at the strike price if the put is exercised Payoff = -max(0, K - S) Profit = Payoff + AV(Prem) Max loss = K - AV(Prem) Max gain = AV(Prem) Strategy = sells insurance against low underlying price
Short forward
Obligation to sell at the forward price Payoff = F - S Profit = Payoff Max loss = infinity Max gain = F Strategy = guarantee/lock in sale price of underlying
Short call
Obligation to sell at the strike price if the call is exercised Payoff = -max(0, S - K) Profit = Payoff + AV(Prem) Max loss = infinity Max gain = AV(Prem) Strategy = sells insurance against high underlying price
Measures of Investment Risk - Semi-variance / Downside Semi-variance
Only cares about downside risk; ignores upside variability. The average of the squared deviations below the mean: Semi-variance = E[min(0, R - E[R])^2] The sample semi-variance is: Semi-variance = 1/n * the sum of min(0, R(i) - E[R])^2 Semi-variance is less than or equal to variance. For a symmetric distribution, semi-variance = 0.5 * variance
Option pricing: risk -neutral valuation
P-star = (e^(r - delta)*h - d) / (u - d) V0 = e^-rh - E*[payoff] = e^-rh[p-star*Vu + (1 - p-star)*Vd] S0*e^(r - delta)*h = p-star*Su + (1 - p-star)*Sd
Outright purchase
Payment time: 0 Receive stock at time: 0 Payment: S(0)
Prepaid forward contract
Payment time: 0 Receive stock at time: T Payment: FP(S)
Fully leveraged purchase
Payment time: T Receive stock at time: 0 Payment: S(0)*e^rt
Forward contract
Payment time: T Receive stock at time: T Payment: F(S)
Rebate Option
Pays fixed amount if barrier is reached
Earnings-Enhanced Death Benefit
Pays the beneficiary an amount an amount based on the increase in the account value over the original amount invested, e.g. 40% * max(ST - K, 0) The embedded option is a call option. Its value is: E[C(Tx)] = the integral from 0 to infinity of C(t)*fTx(t) dt
Out-of-the-money
Produce a negative payoff if the option is exercised immediately
In-the-money
Produce a positive payoff (not necessarily positive profit) if the option is exercised immediately
Corporate Debt: Public Debt
Public debt trades on public exchanges. The bond agreement takes the form of an indenture, which is a legal agreement between the bond issuer and a trust company.
Annual Realized Returns
R = (1 + R(Q1)) * (1 + R(Q2)) * (1 + R(Q3)) * (1 + R(Q4)) - 1
Average Annual Returns
R bar = 1/T * sum from 1 to T of Rt Var[R] = 1/(T - 1) * sum from 1 to T of (Rt - R bar)^2 Standard error = SD(R) / sqrt(# of observations) 95% confidence interval for expected return = Average return +- 2 * standard error = R bar +- 2 * (SD(R) / sqrt(# of observations))
Risk and Return for a 2-stock portfolio
R(P) = x1 * R1 + x2 * R2 sigma(P)^2 = x1^2 * sigma(1)^2 + x2^2 * sigma(2)^2 + 2*x1*x2*Cov[R1, R2]
Risk and Return of a Portfolio
R(P) = x1*R1 + x2*R2 + ... + xn*Rn E[R(P)] = x1*E[R1] + x2*E[R2] + ... + xn*E[Rn] x(i) = value of investment i / total portfolio value; sum of x(i) = 1 Cov[Ri, Rj] = E[(Ri - E[Ri])(Rj - E[Rj])] = 1 / (T - 1) * sum of (Ri - R bar i)(Rj - R bar j) = rho(i, j) * sigma(i) * sigma(j)
Risk and Return of a 3-stock portfolio
R(P) = x1*R1 + x2*R2 + x3*R3 sigma(P)^2 = x1^2 * sigma(1)^2 + x2^2 * sigma(2)^2 + x3^2 * sigma(3)^2 + 2*x1*x2*Cov[R1, R2] + 2*x1*x3*Cov[R1, R3] + 2*x2*x3*Cov[R2, R3]
Realized Returns
R(t + 1) = Capital Gain + Div Yield = [P(t + 1) - Pt] / Pt + D(t + 1) / Pt
Real Options
Real options are capital budgeting options that give managers the right, but not the obligation, to make a particular business decision in the future after new information becomes available.
Other Anomalies for EMH - Stock Split Effect
Returns are higher before and after the company announces the stock split.
EMH Calendar/Time Anomalies - Time-of-day Effect
Returns are more volatile close to the opening and closing hours for the market. Also, the trading volumes are higher during these times.
