Bond Valuations 1. Definition and Example of a Bond 2.How to Value Bonds 3.Bond Concepts

Valuation of Bonds and Stock First Principles: –Value of financial securities = PV of expected future cash flows To value bonds we need to: – Estimate future cash flows: Size (how much) and Timing (when) – Discount future cash flows at an appropriate rate

 PV= CF 1+r r 1n r n. 012n r CF 1 CF n CF 2 Value Financial Asset Valuation

Definition and Example of a Bond A bond is a legally binding agreement between a borrower and a lender: –Specifies the principal amount of the loan. –Specifies the size and timing of the cash flows: In dollar terms (fixed-rate borrowing) As a formula (adjustable-rate borrowing)

How to Value Bonds Identify the size and timing of cash flows. Discount at the correct discount rate. –If you know the price of a bond and the size and timing of cash flows, the yield to maturity is the discount rate.

Example of a Bond Consider a U.S. government bond listed as 6 3/8 of December –The Par Value of the bond is $1,000. –Coupon payments are made semi-annually (June 30 and December 31 for this particular bond). –Since the coupon rate is 6 3/8 the payment is $ –On January 1, 2002 the size and timing of cash flows were:

Pure Discount Bonds Information needed for valuing pure discount bonds: –Time to maturity (T) = Maturity date –Face value (F) –Discount rate (r) Present value of a pure discount bond at time 0:

Pure Discount Bonds: Example Find the value of a 30-year zero-coupon bond with a $1,000 par value and a YTM of 6%.

Level-Coupon Bonds Information needed to value level-coupon bonds: –Coupon payment dates and time to maturity (T) –Coupon payment (C) per period and Face value (F) –Discount rate Value of a Level-coupon bond= PV of coupon payment annuity + PV of face value

Level-Coupon Bonds: Example Find the present value (as of January 1, 2002), of a 6-3/8 coupon T-bond with semi-annual payments, and a maturity date of December 2009 if the YTM is 5-percent. –On January 1, 2002 the size and timing of cash flows were:

NI/YR PV PMTFV -1,000 $ $1, PV annuity PV maturity value Value of bond ====== INPUTS OUTPUT The bond consists of a 10-year, 10% annuity of $100/year plus a $1,000 lump sum at t = 10: Example Using Financial Calculator

When r rises, above the coupon rate, the bond’s value falls below par, so it sells at a discount NI/YR PV PMTFV INPUTS OUTPUT What would happen if expected inflation rose by 3%, causing r = 13%?

What would happen if inflation fell, and r d declined to 7%? If coupon rate > r, price rises above par, and bond sells at a premium NI/YR PV PMTFV -1, INPUTS OUTPUT

Example of Bond valuation for different Yields Suppose the bond was issued 20 years ago and now has 10 years to maturity. What would happen to its value over time if the required rate of return remained at 10%, or at 13%, or at 7%?

M 1,372 1,211 1, r = 7%. r = 13%. r = 10%. Bond Value ($) vs Years remaining to Maturity

Bond Concepts 1.Bond prices and market interest rates move in opposite directions. 2.When coupon rate = YTM, price = par value. When coupon rate > YTM, price > par value (premium bond) When coupon rate < YTM, price < par value (discount bond) 3.A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical. 4. A lower coupon bond has a higher relative price change than a higher coupon bond when YTM changes. All other features are identical.

Rate of Return on a Bond Current yield = Capital gains yield = = YTM = + Annual coupon pmt Current price Change in price Beginning price Exp total return Exp Curr yld Exp cap gains yld

YTM and Bond Value $ Discount Rate Bond Value 6 3/8 When the YTM < coupon, the bond trades at a premium. When the YTM = coupon, the bond trades at par. When the YTM > coupon, the bond trades at a discount.

Maturity and Bond Price Volatility C Consider two otherwise identical bonds. The long-maturity bond will have much more volatility with respect to changes in the discount rate Discount Rate Bond Value Par Short Maturity Bond Long Maturity Bond

Coupon Rate and Bond Price Volatility Consider two otherwise identical bonds. The low-coupon bond will have much more volatility with respect to changes in the discount rate Discount Rate Bond Value High Coupon Bond Low Coupon Bond

Semiannual Bonds 1.Multiply years by 2 to get periods = 2n. 2.Divide nominal rate by 2 to get periodic rate = r/2. 3.Divide annual INT by 2 to get PMT = INT/2. 2n r/2 OK INT/2OK NI/YR PV PMTFV INPUTS OUTPUT

2(10) 13/2 100/ NI/YR PV PMTFV INPUTS OUTPUT Value of 10-year, 10% coupon, semiannual bond if rd = 13%.

r d = r* + IP + DRP + LP + MRP. Here: r d =Required rate of return on a debt security. r*= Real risk-free rate. IP= Inflation premium. DRP= Default risk premium. LP= Liquidity premium. MRP= Maturity risk premium.

What is the nominal risk-free rate? r RF = (1+r*)(1+IP)-1 = r*+ IP + (r*xIP) ≈ r*+ IP. (Because r*xIP is small) r RF = Rate on Treasury securities.

Bond Ratings Provide One Measure of Default Risk Investment GradeJunk Bonds Moody’sAaaAaABaaBaBCaaC S&PAAAAAABBBBBBCCCD

Bond Ratings and Bond Spreads (YahooFinance, 2006) Long-term BondsYieldSpread U.S. Treasury5.25% AAA % AA A BBB BB B CCC

What factors affect default risk and bond ratings? Financial performance –Debt ratio –Coverage ratios, such as interest coverage ratio or EBITDA coverage ratio –Current ratios (More…)

What factors affect default risk and bond ratings? Other factors –Earnings stability –Regulatory environment –Potential product liability –Accounting policies

The Maturity Risk Premium Long-term bonds: High interest rate risk, low reinvestment rate risk. Short-term bonds: Low interest rate risk, high reinvestment rate risk. Nothing is riskless! Yields on longer term bonds usually are greater than on shorter term bonds, so the MRP is more affected by interest rate risk than by reinvestment rate risk.

Term Structure Yield Curve Term structure of interest rates: the relationship between interest rates (or yields) and maturities. A graph of the term structure is called the yield curve.

Hypothetical Treasury Yield Curve