CFAspace. CFA Level I. Provided by APF. Academy of Professional Finance 专业金融学院 FIXED INCOME: Lecturer: Nan Chen

CFAspace Provided by APF CFA Level I FIXED INCOME: Introduction to the Valuation of Debt Securities Lecturer: Nan Chen

Framework Estimate CFs: Coupon and Principal 1. Steps in Bond Valuation Process Determine appropriate discount rate Calculate Present Value Defaults and potential credit problems 2. Difficulties in estimating CFs Embedded options-uncertain principal repayment Floating rate securities- uncertain coupons Convertible or Exchangeable bonds 3. Value of Bonds and Zero Coupon Bonds 4. Time and Value of Bond 5. Discount Rate and Value of Bond 6. Arbitrage-free Valuation

1. Steps in Bond Valuation Process General Rule of Asset Valuation: the sum of the present values of all the expected cash flows Step 1 Estimate the cash flows over the life of the security: coupon and principal Step 2 Determine the appropriate discount rate Step 3 Calculate the present value of the estimated cash flows

2. Difficulties in Estimating CFs *Defaults and potential credit problems make receipt of future cash flows uncertain *Uncertain principal repayment: embedded options (put, call, prepayment option, accelerated sinking fund provisions) *Uncertain coupon payments: floating rate securities *Uncertain cash flows of convertible bonds

2. Difficulties in Estimating CFs EX1: An investor is evaluating a set of bonds from which he will select two issues. The investor s objective is to find bonds with cash flows that will precisely match a known stream of future obligations. Which of the following two issues will most likely to meet the investor s objective? A. A putable bond and a floating-rate bond. B. A mortgage-backed security and a convertible bond. C. A zero-coupon bond and a Treasury strip.

3. Value of Bonds and Zero Coupon Bonds Valuation with a single yield and annual cash flows EX2: Assuming an annual discount rate of 8%, estimate the value of a 10-year, 10% coupon, $1000 par value bond with annual payments. *Formula solution PV= =$1,134.20 *Calculator solution N=10, PMT=100, 1/Y=8, FV=1000, CPT -> PV=-1,134.20 N= Number of years; PMT= the annual coupon payment; 1/Y= the annual discount rate FV= the par value or selling price at the end of an assummed holding period

3. Value of Bonds and Zero Coupon Bonds Valuation with a single yield and semi-annual cash flows EX3: Assuming an annual discount rate of 8%, estimate the value of a 10-year, 10% coupon, $1000 par value bond with semi-annual payments. *Formula solution PV= 50 50 50 50 50 1000... =$1,135.90 2 3 4 20 20 1.04 1.04 1.04 1.04 1.04 1.04 *Calculator solution N=20, PMT=50, 1/Y=4, FV=1000, CPT -> PV=-1,135.90 N= Number of semiannual periods; PMT= the semiannual coupon payment; 1/Y= the semiannual discount rate; FV= par value

3. Value of Bonds and Zero Coupon Bonds Valuation of a Zero-Coupon Bond EX4:Assuming an annual discount rate of 8%, estimate the value of a 10-year, zero coupon, $1000 par value bond. Value of a zero coupon bond =Present Value of the Par Value 1, 000 = 0. 08 10 2 (1 ) 2 =$456.39 *The value of a zero-coupon bond is the present value of the par. *It is customary to value zero-coupon bonds using semiannual discount rates.

EX5: Given the following information in the table, which of the following gives the closest value for a 0% coupon, 5-year, $100 par value option-free bond? Years A. 78.12 B. 80.05 C. 79.94 3. Value of Bonds and Zero Coupon Bonds U.S. Treasury Spot Rate (%) 1 2.5 0.1 2 3.0 0.2 3 3.5 0.3 4 4.0 0.4 5 4.5 0.5 Credit Spread (%) *Annual Discount Rate = Spot Rate + Credit Spread = 4.5% +0.5%=5% *Semiannual Discount Rate = 5%/2 = 2.5% $100 (1 2.5%) *The Value of the Corporate Bond = 10 =$78.12