EMH Calendar/Time Anomalies - January Effect
Returns have been higher in January (and lower in December) than in other months.
EMH Calendar/Time Anomalies - Monday Effect
Returns have been lower on Monday and higher on Friday than on other days of the week.
Bounds of option prices: call and put
S >= C(amer) >= C(eur) >= max(0, FP(S) - Ke^-rT) K >= P(amer) >= P(eur) >= max(0, Ke^-rT - FP(S))
Prepaid forward - no dividends
S(t)
Prepaid forward - discrete dividends
S(t) - PV(Divs)
Forward - continuous dividends
S(t)*e^(r-delta)(T-t)
Prepaid forward - continuous dividends
S(t)*e^-delta(T-t)
Forward - no dividends
S(t)*e^r(T-t)
Forward - discrete dividends
S(t)*e^r(T-t) - AV(Divs)
Shout option payoff
S* is the value of the stock at the time when the option owner shouts to the option writer. Payoff for a shout call = max[ST - K, S* - K, 0] if exercised and max[ST - K, 0] if not exercised. Payoff for a shout put = max[K - ST, K - S*, 0] if exercised and max[K - ST, 0] if not exercised.
Option on currencies
S0 = x0, r = rd, delta = rf u = e^(rd - rf)*h + sigma*sqrt(h) d = e^(rd - rf)*h - sigma*sqrt(h) p-star = (e^(rd - rf)*h - d) / (u - d)
Semi-Strong Form EMH
Semi-strong form EMH says that market prices reflect past market data and public info. It is impossible to consistently attain superior profits by analyzing past returns. Prices will adjust immediately upon the release of any public announcements (earnings, mergers, etc.). A semi-strong form efficient market is also weak form efficient.
The Agency Costs of Leverage - Debt Overhang or Underinvestment
Shareholders may be unwilling to finance new, positive-NPV projects.
Auction IPOs
Shares sold through an auction system and directly to the public.
Best-efforts
Shares will be sold at the best possible price. Usually used in smaller IPOs.
Interest rate on haircut is called
Short rebate in the stock market Repo rate in the bond market
Equity Funding for Private Companies
Sources of funding for private companies: angel investors, venture capital firms, private equity firms, institutional investors, corporate investors When a private company first sells equity, it typically issues preferred stock instead of common stock.
Reasons for short-selling assets
Speculation - to speculate that the price of a particular asset will decline Financing - to borrow money for additional financing of a corporation Hedging - to hedge the risk of a long position on the asset
Lookback Option Payoffs
Standard lookback call payoff: ST - min(S) Standard lookback put payoff: max(S) - ST Extrema lookback call payoff: max[0, max(S) - K] Extrema lookback put payoff: max[0, K - min(S)]
Strong Form EMH
Strong form EMH says that market prices reflect past market data, public info, and private info. There are only lucky and unlucky investors. No one (not even company insiders) can consistently attain superior profits. Passive strategy is the best. A semi-strong form efficient market is also semi-strong and weak form efficient.
Geometric Series
Sum = (first term - first omitted term) / (1 - common ratio)
Measures of Investment Risk - Tail Value-at-Risk (TVaR)
TVaR focuses on what happens in the adverse tail of the probability distribution. Also known as the conditional tail expectation or expected shortfall. If X represents GAINS, then the risk we are concerned about comes from the LOW end of the distribution: TVaR alpha = E[X | X is less than or equal to pi(alpha)] = 1 / alpha * integral from - infinity to pi(alpha) of x * fx(x) dx If X represents LOSSES, then the risk we are concerned about comes from the HIGH end of the distribution: TVaR alpha = E[X | X > alpha] = 1 / (1 - alpha) * integral from pi(alpha) to infinity of x * fx(x) dx If the risk we are concerned about is unclear, then use the following rule of thumb: if alpha < 0.5, then presumably the risk of concern comes from the low end. if alpha > 0.5, then presumably the risk of concern comes from the high end TVaR will provide a more conservative number than VaR.
Prediction interval
The 100%(1 - p) prediction interval is given by SLT and SUT such that Pr[SLT < ST < SUT] = 1 - p. Pr[Z < zL] = p/2 => zL = N^-1(p/2) zU = -zL = -N^-1(p/2) SLT = St*e^(alpha - delta - 0.5*sigma^2)(T-t) + sigma*sqrt(T-t)*zL SUT= St*e^(alpha - delta - 0.5*sigma^2)(T-t) + sigma*sqrt(T-t)*zU
Cash-and-carry
The actual forward is overpriced. Short actual forward + long synthetic forward
Reverse cash-and-carry
The actual forward is underpriced. Long actual forward + short synthetic forward
Rational Expectations
The assumption of rational expectations is less rigid than that of homogenous expectations. If we assume investors have rational expectations, then all investors correctly interpret and use their own information, along with information from market prices and the trades of others.
