Copyright © 1999 Addison Wesley Longman CHAPTER CHAPTER 8 Asset Valuation: Bonds Copyright © 1999 Addison Wesley Longman 1

Objectives 1. Understand the valuation concept. 2. Show how bonds are valued 3. Compute yield to maturity

The Bond Components 1.Bond: a contractual obligation of a borrower to make periodic cash interest payments to a lender for a fixed number of years 2.Coupon (CF): the periodic interest payment 3.Term to maturity (n): the number of years over which the bond contract extends

4. Par Value (F): the number of dollars paid to the lender upon maturity 5. Coupon Rate (C/F): the coupon payment expressed as a percentage of a principal amount (Par value) 6. Market Rate or Yield (i): the actual rate paid on the bond by the market

The Value of Any Investment: Is the present value of all future cash flows discounted at an interest rate that reflects the risk of the investment. Bonds are valued like any other asset

The cash flows from a bond: The equal annual interest payments (an annuity) C = cF .08 x $1000 = $80 And the final lump sum payment at maturity (F) $1000

The Coupon Bond Price Formula PB = C1 + C2 + . . . + Cn + Fn (1+i)1 (1+i)2 (1+i)n PB = Price of bond = PV of future cash flows Ct = Coupon pmt. in period t Fn = par value (amount to be paid at maturity) i = interest rate (discount rate) or YTM n = number of periods to maturity

Example: Calculate the price of a 3 year bond with a par value of $1000, a coupon rate of 8% and a discount or market rate of 10% Step 1: Calculate coupon pmt. C = cF

Step 2: Find the PV of the cash flows which equals the price t=0 t=1 t=2 t= 3 |--------------------|--------------------|--------------------| $80 $80 $80+1000   PB = ________ + ________ + ________ ( ) ( ) ( )

Using the Calculator [N] = [I/Y] = [PV] = [PMT] = [FV] =

Properties of the Bond Price Formula 1. Case 1: coupon = market rate = PB = par value bond is selling at par 2. Case 2: coupon < market rate = PB < par value bond is selling at a discount 3. Case 3: coupon > market rate = PB > par value bond is selling at a premium

The Zero Coupon Bond Price Formula Zero Coupons: 1. trade at a discount 2. have no coupon reinvestment risk 3. examples a. t-Bills b. savings bonds

4. PB = Fn [ 1/(1+i/k)kn] PB = Price of bond Fn = value at maturity i = interest yield for n periods n = # of years until maturity k = # of compoundings annually

Example: Find the price of a 10 yr. Zero coupon bond with a par value of $1000 and a market rate of 12% compounded semi-annually     Using Calculator [N] = [I/Y] = [PV] = [PMT] = [FV] =

Example: What is the price of a bond if it pays interest semi-annually on a 7% coupon and if it has 5 years to maturity, a $1000 face amount and a required return of 10%? Using Calculator [N] = [I/Y] = [PV] = [PMT] = [FV] =

As market interest rates rise and fall: If interest rates increase, the value of the bond falls. If interest rates decrease, the valud of the bond rises. As maturity decreases, the valud of the bond approaches par.

Market Rate Term 9% 10% 11% 1 $1,009.17 $1,000.00 $991.00 TABLE 8.2 Price of $1,000 Par, 10% Coupon Bond with Different Maturities and Market Interest Rates Market Rate Term 9% 10% 11% 1 $1,009.17 $1,000.00 $991.00 10 1,064.18 1,000.00 941.11 20 1,091.28 1,000.00 920.37

FIGURE 8.1 10% Coupon Bond at 9% and 11% Market Rates 2

Bond Yield Measures A. The actual return on the bond depends on 1. default risk 2. reinvestment risk 3. price risk B. The ideal measure tries to capture 1. coupon payments 2. reinvestment rate 3. capital gains and losses

Yield to Maturity The investors expected yield if the bond is held to maturity and the cash flows are reinvested at the “Yield to Maturity” for the life of the bond. 1. Yield to maturity varies inversely to bond prices change in BP change in YTM 2. When bond is selling at par, the coupon rate should equal the market rate of interest. (Par Bond) < 0

Example: Given a 3 year bond with par=$1000, coupon rate=8% and a current price = $950.27, what is the bonds YTM? Using Calculator [N] = [I/Y] = [PV] = [PMT] = [FV] =

Expected Yield The ex-ante yield assuming a forecasted sale price prior to maturity based on expected interest rates.

Realized Yield The ex-post actual rate of return given the cash flows actually received and their timing Realized Yields affected by: 1. change in timing of promised cash flows 2. change in the interest rates 3. sell bond before maturity at different interest rate

The Current Yield: The percentage return earned in a year from interest payments. CY = Coupon Payment / Market Price

Example: If the current market price is $900, and the coupon payment is $80, then what is the current yield?

Even though bonds are called “fixed income securities,” They are subject to significant price changes when interest rates are volatile.

Bond Price Volatility The % change in a bond price for a given change in yield 1. volatility increases as time to maturity increases 2. volatility decreases as coupon rates increase 3. for any change in a bond price, short term rates vary more than term rates

Interest Rate Risk 1. Price risk 2. Reinvestment risk 3. Price risk and reinvestment risk partially offset each other. a. PR up RR down b. PR down RR up

Duration 1. Duration is the investment period necessary for price risk and reinvestment risk to be exactly offset. 2. The weighted average # of years needed to fully recover the present value of principle and coupon payments.

Uses of Duration 1. The most important use of duration is for reducing or eliminating interest risk over a given holding period. 2. Compare sensitivity of bonds with different coupons and maturities to changes in interest rates.

Duration Equation n CFt(t) D = S t=1 (1+i)t PB PB = Price of bond   D = Duration PB = Price of bond CF = Cash flows or coupon payments t = Years  

Example: Calculate the duration of a bond with 3 years to maturity, a price of $840.17, a coupon rate of 8%, and a current market rate of 15%.   __________ + _________ + __________ D = ( ) ( ) ( ) ________________________________ ( )

Results 1. high coupon rates => shorter duration low coupon rates => longer duration 2. generally a (+) relationship exists between duration and term to maturity 3. for a zero coupon bond, duration = maturity 4. ceteris paribus : The higher the market rate of interest, the shorter the duration of the bond.