Ch. 7 Bond Valuation  1999, Prentice Hall, Inc.

Characteristics of Bonds  Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the par value at maturity.

Characteristics of Bonds  Bonds pay fixed coupon (interest) payments at fixed intervals (usually every 6 months) and pay the par value at maturity n $I $I $I $I $I $I+$M

example: AT&T 9s of 2018 n par value = $1000 n coupon = 9% of par value per year. = $90 per year ($45 every 6 months). = $90 per year ($45 every 6 months). n maturity = 20 years. n issued by AT&T.

example: AT&T 9s of n $45 $45 $45 $45 $45 $45+$1000 n par value = $1000 n coupon = 9% of par value per year. = $90 per year ($45 every 6 months). = $90 per year ($45 every 6 months). n maturity = 20 years. n issued by AT&T.

Types of Bonds n Debentures - unsecured bonds. n Subordinated debentures - unsecured “junior” debt. n Mortgage bonds - secured bonds. n Zeros - bonds that pay only par value at maturity; no coupons. n Junk bonds - speculative or below- investment grade bonds; rated BB and below.

Types of Bonds n Eurobonds - bonds denominated in one currency and sold in another country. (Borrowing overseas). n example - suppose Disney decides to sell $1,000 bonds in France. These are U.S. denominated bonds trading in a foreign country. Why do this? n If borrowing rates are lower in France, n To avoid SEC regulations.

Convertibility n Some bonds may be converted to common stock. n Is this a benefit to the investor? Yes! Yes!

The Bond Indenture n The bond contract. n Lists all of the bond’s features: coupon, par value, maturity, etc. coupon, par value, maturity, etc. n Lists covenants which are designed to protect bondholders. n Describes repayment provisions.

Security Valuation n In general, the intrinsic value of an asset = the present value of the stream of expected cash flows discounted at an appropriate required rate of return. n Can the intrinsic value of an asset differ from the market value?

Bond Valuation n Simply discount the cash flows at the investor’s required rate of return. 1) the coupon payment stream (an annuity). 2) the par value payment (a single sum).

Bond Valuation n $I = the coupon interest payment. n k b = the investor’s required rate of return (which depends on the riskiness of the bond). n V b = the intrinsic value of the bond. V b = $I $M (1 + k b ) t = 1 n t n + 

Bond Valuation V b = $I (PVIFA i, n ) + $M (PVIF i, n ) + V b = $I $M (1 + k b ) t = 1 n t n 

Bond Example n S’pose our firm decides to issue 20-year bonds with a par value of $1,000 and annual coupon payments. The return on other bonds of similar risk is currently 12%, so we decide to offer a 12% coupon interest rate. n What would be a fair price for these bonds?

P/YR = 1 N = 20 I%YR = 12 FV = 1,000 PMT = 120 Solve PV = -$1,000 Note: If the coupon rate = discount rate, the bond will sell for par value.

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 )

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 ) PV = 120 (PVIFA.12, 20 ) (PVIF.12, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i 1 PV = (1.12 ) / (1.12) 20 = $

n Suppose interest rates fall immediately after we issue the bonds. The required return on bonds of similar risk drops to 10%. n What would happen to the bond price?

P/YR = 1 Mode = end N = 20 I%YR = 10 PMT = 120 FV = 1000 Solve PV = - $1,170.27

P/YR = 1 Mode = end N = 20 I%YR = 10 PMT = 120 FV = 1000 Solve PV = - $1, Note: If the coupon rate > discount rate, the bond will sell for a premium.

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 )

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 ) PV = 120 (PVIFA.10, 20 ) (PVIF.10, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i 1 PV = (1.10 ) / (1.10) 20 = $1,

n Suppose interest rates rise immediately after we issue the bonds. The required return on bonds of similar risk rises to 14%. n What would happen to the bond price?

P/YR = 1 Mode = end N = 20 I%YR = 14 PMT = 120 FV = 1000 Solve PV = -$867.54

P/YR = 1 Mode = end N = 20 I%YR = 14 PMT = 120 FV = 1000 Solve PV = -$ Note: If the coupon rate < discount rate, the bond will sell for a discount.

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 )

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 120 (PVIFA.14, 20 ) (PVIF.14, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i 1 PV = (1.14 ) / (1.14) 20 = $

Suppose coupons are semi-annual P/YR = 2 Mode = end N = 40 I%YR = 14 PMT = 60 FV = 1000 Solve PV = -$866.68

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 )

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 ) PV = 60 (PVIFA.14, 20 ) (PVIF.14, 20 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i 1 PV = (1.07 ) / (1.07) 40 = $

Yield To Maturity n The average annual rate of return investors expect to receive on a bond if they hold it to maturity.

Yield To Maturity n The average annual rate of return investors expect to receive on a bond if they hold it to maturity. V = $I (PVIFA i, n) + $M (PVIF i, n) b Just solving for i !!!

YTM Example Suppose we paid $ for a $1,000 par 10% coupon bond with 8 years to maturity and semi-annual coupon payments. What is our yield to maturity?

Using the Financial Calculator P/YR = 2 Mode = end N = 16 PV = PMT = 50 FV = 1000 Solve I%YR = 12%

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) = 50 (PVIFA i, 16 ) (PVIF i, 16 )

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i = (1 + i ) / (1 + i) 16 i i

Bond Example Mathematical Solution: PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) PV = PMT (PVIFA i, n ) + FV (PVIF i, n ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) = 50 (PVIFA i, 16 ) (PVIF i, 16 ) 1 1 PV = PMT 1 - (1 + i) n + FV / (1 + i) n i = (1 + i ) / (1 + i) 16 isolve using trial and error isolve using trial and error

Zero Coupon Bonds No coupon interest payments. The bond holder’s return is determined entirely by the price discount.

Zero Example Suppose you pay $508 for a bond that has 10 years left to maturity. What is your yield to maturity?

Zero Example Suppose you pay $508 for a bond that has 10 years left to maturity. What is your yield to maturity? $508 $1000

Zero Example P/YR = 1 Mode = End N = 10 PV = -508 FV = 1000 Solve: I%YR = 7%

Mathematical Solution: PV = FV (PVIF i, n ) PV = FV (PVIF i, n ) 508 = 1000 (PVIF i, 10 ) 508 = 1000 (PVIF i, 10 ).508 = (PVIF i, 10 ) [use PVIF table].508 = (PVIF i, 10 ) [use PVIF table] PV = FV /(1 + i) 10 PV = FV /(1 + i) = 1000 /(1 + i) = 1000 /(1 + i) = (1 + i) = (1 + i) 10 i = 7% i = 7% Zero Example PV = -508 FV = 1000

The Financial Pages: Corporate Bonds Cur Net Bonds Yld Vol Close Chg Eckerd 9 1 / / 2... What is Eckerd’s yield to maturity? n P/YR = 2, N = 12, FV = 1000, PV = $-1,075, n PMT = n Solve: I/YR = 7.67%

The Financial Pages: Corporate Bonds Cur Net Bonds Yld Vol Close Chg AlldC zr / 8 +2 What is Allied Chemical’s yield to maturity? n P/YR = 1, N = 11, FV = 1000, PV = $ , n PMT = 0 n Solve: I/YR = 7.83%

The Financial Pages: Treasury Bonds Maturity Ask Maturity Ask Rate Mo/Yr Bid Asked Chg Yld 9 Nov 18128:18 128: Nov 18128:18 128: n What is the yield to maturity of this Treasury bond? n P/YR = 2, N = 40, FV = 1000, PMT = 45, PV = - 1, (128.75% of par) PV = - 1, (128.75% of par) n Solve: I/YR = 6.43%