4 IPO Puzzles
The average IPO seems to be priced too low. New issues appear cyclical. The transaction costs of an IPO are high. Long-run performance after an IPO is poor on average.
Measures of Investment Risk - Variance
The average of the squared deviations above and below the mean: Variance = E[(R - E[R])^2]
Mortgage-backed Security
The biggest sector is the ABS market is the mortgage-backed security (MBS) sector. An MBS has its cash flows backed by home mortgages. Because mortgages can be repaid early, the holders of an MBS face prepayment risk.
Alternatives to Bankruptcy - Workout
The company negotiates directly with creditors and works out an agreement.
Alpha
The difference between a security's expected return and the required return (as predicted by CAPM) is called ALPHA: alpha(i) = E[R(i)] - r(i) = E[R(I)] - (rf + beta(i) * {E[R(mkt)] - rf}) If the market portfolio is EFFICIENT, then all securities are on the SML and E{R(i)] is equal to r(i) and alpha(i) equals 0. If the market portfolio is NOT EFFICIENT, then not all securities are on the SML and E{R(i)] is not equal to r(i) and alpha(i) does not equal 0. Investors can improve the market portfolio by: buying stocks whose E[R(i)] > r(i), (i.e. alpha(i) > 0) and selling stocks whose E[R(i)] < r(i), (i.e. alpha(i) < 0)
Diversification observations
The diversification effect is most significant initially. Even with a very large portfolio, we cannot eliminate all risk. The remaining risk is systematic risk that cannot be avoided through diversification. For a portfolio with n individual stocks with arbitrary weights: sigma(P) = the sum of i = 1 to n of x(i) * sigma(i) * rho(i, P) Each security contributes to the portfolio volatility according to its total risk scaled by its correlation with the portfolio, which adjusts for the fraction of the total risk that is common to the portfolio. As long as the correlation is not +1, the volatility of the portfolio is always less than the weighted average volatility of the individual stocks.
Debt cost of capital / cost of debt / required return on debt
The rate of return that the debt holders require in order for them to contribute their capital to the firm.
Leverage Ratchet Effect
The leverage ratchet effect explains that once existing debt is in place: Equity holders may have an incentive to take on more debt even if it reduces the firm value. Equity holders will not have an incentive to decrease leverage by buying back debt even if it will increase the firm value.
The market portfolio can be inefficient when...
The market portfolio can be inefficient (and thus it is possible to beat the market) only if a significant number of investors: do not have rational expectations (thus info is misinterpreted) and care about the aspects of their portfolio other than expected return and volatility (thus they are willing to hold portfolios that are mean-variance inefficient).
The Performance of Fund Managers
The median mutual fund actually destroys value. The mutual fund industry still have positive value added because skilled managers manage more money and add value to the whole industry. On average, an investor does not profit more from investing in an actively managed mutual fund compared to investing in passive index funds. The value added by a fund manager is offset by the mutual fund fees. Superior past performance of funds was not a good predictor of future ability to outperform the market.
Net Present Value (NPV) and Cost of Capital
The net present value (NPV) of a project equals the present value of all expected net cash flows from the project. The discount rate for a project is its cost of capital.
Subordinated Debenture
The new debt that has lower seniority than existing debenture issues is called a subordinated debentures.
Optimal Portfolio Choice
The optimal risky portfolio to combine with the risk-free asset is the one with the highest Sharpe ratio, where the capital allocation line just touches (i.e. tangent to) the efficient frontier of risky investments. The portfolio that generates this tangent line is known as the tangent portfolio. The tangent line will always provide the best risk and return tradeoff available to investors. All portfolios on the tangent line (i.e. all portfolios that are combinations of the risk-free asset and the tangent portfolio) are efficient portfolios. The tangent portfolio is the optimal risky portfolio that will be selected by a rational investor regardless of risk preference.
Bid price
The price at which market-makers will buy and end-users will sell.
Ask/offer price
The price at which market-makers will sell and end-users will buy.
Implications for Equity from Adverse Selection
The stock price declines on the announcement of an equity issue. The stock price tends to rise prior to the announcement of an equity issue. Firms tend to issue equity when information asymmetries are minimized, such as immediately after earnings announcement.
