Module 3 (Curriculum 4) (Math: Traditional and spot rates, Pv full and flat, Conversion between EAR and SAR, Yield to call, Bond equivalent Yield, Quoted & required margin, forward rates/spot rates, G I Z spread)

Calculation: Example 4.4 Calculating Multiperiod Forward Rates Calculate the 4-year forward rate 12 years from today if the 12-year spot rate is 4.5% and the 16-year spot rate is 4.6%. (important)

(1.046)^16 = (1.045)^12 *(1 +x)^4 = (1.210921) ^ 1/4 = 1.049, -1 = 4.9%

Definitions: Par curve (I don't think you need to know the calculation, just know the definition)

- A sequence of yields-to-maturity such that each bond is priced at par value. The bonds are assumed to have the same currency, credit risk, liquidity, tax status, and annual yields stated for the same periodicity. - The par curve is derived from the spot rate curve -Coupon rate = YTM -Calculation" Remember when we calculate, par curve is derived from spot rates. so we use spot rates. Also, we already know PV = FV = 100 or 1000 for exam purpose. Now, we plug in the spot rates calculation to find missing PMT which is coupon = YTM. - Example (check picture): tip: there're 2 spot rates, and we're most likely looking for PMT in numerator. Thus, we can use the 2 spot rates as a quick eyeballing technique. The number will be between the 2 spot rates, anything outside the rates can assume to be wrong. Thus, we can plug and go with the given multiple choice.

Summarization yield-to-maturity into two components: the benchmark yield and the spread. (Job on the exam is to identify, not memorize)

- Benchmark yield is the base rate, and is also referred to as the risk-free rate of return. It captures macroeconomic factors, such as the expected rate of inflation, currency denomination, and the impact of monetary and fiscal policy. Changes in these factors impact all bonds in the market. The benchmark can also be broken down into (1) the expected inflation rate and (2) the expected real rate. (Nominal risk free rate) - The spread refers to the difference between the yield-to-maturity on a bond and on the benchmark. It captures all microeconomic factors specific to the issuer, such as credit risk of the issuer, changes in the issue's credit rating, liquidity, and tax status of the bond. The spread is also known as the risk premium over the risk-free rate of return. + For example, consider a 5-year corporate bond that offers a yield-to-maturity of 5.50%. The benchmark bond is a 5-year U.S. Treasury, which offers a yield of 4.50%. This means that the corporate bond offers a spread of 100 bps. Now suppose that the yield on the corporate bond increases from 5.50% to 6.00%. + If the yield on the benchmark has also increased by 50 bps, we can infer that the change in the bond's yield is caused by macroeconomic factors that affect all bond yields. + However, if the yield on the benchmark has remained the same, we can infer that the change in the bond's yield was caused by firm-specific factors, such as changes in the issuer's creditworthiness.

- Implied forward rates (also known as forward yields) can be computed from spot rates. - Forward rates: - Forward Curves -tips: Law of One Price (important) -2 identical CF and identical risk of 2 portfolios should have the same value today (Derivative rule, must know!!)

- Forward rates: described as the market's current estimate of future spot rates. - Forward Curves represents a series of forward rates , each having the same horizon

Definitions: Spot Rate Curve (Job on the exam is to identify, not memorize)

- Ideal data set would be yields-to-maturity on zero-coupon government bonds or "spot rates" for a range of maturities. This data set is called "Spot Rate Curve" They have no reinvestment risk. ( a graph) Note: spot rates are yields-to-maturity on zero-coupon bonds. (YTM on 'zero coupon bond, and it's government bond)

Definitions: Accrued Interest

- Interest earned by the seller, which has been accrued from the last payment date till the transaction date

Definitions/Components: Relationship in Bonds: - Inverse effects - Convexity Effect - Coupon Effect -Maturity Effect (Very Important, must understand Convexity and Duration, I'm not familiar with them and further detail will disclose in the next module)

- Inverse effects = bond price vs market discount rates (common knowledge) - Convexity effect***= (see more in convexity lesson) given same coupon rate and term to M, % change is greater when discount rate is decreasing than it increases. - Coupon effect & Term to Maturity effect= (regarding Duration) + Duration = 1. price sensitivity to changes in interest rates 2. term to recoup initial cost of the bond + Higher the duration, more sensitive bond is to changes in interest rates + When do we have higher duration? longer the term + When are we very sensitive to interest rates: low coupon (see convexity and duration in its lesson)

