# 2. Determine the appropriate discount rate based on the risk of the security

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1 Fixed Income Instruments III Intro to the Valuation of Debt Securities LOS 64.a Explain the steps in the bond valuation process 1. Estimate the cash flows coupons and return of principal 2. Determine the appropriate discount rate based on the risk of the security 3. Calculate the PV of the cash flows LOS 64.b Identify the types of bonds for which estimating the expected cash flows is difficult, and explain the problems encountered when estimating the cash flows for these bonds. Principal repayments are not known with certainty. 1. This includes bonds with embedded options that effect the potential timing of the return of principal. These bonds are somewhat dependent on the path that market interest rates take. 2. The coupon payments are not known with certainty. Floating rate securities are dependent on the path that market interest rates take. 3. The bond is convertible or exchangeable into another security. Without knowledge of what the future price of the underlying stock or interest rates it is impossible to know when the cash flows will occur and how large or small they will be. LOS 64.c Compute the value of a bond and the change in value that is attributable to a change in the discount rate. Treasury bonds Non-Treasuries a risk premium mist be added to the risk free discount rate. Computing the value of a bond - Valuation with a single discount rate: essentially similar to valuing an annuity with principal payment entered as the Future Value. Consider a 10% annual pay, par bond with 10 years to maturity with an 8% YTM.

2 The value is the sum of all of the present values. On your TI BAII N = I/Y = PMT = FV = Compute PV = LOS 64.d Explain how the price of a bond changes as the bond approaches maturity, and compute the change in value that is attributable to the passage of time. Prior to maturity a bond can trade at a premium or a discount to par, but it will always converge to its par value at maturity. 3 year, annual pay, 5% coupon bond Years to Maturity YTM = 4% YTM = 5% YTM = 8% 3.0 \$1, \$1, \$ \$1, \$1, \$ \$1, \$1, \$ \$1, \$1, \$ \$1, \$1, \$ \$1, \$1, \$ \$1, \$1, \$1, LOS 64.e Compute the value of a zero coupon bond. Zeros have only one cash flow the return of principal, so its value is the PV of its par value at maturity. Par Value = \$1,000

3 Semi-annual discount rate = i/2 or years*2 Life of the bond in years LOS 64.f Explain the arbitrage free valuation approach and the market process that forces the price of a bond toward its arbitrage free price, and explain how a dealer can generate an arbitrage profit if it is mispriced. YTM Arbitrage Free Approach The first step in checking arb-free valuation is to value the coupon bond with the correct spot rates. The second is to compare this value to the market price. Example: 6% bond with 1.5 years to maturity and spot rates are: 6 months = 3%, 1 year = 5%, 1.5 years = 7%, the bond is priced at \$975. Compute the arb profit and explain the arb process. Answer: First the cash flows received are \$30, \$30, (\$30+\$1,000) Second: This value is greater than the market price, so we want to buy the bond on the market and disaggregate the individual cash flows and sell them as individual zero coupon bonds for a \$12.11 arbitrage profit. Dealers can do this according to the STRIPS program which allows dealers to divide bonds into their individual cash flows and also reconstitute them into whole bonds. Ignoring the costs of doing the dividing and reconstituting this process assures that arbitrage free pricing is met.

4 YIELD MEASURES, SPOT RATES AND FORWARD RATES LOS 65.a Explain the sources of return from bond investing. There are three sources: LOS 64.b Compute and interpret the traditional yield measures for fixed-rate bonds and explain their limitations and assumptions. Current yield is the simplest return measure. CY = Annual Coupon Payment / Current Bond Price Yield to Maturity (YTM) annualized IRR, based on a bond s current price and promised coupon payments. The formula is: Example: A 15 year, semi-annual pay, 7% coupon priced at \$1,050. N = FV = PMT = PV = Compute I/Y = Bond Equivalent Yield (BEY) semi-annual YTM. A BEY must be divided by 2 to get the S/A discount rate. Yield to Call (YTC) used to calculate the yield on callable bonds that are selling at a premium, which may be less than the YTM if the call price is below the current market price. The calculation is the same as for YTM except that the call price is substituted for the par value and the number of S/A periods until call is swapped in for periods until maturity. If there is a period of call protection then we calculate the yield to first call until the first possible call date and the call price is the FV. Examples: 20 year, 10% S/A pay bond at 112 that can be called in 5 years at 102 and in 7 years at par. What is the YTM, YTC, and yield to first par call?

5 YTM YTC YTFC N = 40 N = 10 N = 14 CPT I/Y =.04361x2 = 8.72% CPT I/Y =.0371x2 = 7.42% CPT I/Y =.03873x2 = 7.746% PMT = \$50 PMT = \$50 PMT = \$50 FV = \$1,000 FV = \$1,020 FV = \$1,000 PV = -\$1,120 PV = -\$1,120 PV = -\$1,120 Yield to Worst is the worst possible outcome of any of the given call provisions of the bond. In the examples above the YTW is the YTC at 7.42%. Yield to refunding is Yield to Put (YTP) Cash Flow Yield Used for MBS and other amortizing ABS with monthly CFs. BEY = {[(1 + monthly CFY)^6] 1} x2 Assumptions and Limitations of Traditional Yield Measures The main one is that YTM doesn t tell us the compound rate of return we will realize over the bond s life. This is because we don t know the reinvestment rate of the coupons. The realized yield on a bond is the actual compound return that was earned on the initial investment. For a bond to realize its actual YTM all cash flows must be reinvested at the YTM rate and it must be held to maturity. If the reinvestment rate is below the YTM, the realized yield will be below the YTM. The other yield measures YTC, TYP suffer from the same shortcomings and don t account for reinvestment income. LOS 65.c Explain the importance of reinvestment income in generating the yield computed at the time of purchase, calculate the amount of income required to generate that yield and discuss factors that affect reinvestment risk.