4. Time and Value of Bond Consider a bond with a par value of $1,000, coupon rate of 6% and 3 years to maturity. Semiannual pay. Time to Maturity YTM=3% YTM=6% YTM=12% 3.0 $1,085.46 $1,000 $852.48 *When Coupon Rate= Market Yield Coupon Rate>Market Yield Coupon Rate <Market Yield Trade at par Trade at premium Trade at discount

4. Time and Value of Bond Consider a bond with a par value of $1,000, coupon rate of 6% and 3 years to maturity. Semiannual pay. Time to Maturity YTM=3% YTM=6% YTM=12% Decrease to par value 3.0 $1,085.46 $1,000 $852.48 2.5 $1,071.74 $1,000 $873.63 2.0 $1,057.82 $1,000 $896.05 1.0 $1,029.34 $1,000 $945.00 0.0 $1,000 $1,000 $1,000 Increase to par value Time to Maturity=3 YR: FV=1000, PMT=30, N=6,1/Y=1.5 CPT->PV=-1,085.46 Time to Maturity=2.5 YR:FV=1000, PMT=30 1/Y=1.5,N=5 CPT->PV=-1,071.74 Time to Maturity=2.0 YR:FV=1000, PMT=30 1/Y=1.5,N=4 CPT->PV=-1,057.82 *Regardless of its required yield, the price will converge to par value as maturity approaches.

4. Time and Value of Bond EX6:An 8% coupon, $100 par value bond matures in 2 years and is selling at $98.79 to yield 8 percent. One year ago this bond was sold at a price of $97.02 to yield 9 percent. The bond pays interest annually. The change in price attributable to the change in maturity is closest to: A. 4.97 B. 3.21 C. 1.22 Time to Maturity YTM=9% 3 years $97.02 2 years? N=2, PMT=8, FV=100, 1/Y=9, CPT->PV=-98.24 Change in Price attributable to the change in maturity = 98.24-97.02=1.22

5. Discount Rate and Value of Bond *Discount Rate= YTM *Bond prices and market yields are inversely related.

4. Time and Value of Bond EX7: Consider a $100 par value bond. It has a 6% coupon paid annually and 10 years to maturity. The bond is valued at $102.08 today with a discount rate of 5.5%. One day later, the discount rate increases to 6.5%. Assuming the discount rate remains at 6.5% over the remaining life of the bond, the price of the bond between today and maturity will most likely: A. Decline then remain unchanged. B. Decline then rise. C. Rise then decline B is correct because if the discount rate rises to 6.5% from 5.5%, the price of a bond declines. At a discount rate of 6.5%, the bond sells at a discount to face value. As a discount bond approaches maturity, it will rise in price over time until it reaches par at maturity.

6. Arbitrage-free Valuation *Arbitrage-free Valuation: discount each cash flow with a discount rate that is specific to the maturity of each cash flow (spot rate) EX8.1: Consider a 6% Treasury note with 1.5 years to maturity with $1,000 par value. Spot rates are: 6 months= 5%, 1 year = 6%, 1.5 years=7%. What is the arbitrage-free price of the Treasury note? 30 30 1030 Present Value using spot rates= 2 3 =$986.55 1.025 1.03 1.035 *If Market Value of the Bond arbitrage-free value, there is arbitrage opportunity. Market Value > Arbitrage-free Value Market Value < Arbitrage-free Value Buy the pieces of expected cash flows, package them to make a bond, and then sell the bond package to earn an arbitrage profit. Buy the bond and sell the pieces of expected cash flows EX8.2: If the market price of the Treasury note is $992, since $992 > $986.55, there is arbitrage opportunity: buy the individual cash flows (zero-coupon bonds), combine them into a 1.5 year note, and sell the package for the market price of the note.

6. Arbitrage-free Valuation EX9: If the price of a U.S. Treasury security is higher than its arbitragefree value, an arbitrage profit can be generated by: A. buying the U.S Treasury security, stripping it and selling the strips. B. shorting the U.S. Treasury security and calling it from the issuer. C. shorting the U.S. Treasury security and reconstructing it from strips C is correct because strips can be purchased to create a synthetic U.S. Treasury security to cover the short at a price lower than the price at which the U.S Treasury security was shorted, generating an arbitrage profit.