Definitions: yield curve for coupon bonds (Job on the exam is to identify, not memorize)

- Shows the YTM for coupon-paying bonds of different maturities. (A graph) We would have reinvestment risk. - [Recently issued (on-the-run), actively traded government bonds, reinvestment risk] - To build the Treasury yield curve, analysts use only the most recently issued and actively traded government bonds, as they have similar liquidity and tax status. Even though Treasury securities are not available for every single maturity, YTMs for maturities where there are gaps can be estimated through a variety of interpolation methods, the simplest of which is linear interpolation -For example, if the yield on the 7-year Treasury is 4% and that on the 10-year Treasury is 5%, linear interpolation would compute the 8-year yield to be 4.33%, and the 9-year yield to be 4.67%. (Check notebook for its example)

Definitions: Matrix Pricing: ( I voluntarily not going to pay heavy mind on its calculation, I will focus on what it is for) (important)

- a method used to estimate the market discount rate and price of bonds that are not actively traded.

Definitions: z-spread (Important)

- practitioners tend to favor use of the z-spread over the G- and I-spreads. The z-spread (or zero-volatility spread or static spread) of a bond is a constant spread over the government (or interest rate swap) spot rate curve. ( They say Z spread includes spot rates which is more realistic)[From LV2 lecture: They're also sort of hypothetical corporate bond spot rate curve] (Check textbook pg 68 or prior slides for understanding, this slide is added after the prior slide which thoroughly explain Z spread)

Definitions: Yield Curve (Check notebook if there's confusion!) (Job on the exam is to identify, not memorize)

- relationship between yields-to-maturity and terms-to-maturity( A graph)

Example 5.1 Illustrating the G-spread A 7% annual-pay corporate bond with 2 years remaining to maturity is trading at a price of 107.75. The 2-year, 6% annual-pay government benchmark bond is trading at a price of 106.50. The 1-year and 2-year government spot rates are 1.085% and 2.67%, respectively, stated as effective annual rates. Calculate the G-spread on the corporate bond. Demonstrate that the z-spread is 33.5 bps. (Very Important)

1) + The yield-to-maturity for the corporate bond is calculated as: N = 2; PMT = -$7; FV = -$100; PV = $107.75; CPT I/Y; I/Y = 2.953% + The yield-to-maturity for the government benchmark bond is calculated as: N = 2; PMT = -$6; FV = -$100; PV = $106.5; CPT I/Y; I/Y = 2.622% Therefore, the G-spread equals 33.1 bps (= 0.02953 − 0.02622). 2) We solve for the value of the corporate bond using z1 = 0.01085, z2 = 0.0267, and Z = 0.00335. The resulting value must equal 107.75 if the z-spread is indeed 33.5 bps. =7 / (1+0.01085+0.00335) + 107 (1+0.0267+0.00335)^2 = 7 / 1.0142 + 107 / (1.03005)^2 =107.75

Calculation: Computing Forward Rates The current 1-year spot rate is 5%, 2-year spot rate is 5.25%, and 3-year spot rate is 5.55%. Calculate the 1-year forward rate 1 year from now and 2 years from now. (important)

1) Calculate 1 year forward rate 1 year from now: => 1 year rate + one year from now( given 5 %) = 2 year spot rate (given 5.25%) = (1.0525)^2 = (1.05)^1 * (1 +x)^1 =(1.0525)^2 / (1.05) = 1.055 =1.055 - 1 = 5.5% 1 year rate 2 year from now (1.055)^3 = (1 + x) (1.0525)^2 1.17591 / 1.107756 = 1.0614, - 1 = 6.14 % - So whether you invest your money for 5.25% for two years and then roll your money over for one year, or you invest in a three year spot, we get virtually the same cumulative return, 17.59% approximately.