6 Reinvestment income important because if reinvestment rate is below YTM the realized yield will be below YTM. Example: A 9%, 8 year bond at par, how much reinvestment income is needed to give the holder a 9% compound return? 1) Par is \$1,000 2) Calc the total value needed over 8 years (16 periods) from now 1,000 x (1+.045)^16 = \$2, There are 16 coupons of \$45 each, totaling \$720 and principal of \$1,000, the required reinvestment income is: \$2, \$720 - \$1,000 = \$ All else equal reinvestment risk increases when: Higher coupons - because there is more cash to reinvest Longer maturities because more of the total bond value is coupons LOS 65.d Compute and interpret the BEY of an annual bond and the annual pay yield of a S/A bond. ICONV button A firm has a S/A bond with a YTM of 6.25% and an annual bond with a YTM of 6.3%, which is better? ICONV, NOM = 6.25, C/Y = 2, EFF = LOS 65.e Describe the methodology for computing the theoretical Treasury spot rate curve and compute the value of a bond using spot rates. Bootstrapping Example: Maturity Coupon YTM Price 1 year 4.0% 4.0% \$ 1,000 2 years 5.0% 5.0% \$ 1,000 3 years 6.0% 6.0% \$ 1,000 All of the above bonds are at par. Begin with the 1 year bond, is has one CF of \$1,040 and it is priced at \$1,000, so its spot rate must be 4%. The next bond has 2 CFs, and we already know the first spot rate is 4% and we also know that the no-arb principle tells us that the PV of the CFs must equal its price we can calc the year 2 spot rate.

7 Now we can solve for the 2 year spot rate by: Step 1 Step 2 Step 3 Now we have the 1 and 2 year spot rates, we can get the 3 years spot rate. Step 1 Step 2 Step 3 LOS 65.f Differentiate between the nominal spread, the zero volatility spread, and the OAS. Nominal spread is the issue s YTM minus the YTM on a similar maturity Treasury. Zero volatility spread Z spread the equal amount that must be added to each rate on the Treasury spot rate curve in order to make the PV of the risky bond s cash flows equal to its market price. Instead of measuring the spread to YTM the Z spread

9 LOS 65.h Explain a forward rate, and compute spot rates from forward rates, forward rates from spot rates and the value of a bond using forward rates. A forward rate is an interest rate for a loan to be made in the future. Notation demonstrates the length and when in the future the loan is to be made. 1 f 1 is notation for a one year loan to be made in one year from today and 1 f 2 is a one year loan to be made 2 years from now. The main idea is that borrowing for 3 years and borrowing for 1 year then rolling it over for a year, then rolling it over again for a third year should yield the same end results. (1 + S 3 ) 3 = (1+ 1 f 0 ) + (1+ 1 f 1 )+ (1+ 1 f 2 ) Example: The current 1 year rate is 3%, the one year forward rate is 4% and the 2 year forward rate is 5%, what is the 3 year spot rate? S 3 = [(1.03)x(1.04)x(1.05)] 1/3-1 = 3.997% This implies that a dollar compounded at 3.997% for 3 years would have the same ending value as a dollar that earns compound interest of 3% in year 1, 4% in year 2 and 5% in year 3. Going the other direction: (1 + S 2 ) 2 = (1+ 1 f 0 ) + (1+ 1 f 1 ) The 2 period spot rate is 10% and the 1 period spot rate is 7%, calculate the forward rate for one period, one period from now, 1 f 1. 2 year bond, S 2 = 10% From here: 1 year (1 + Sbond, 2 ) 2 = (1+ S 1 = 1 f7% 0 ) + (1+ 1 f 1 ) 1 year bond, in 1 year 1 f 1 =??% We can get Since both sides have the same payoff in two years. (1.10) 2 = (1.07) x (1+ 1 f 1 )

10 (1+ 1 f 1 ) = (1.10) 2 / (1.07) (1+ 1 f 1 ) = so, the forward rate in one year is % This boils down to the idea that investors will accept either 10% for two consecutive years or 7% for one year so long as they can then invest for the second year at %. NOTE: Simple averages give close approximations of forward rates from spot rates. In the last example we had spot rate of 7% for 1 year or 10% for two years. Given that two years at 10% is 20%, so if the first year rate is 7% then the second year rate is close to 20% - 7% = 13%, in reality it is % Valuing a bond using forward rates The current 1 year rate 1 f 0 is 5% and the one year forward rate 1 f 1 is 6%. The one year rate for lending from period 2 to period 3 is 1 f 2 7%. Value a 3 year annual pay bond with a 10% coupon and par of \$1,000. Value = [100/(1+ 1 f 0 )] + [100/(1+ 1 f 0 ) x (1+ 1 f 1 )] + [1,100/(1+ 1 f 0 ) x (1+ 1 f 1 ) x (1+ 1 f 2 )] Value = [100/(1+.05)] + [100/(1+.05) x (1+.06)] + [1,100/(1+.05) x (1+.06) x (1+.07)] Value = = \$1,108.84

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