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**Bond Valuation and Risk**

CHAPTER 8 Bond Valuation and Risk

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**CHAPTER 8 OVERVIEW This chapter will:**

A. Explain how debt securities are priced B. Identify the factors that affect bond prices C. Explain how the sensitivity of bond prices to interest rates is dependent on particular bond characteristics

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**Preparing BAII Plus for use**

Press ‘2nd’ and [Format]. The screen will display the number of decimal places that the calculator will display. If it is not eight, press ‘8’ and then press ‘Enter’. Press ‘2nd’ and then press [P/Y]. If the display does not show one, press ‘1’ and then ‘Enter’. Press ‘2nd’ and [BGN]. If the display is not END, that is, if it says BGN, press ‘2nd’ and then [SET], the display will read END.

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Magic Tools Time Line PV, FV, I/Y, PMT, N

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**Zero-coupon bond (ZCB) 1**

FI Corporate Finance Leng Ling Zero-coupon bond (ZCB) 1 Zero coupon rate, no coupon paid during bond’s life. Bond holder receives one payment at maturity, the face value Price of a ZCB, PZCB F = face value of the bond N = number of years to maturity cost of ZCB debt capital (in decimals)

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**Zero-coupon bond (ZCB) 2**

As long as interest rates are positive, the price of a ZCB must be less than its face value. Why? With positive interest rates, the present value of the face value (i.e., the price) has to be less than the face value.

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**FI 3300 - Corporate Finance Leng Ling**

ZCB Problems 1) Find the price of a ZCB with 20 years to maturity, par value of $1000 and a required rate of return of 15% p.a. N=20, I/Y=15, FV=1000, PMT=0. Price = $61.10 2) XYZ Corp.’s ZCB has a market price of $ 354. The bond has 16 years to maturity and its face value is $1000. What is the cost of debt for the ZCB (i.e., the required rate of return). PV=-354, FV=1000, N=16, PMT=0. Required rate of return/ Cost of debt =6.71% p.a.

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**Fixed-coupon bond (FCB) 1**

Firm pays a fixed amount (‘coupon’) to the investor every period until bond matures. At maturity, firm pays face value of the bond to investor. Face value also called par value. Unless otherwise stated, always assume face value to be $1000. Period: can be year, half-year (6 months)

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**Fixed-coupon bond (FCB) 2**

FCB gives you a stream of fixed payments plus a single payment (face value) at maturity. This cash flow stream is just an annuity plus a single cash flow at maturity. We use the financial calculator to compute the price of the FCB.

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**Fixed-coupon bond (FCB) 3**

Pricing formula: C = coupon payment paid in each period Par = par value r = required rate of return n = number of period to maturity

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Example A bond has a par value of $1,000, pays $100 at the end of each year in coupon payment, and has thee years remaining until maturity. Assume the annualized yield required is 12%, what is the price?

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**Exhibit 8.1 Valuation of a Three-Year Bond**

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**FI 3300 - Corporate Finance Leng Ling**

Find FCB price A $1,000 par value bond has coupon rate of 5% and the coupon is paid semi-annually. The bond matures in 20 years and has a required rate of return of 10%. Compute the current price of this bond. PMT = 25. Why? FV=1000, PMT =25, I/Y=5, N=40. CPT, then PV. PV = Thus, price = $ < par value

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**Useful property 1 Go back to the bond in the last problem.**

Suppose annual coupon rate = 10%. Verify that price = $1000 = par value Suppose annual coupon rate = 12% Verify that price = $1, > par value. It turns out that the following property is true.

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**FI 3300 - Corporate Finance Leng Ling**

Useful property 2 Coupon rate < discount rate Price < face value selling at discount Coupon rate = discount rate Price = face value selling at par Coupon rate > discount rate Price > face value selling at premium

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**FI 3300 - Corporate Finance Leng Ling**

Apply what we learnt A 10-year annual coupon bond was issued four years ago at par. Since then the bond’s yield to maturity (YTM) has decreased from 9% to 7%. Which of the following statements is true about the current market price of the bond? The bond is selling at a discount The bond is selling at par The bond is selling at a premium The bond is selling at book value Insufficient information

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**FI 3300 - Corporate Finance Leng Ling**

Try one more One year ago Pell Inc. sold 20-year, $1,000 par value, annual coupon bonds at a price of $ per bond. At that time the market rate (i.e., yield to maturity) was 9 percent. Today the market rate is 9.5 percent; therefore the bonds are currently selling: at a discount. at a premium. at par. above the market price. not enough information.

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**FI 3300 - Corporate Finance Leng Ling**

Find YTM, Coupon rate 1)A $1,000 par value bond sells for $ It matures in 20 years, has a 10 percent coupon rate, and pays interest semi-annually. What is the bond’s yield to maturity on a per annum basis (to 2 decimal places)? Verify that YTM = 11.80% 2) ABC Inc. just issued a twenty-year semi-annual coupon bond at a price of $ The face value of the bond is $1,000, and the market interest rate is 9%. What is the annual coupon rate (in percent, to 2 decimal places)? Verify that annual coupon rate = 6.69% What happens if bond pays coupon annually?