Quick Recap: 3 assumptions when we calculate Bond in TVM( when we use the YTM, which is internal rate of return)

1) The investor holds on to the bond until maturity. 2.) The issuer makes all promised payments on time in their full amount. 3.) The investor is able to reinvest all coupon payments received during the term of the bond at the stated yield-to-maturity until the bond's maturity date

Definitions: G-spread: (Important)

A yield spread over an actual government bond. (From example in prior slides: For example, consider a 5-year corporate bond that offers a yield-to-maturity of 5.50%. The benchmark bond is a 5-year U.S. Treasury, which offers a yield of 4.50%. The 1% or 100 bps differences are the G-spread)

Example 2.1 Computing Full Price, Accrued Interest, and Flat Price A 5% U.S. corporate bond is priced for settlement on July 20, 2015. The bond makes semiannual coupon payments on April 21 and October 21 of each year and matures on October 21, 2021. The bond uses the 30/360 day-count convention for accrued interest. Calculate the full price, the accrued interest, and the flat price per USD100 of par value for three stated annual yields-to-maturity: (A) 4.7%, (B) 5.00%, and (C) 5.30%. (important)

A) (Sort of instructor/my own order to solve this questions) 1. We find the time frames. Using 360 days, 4/21 /2015 - 10/21/2021. We have 6.5 years left until maturity. 4/21-7/20 = 89 days for accrued interest time frame. Accrued interest = 5/2 * 89/180 = 1.2361 2. we find PV of the bond: N: 6.5 * 2, I/Y: 4.7%/2, FV = 100, PMT = 5/2, CPT PV = 101.6636 3. Full price = PV * (1 + r)^t/T 101.6636 * (1.0235)^89/180 = 102.838 4. Flat price = PV of Full - Accrued Interest 102.838 - 1.2361 = 101.6019 B) and C) are the same.

[Note: I added here afterward, Olinto way is very confusing. From now on, we use this formula. Nice and clear] Add-on rate basis: (bond equivalent yield) Step 1) AOR= (Year / Days) × (FV−PV / PV) Note that PV is calculated using an assumed 360-day year and AOR (bond equivalent yield) is calculated using a 365-day year. Ex: A 365-day year bank certificate of deposit has an initial principal amount of USD96.5 million and a redemption amount due at maturity of USD100 million. The number of days between settlement and maturity is 350. The bond equivalent yield is closest to: 3.48%. 3.65%. 3.78%.

AOR= (Year / Days) × (FV−PV / PV) AOR=(365 / 350)×(100 − 96.5 / 96.5) AOR = 1.04286 × 0.03627. AOR = 0.03783, or approximately 3.78%.

CFA Practice question: Which of the following statements describing a par curve is incorrect? A par curve is obtained from a spot curve. All bonds on a par curve are assumed to have different credit risk. A par curve is a sequence of yields-to-maturity such that each bond is priced at par value. ---- A yield curve constructed from a sequence of yields-to-maturity on zero-coupon bonds is the: par curve. spot curve. forward curve.

All bonds on a par curve are assumed to have different credit risk. B is correct. All bonds on a par curve are assumed to have similar, not different, credit risk. Par curves are obtained from spot curves, and all bonds used to derive the par curve are assumed to have the same credit risk, as well as the same periodicity, currency, liquidity, tax status, and annual yields. A par curve is a sequence of yields-to-maturity such that each bond is priced at par value. ------ spot curve.

Important notes from Lesson's Assessment: Which of the following statements is most likely? All other factors constant, a bond with a lower coupon rate has higher interest rate risk and higher reinvestment risk than a bond with a higher coupon rate. All other factors constant, a bond with a lower coupon rate has higher interest rate risk and lower reinvestment risk than a bond with a higher coupon rate. All other factors constant, a bond with a lower coupon rate has lower interest rate risk and lower reinvestment risk than a bond with a higher coupon rate. (My concern is with "interest rate risk" instead of "reinvestment risk")

Ans: All other factors constant, a bond with a lower coupon rate has higher interest rate risk and lower reinvestment risk than a bond with a higher coupon rate. Explanation: 1) Reinvestment risk: is the risk that you may reinvest coupon payments at lower yielding securities. So it makes sense here if higher yield has higher reinvestment risk. 2) Interest Rate Risk: Interest rate risk is specifically the exposure to changes in the value of the bond. Using a bond example: a zero coupon bond. No return comes from coupons and hence there is no reinvestment risk. All the return comes from the fact that you buy the bond at a discount to its face value. So you are reliant upon the value of the bond when you sell it. If interest rates move, the price of the bond changes, and this what we are referring to as interest rate risk (as meaured by duration).

CFA practice A two-year spot rate of 5% is most likely the: yield to maturity on a zero-coupon bond maturing at the end of Year 2 coupon rate in Year 2 on a coupon-paying bond maturing at the end of Year 4. yield to maturity on a coupon-paying bond maturing at the end of Year 2.