Pre-Money and Post-Money Valuation
The value of a firm BEFORE a funding round is called the pre-money valuation. The value of a firm AFTER a funding round is called the post-money valuation. Post-money valuation = pre-money valuation + amount invested = # of shares after the funding rounds * pre-money price per share Percentage ownership = amount invested / post-money valuation = (# of shares owned * pre-money price per share) / post-valuation = # of shares owned / total # of shares
Compound Option
The value of the underlying option at time t1 = V[S(t1), K, T - t1] The value of the compound call at time t1 = max[0, V(S(t1), K, T - t1) - x] The value of the compound put at time t1 = max[0, x - V(S(t1), K, T - t1)] where K is the strike price of the underlying option, x is the strike of the compound option, T is the maturity of the underlying option, t1 is the maturity if the compound option
EMH Underreaction/Overreaction Anomalies - Reversal Effect
There is a negative serial correlation in stock prices as investors overreact to new information.
EMH Underreaction/Overreaction Anomalies - Momentum Effect
There is a positive serial correlation in stock prices as investors underreact to new information.
Foward start options
They are useful for hedging guarantees that will come into effect during the payout period of a GMWB while the variable annuity is still in the accumulation period.
Fama-French-Carhart
This model consists of 4 self-financing factor portfolios: market portfolio, small-minus-big portfolio, high-minus-low portfolio, and momentum. The FFC estimates the expected return as E[Ri] = rf + beta(i, mkt) * (E[Rmkt] - rf) + beta(i, smb) * E[Rsmb] + beta(i, hml) * E[Rhml] + beta(i, pr1yr) * E[Rpr1yr]
Shout options
Useful for hedging variable annuity guarantees in situations where the guarantee value is recalculated as the discretion of the policyholder.
Lookback options
Useful for hedging variable annuity guarantees where the guarantee value is periodically recalculated as the greater of the account value and the existing guarantee value.
Chooser options
Useful hedging tools for variable annuities with two-sided guarantees, e.g. a GMDB with a return-of-premium guarantee and an earnings-enhanced death benefit equal to 35% of any account value gains.
Rainbow options
Useful hedging tools when policyholders can hold multiple assets in their accounts and the guarantee applies to the account as a whole rather than individual assets in the account.
Multi-factor Model Key Equations
Using a collection of N factor portfolios: E[Ri] = rf + the sum from n = 1 to N of beta(i, Fn) * (E[R(Fn)] - rf) where: beta(i, F1),...,beta(i, Fn) are the factor betas of asset i that measure the sensitivity of the asset to a particular factor, holding other factors constant. E[R(Fn)] - rf is the risk premium of the expected excess return for a factor portfolio. If all the factor portfolios are self-financing, then we can rewrite the equation as: E[Ri] = rf + the sum from n = 1 to N of beta(i, Fn) * (E[R(Fn)]) For a self-financing portfolio, the portfolio weights sum to zero rather than 1.
Delta-Gamma-Theta Approximation
V(t + h) = Vt + delta * E + 0.5*gamma*E^2 + theta*h E = S(t + h) - St
Measures of Investment Risk - Value-at-Risk (VaR)
VaR of X at the 100% * alpha confidence level is its 100alphath percentile denoted as VaR alpha (X) or pi(alpha): Pr[X is less than or equal to pi(alpha)] = alpha If X is continuous, then: Fx(pi(alpha)) = alpha => pi(alpha) = Fx inverse(alpha) VaR 0.5 (x) is the 50th percentile or the median of X.
Other Anomalies for EMH - Value Effect
Value stocks have consistently outperformed growth stocks.
Coherence and Variance, Semi-Variance, VaR, TVaR
Variance and semi-variance do not satisfy any of the 4 characteristics; not coherent. VaR is usually NOT coherent since it does not satisfy the subadditivity characteristic. If the distributions are shown to be normal, then VaR can be shown to be coherent. TVaR is ALWAYS coherent.
Venture Capital Financing Terms
Venture capitalists typically hold convertible preferred stock, which differs from common stock due to: Liquidity preference (liquidity preference = multiplier * initial investment) Participation rights Seniority Anti-dilution protection Board membership There are two ways to exit from a private company: Acquisition Public offering
The Agency Benefits of Leverage - Free Cash Flow Hypothesis
Wasteful spending is more likely to happen when firms have high levels of cash flow in excess of what is needed.
Weak Form EMH
Weak form EMH says that past market prices reflect only past market data. It is impossible to consistently attain superior profits by analyzing past returns.
The Agency Costs of Leverage - Cashing Out
When a firm faces financial distress, shareholders have an incentive to liquidate assets at prices below their market values and distribute the proceeds as dividends.
Stock Recommendations
When a stock recommendation is given at the same time that news about the stock is released, the initial stock price reaction appears correct. The stock price increases in the beginning, then it flattens out. When a stock recommendation is given without news, the stock price seems to overreact. The stock price surges the following day, then it falls compared to the market.