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**FI 3300 - Corporate Finance Leng Ling**

Long FCB question HMV Inc. needs to raise funds for an expansion project. The company can choose to issue either zero-coupon bonds or semi-annual coupon bonds. In either case the bonds would have the SAME nominal required rate of return, a 20-year maturity and a par value of $1,000. If the company issues the zero-coupon bonds, they would sell for $ If it issues the semi-annual coupon bonds, they would sell for $ What annual coupon rate is HMV Inc. planning to offer on the coupon bonds? State your answer in percentage terms, rounded to 2 decimal places. Verify that annual coupon rate = 7.01%

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**Bond Valuation Impact of the Timing of Payments on Bond Valuation**

a. Timing affects the market price of a bond b. Funds received sooner can be reinvested to earn additional returns c. Most bonds have semiannual coupons

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**Factors that Affect Bond Prices**

Factors That Affect the Risk-Free Rate a. Inflationary Expectations b. Economic Growth c. Money Supply Growth d. Federal Government Budget Deficit

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**Factors that Affect Bond Prices**

Factors That Affect the Credit (Default) Risk Premium a Changes in the Credit Risk Premium over Time b. Changes in Bond Ratings over Time

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**Sensitivity of Bond Prices to Interest Rate Movements**

Duration: measurement of the life of a bond on a present value basis. The longer the duration, the greater its sensitivity to interest rate changes. A measure of interest rate sensitivity. The weighted average of the times to each coupon or principal payment made by the bond.

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**Duration Formula Time until 4th cash flow Weight of 4th cash flow**

Formula for annual coupon payment bond. For semi-annual payment bond, y is the semi-annual YTM, T is the no. of semi-annual periods to maturity, CFt is the semi-annual coupon payment, bond price is computed assuming semi-annual compounding. Cash flow paid at time t

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**Examples: 3-year bond, 8%, annual coupon payments, YTM = 10%**

This slide shows the examples of computing maturity. Part A computes maturity for 8% coupon bond, Part B computes zero-coupon bond. This example shows that duration is shorter when there are cash flows paid before maturity. This is exactly what we want and accords to what we saw earlier. Also note that in general, the Macaulay of a zero coupon bond is exactly equal to its maturity. Tell student that if coupons are paid semi-annually, then the duration you get is the semi-annual duration. To convert to annual duration, divide the semi-annual duration by 2. Use the same example to compute convexity.

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Duration The duration of a bond with $1,000 par value and 7% annual coupon rate, three years remaining to maturity, and a 9% yield to maturity is

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**Modified duration YTM (in decimal)**

If Macaulay’s duration is annual, then modified duration is also annual. If Macaulay’s duration is semi-annual, then modified duration is also semi-annual. Modified durations are also in terms of years/semi-annual periods. If the original compounding basis on the bond was semi-annual, the modified duration must first be calculated on a semi-annual basis and then annualized. YTM (in decimal)

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**Using Modified duration to estimate percentage price change**

Dollar change in bond price Change in YTM in decimal Macaulay’s duration not only tells us the interest rate sensitivity, it can also be used to estimate bond price change for a given change in YTM. To do that, we compute modified duration, which estimates the proportional (percentage) change in bond price for a given change in YTM. We use Δy to represent the change in YTM The equation says that bond price volatility (price change) is proportional to the bond’s duration. For a given change in yield, the bigger the duration, the bigger the price change and the more sensitive is the bond, vice versa. Thus, duration becomes a natural measure of interest rate exposure. Why is there a negative sign? Because of the inverse relationship between bond price and interest rate. Duration is always positive. When yield increases, the second term on the RHS is positive. We need the negative sign to ensure that the number on the RHS is negative. This ensures consistency with the inverse price-yield relationship. Similarly, when yield decreases, the second term on the RHS is negative. This combines with the negative sign ensures that the price change is positive, again consistent with the price-yield relationship. Tell student that this is just an approximation. The approximation is more accurate for small changes in YTM. Turns out that for big changes in yield, using duration will not give good approximation. We need to include another measure, called convexity. Initial price Modified duration

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**Interest rate risk example**

If Bond X’s duration is 8 years, its yield is 10%, what is the percentage change in price for an increase in yield of 0.2 percentage point?

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Interpretation As Macaulay’s duration ↑, interest rate sensitivity ↑ , vice versa. As modified duration ↑, interest rate sensitivity ↑ , vice versa. The higher the Macaulay's duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa. The higher the modified duration, the greater the interest rate sensitivity of a bond, i.e., the bigger the price change for a given change in YTM, vice versa.

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Another example Chapter 8 problem 13.

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**Homework Assignment 5 Chapter 7: Questions and Applications 12,14.**

Problems 1,2,3,5,7,11 (a b),12,17. Now discussion.

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