Ans: yield to maturity on a zero-coupon bond maturing at the end of Year 2

Calculation: Full Price (Dirty Price or Invoice Price ) Flat Price (important)

Full price = PV of Flat price + AI(Accrued Interest) Full Price = (right before the current gap period for PV of bond) PV of Bonds * (1 + r)^t/T (PV of Bonds = price of bonds at last /previous coupon pmt) Flat price = PV of full - AI Accrued interest = t/T * PMT ** (We'll see the difference in the future calculation as well) - For government bonds, the actual/actual day-count convention is usually applied. For example, if the coupon payment dates for a semiannual-pay bond are May 15 and November 15, the accrued interest for settlement on June 27 would be calculated as 43 (the actual number of days from May 15 to June 27) divided by 184 (the number of days from May 15 to November 15) times the coupon payment. - For corporate bonds, the 30/360 day-count convention is often used. For example, if the coupon payment dates for a semiannual-pay bond are May 15 and November 15, the accrued interest for settlement on June 27 would be calculated as 42 (= 30 + 12) divided by 180 (= 30 × 6), times the coupon payment.

Example: Compute the YTM of a 10-year, $1,000 par bond with an 8% coupon rate that makes semiannual coupon payments given that its current price is $925.

PV = -$925; N = 10 × 2 = 20; PMT = $40; FV = $1,000; CPT I/Y; I/Y = 4.581% 4.581% × 2 = 9.16% ( I didn't elaborate why, we know it's simple)

Calculation: Example: (they are together) + Purchase a 1-year T-bill today. The 1-year spot rate today (yield on the zero-coupon 1-year T-bill) is given as 4.6%. + Purchase a 6-month T-bill now and upon its expiration, purchase another 6-month T-bill. The 6-month spot rate today (yield on the first 6-month T-bill) is given as 4%. (important)

Step 1) We are thinking " whats the 6 month rate (annualized) in 6 months?" 4.6%/2 = 2.3% 4%/2 = 2% "So this is my client's total time horizon. Client, how would you like to earn 2.3% every six months for the next two periods? Or you can earn 2% for one six month period and then roll your money over for another six months, six months from now." step 2) (1+2.6%)^2 = (1 + 2%)^1 * (1 + x)^1 we solve for x x = 2.6% or 5.2% annually.

Definitions: I-spread or interpolated spread to the swap curve (Important)

The yield spread of a specific bond over the standard swap rate in that currency of the same tenor.

Add on [After confusion in the practice]: Using Forward rates to value bonds, it's simple. We just use the given rates and calculate it the bond value in a spot rate calculation TVM like what we went through before. But, note, there's a small difference. I realized that in the regular spot rate TVM, we do this : 30/(1.01) + 30/(1.023)^2 + 130/(1.03)^3 We use the same rate in each year. In forward rate valuing bonds, we do this like the following question: 40 / (1.04) + 40/(1.04)(1.0425) + 1040/(1.04)(1.0425)(1.043) There's a difference. Example 4.5 Valuing Bonds Using Forward Rates The current 1-year forward rate is 4%, the 1-year forward rate 1 year from now is 4.25%, and the 1-year forward rate 2 years from today is 4.3%. Calculate the value of a $1,000 par, annual-pay coupon bond that has a coupon rate of 4% and 3 years remaining to maturity. = 40 / (1.04) + 40/(1.04)(1.0425) + 1040/(1.04)(1.0425)(1.043) Listen, it's really simple. You literally find out all the rates each year, then you use the same method as "spot rates calculation" (from MBA and prior lesson) to find the bond price. No biggie) (Must know this due to LOS)

a

Calculation: current yield = Annual coupon PMT/ Bond Price

a bond's annual coupon divided by its price = Annual PMT/ Bond Price

Definitions/Calculation: Yield measures for FRN: 2 things to watch out - Quoted Margin ( a spread remains constant) + MRR = Coupon Payment - Required Margin ( changes over time) + MRR = Discount rates (Required margin is known as discount margin) Thus, when coupon = discount => bond is at par -MRR = LIBOR (for exam purposes) - Quoted Margin = Required Margin = Trading at Par. Quoted Margin > Required Margin = Trading at Premium. Quoted Margin < Required Margin = Trading at Discount - Tips: Don't forget to unannualized it, Check picture. (important)

ex 3.4- It's quite simple, thus look picture of example should be fine