Who bears the agency costs?
When an unlevered firm issues new debt, equity holders will ultimately bear the costs. Once a firm has debt already in place, some of the bankruptcy or agency costs from taking on additional debt can fall on existing debt holders.
Asymmetric Information
When managers have more information about a firm than investors.
Costs of Asymmetric Information - Lemons Principle
When managers have private information about the value of a firm, investors will discount the price they are willing to pay for new equity issue due to adverse selection.
Solving a Decision Tree
Work backwards from the end of the tree. At each decision node, determine the optimal choice by comparing the PV of remaining payoffs along each branch. At each information node, compute the expected PV of the payoffs from the subsequent branches. Discount rate: if the true probability is given, use the cost of capital to discount. If the risk-neutral probability is given, use the risk-free rate to discount. Value of real option = NPV(with option) - NPV(without option)
Normal vs. Lognormal
X ~ N(m, v^2) <=> Y = e^x ~ LogN(m, v^2) E[Y] = e^m + .5*v^2 Var[Y] = (E[Y])^2[e^v^2 - 1] X = m + v*Z, Z ~ N(0,1) N(-a) = 1 - N(a) Two important properties of lognormal: it cannot be negative, and the product of two lognormal is a lognormal
Security Market Line (SML)
a positively sloped straight line displaying the relationship between expected return and beta, a graphical representation of CAPM
PV of an Annuity
a[n] = v + v^2 + ... + v^n = (1 - v^n) / i a[infinity] = v + v^2 + ... = 1 / i The PV of an n-year annuity immediate with payments of 1, (1 + k), (1 + k)^2, ... , (1 + k)^n-1: PV = [1 - (1 + k / 1 + i)^n] / (i - k) The PV of a geometrically increasing perpetuity immediate with payments of 1, (1 + k), (1 + k)^2, ... : PV = 1 / (i - k)
Nonsystematic Risk
also known as firm-specific, independent, idiosyncratic, unique, or diversifiable risk Fluctuations in a stock's return that are due to firm-specific news Total risk = systematic risk + nonsystematic risk
Multiple Greeks Hedging
delta(stock) = 1, all other Greeks of the stock = 0. To hedge multiple Greeks, set the sum of the Greeks you are hedging to zero.
Coherent Risk Measures
g(X) is coherent if it satisfies (for c > 0): Translation variance: g(X + C) = g(X) + C Positive homogeneity: g(cX) = c * g(X) Subadditivity: g(X + Y) is less than or equal to g(X) + g(Y) Monotonicity: If X is less than or equal to Y, then g(X) is less than or equal to g(Y)
Maxima and Minima
max(A, B) = max(0, B - A) + A max(A, B) = max(A - B, 0) + B max(cA, cB) = c * max(A, B) where c > 0 max(cA, cB) = c * min(A, B) where c < 0 max(A, B) + min(A, B) = A + B => min(A, B) = -max(A, B) + A + B
Elasticity
omega = (% change in option price) / (% change in stock price) = (delta * S) / V
Synthetic short forward is created by
selling a stock and lending money (buying a bond) or selling a call and buying a put at the same strike
For an equally weighted n-stock portfolio:
sigma(P)^2 = 1/n * Var(bar) + (1 - 1/n) * Cov(bar) In a very large portfolio (n => infinity), the covariance among the stocks account for the bulk of portfolio risk: sigma(P)^2 = Cov(bar) If the stocks are independent and have identical risks, then Cov(bar) = 0, and sigma(P)^2 = 1/n * Var(bar) As n => infinity, sigma(P)^2 => 0. Thus, a very large portfolio with independent and identical risks will have zero risk.
Risk and Return for an n-stock portfolio
sigma(P)^2 = the sum from i = 1 to n of x(i) * Cov[R(i), R(P)] = the sum from i = 1 to n and j = 1 to n of x(i) * x(j) * Cov[R(i), R(j)] In the covariance matrix, we have: n *n = n^2 total elements n variance terms n^2 - n true covariance terms (n^2 - n) / 2 unique true covariance terms Note that: Cov[R(i), R(j)] = Cov[R(j), R(i)] = rho(i,j) * sigma(i) *sigma(j) Cov[R(i), R(i)] = Var(Ri) Cov[aR1+ bR2, cR1 + dR2] = acCov[R1, R1] + adCov[R1, R2] + bcCov[R2, R1] + bdCov[R2, R2] = acVar[R1] + adCav[R1, R2] + bcCov[R2, R1] + bdVar[R2]
