Valuing Government Bonds Principles of Corporate Finance Eighth Edition Chapter 23 Valuing Government Bonds Slides by Matthew Will McGraw-Hill/Irwin Copyright © 2006 by The McGraw-Hill Companies, Inc. All rights reserved

Topics Covered Real and Nominal Rates of Interest The Term Structure and YTM How Interest Rate Changes Affect Bond Prices Explaining the Term Structure

UK Bond Yields

Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher Nominal Interest Rate = The rate you actually pay when you borrow money

Debt & Interest Rates Classical Theory of Interest Rates (Economics) developed by Irving Fisher Nominal Interest Rate = The rate you actually pay when you borrow money Real Interest Rate = The theoretical rate you pay when you borrow money, as determined by supply and demand r Supply Real r Demand $ Qty

Debt & Interest Rates Nominal r = Real r + expected inflation Real r is theoretically somewhat stable Inflation is a large variable Q: Why do we care? A: This theory allows us to understand the Term Structure of Interest Rates. Q: So What? A: The Term Structure tells us the cost of debt.

Present Value of a Loan The Term Structure can be reflected in using various “r” terms for different time periods

Term Structure and Yields The Return on US Treasury Bills and the Inflation rate (1953-2003)

Valuing a Bond

Valuing a Bond Cash Flows Sept 03 04 05 06 07 115 115 115 115 1115 Example If today is October 2005, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2010 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%) Cash Flows Sept 03 04 05 06 07 115 115 115 115 1115

Valuing a Bond Example continued If today is October 2005, what is the value of the following bond? An IBM Bond pays $115 every Sept for 5 years. In Sept 2010 it pays an additional $1000 and retires the bond. The bond is rated AAA (WSJ AAA YTM is 7.5%)

Term Structure (Feb 2004) Nov 2014 Feb 2004

Bond Prices and Yields Price Yield

Duration Calculation

Duration Example (Bond 1) Calculate the duration of our 6 7/8 % bond @ 4.9 % YTM Year CF PV@YTM % of Total PV % x Year 1 68.75 65.54 .060 0.060 2 68.75 62.48 .058 0.115 3 68.75 59.56 .055 0.165 4 68.75 56.78 .052 0.209 5 68.75 841.39 .775 3.875 1085.74 1.00 Duration 4.424

Duration Example (Bond 2) Given a 5 year, 9.0%, $1000 bond, with a 8.5% YTM, what is this bond’s duration? Year CF PV@YTM % of Total PV % x Year 1 90 82.95 .081 0.081 2 90 76.45 .075 0.150 3 90 70.46 .069 0.207 4 90 64.94 .064 0.256 5 1090 724.90 .711 3.555 1019.70 1.00 Duration= 4.249

Duration & Bond Prices Bond Price, percent Interest rate, percent

Spot/Forward rates Example 1000 = 1000 (1+R3)3 (1+f1)(1+f2)(1+f3)

Spot/Forward rates Forward Rate Computations (1+ rn)n = (1+ r1)(1+f2)(1+f3)....(1+fn)

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995 3 year zero treasury YTM = 9.660

Spot/Forward rates Example What is the 3rd year forward rate? 2 year zero treasury YTM = 8.995 3 year zero treasury YTM = 9.660 Answer FV of principal @ YTM 2 yr 1000 x (1.08995)2 = 1187.99 3 yr 1000 x (1.09660)3 = 1318.70 IRR of (FV1318.70 & PV=1187.99) = 11%

Spot/Forward rates Example Two years from now, you intend to begin a project that will last for 5 years. What discount rate should be used when evaluating the project? 2 year spot rate = 5% 7 year spot rate = 7.05%

Spot/Forward rates Coupons paying bonds to derive rates Bond Value = C1 + C2 (1+r) (1+r)2 Bond Value = C1 + C2 (1+R1) (1+f1)(1+f2) d1 = C1 d2 = C2 (1+R1) (1+f1)(1+f2)

Spot/Forward rates Example 8% 2 yr bond YTM = 9.43% What is the forward rate? Step 1 value bonds 8% = 975 10%= 1010 Step 2 975 = 80d1 + 1080 d2 -------> solve for d1 1010 =100d1 + 1100d2 -------> insert d1 & solve for d2

Spot/Forward rates Example continued Step 3 solve algebraic equations d1 = [975-(1080)d2] / 80 insert d1 & solve = d2 = .8350 insert d2 and solve for d1 = d1 = .9150 Step 4 Insert d1 & d2 and Solve for f1 & f2. .9150 = 1/(1+f1) .8350 = 1 / (1.0929)(1+f2) f1 = 9.29% f2 = 9.58% PROOF

Term Structure Year Spot Rate - The actual interest rate today (t=0) YTM (r) 1981 1987 & Normal 1976 Year 1 5 10 20 30 Spot Rate - The actual interest rate today (t=0) Forward Rate - The interest rate, fixed today, on a loan made in the future at a fixed time. Future Rate - The spot rate that is expected in the future Yield To Maturity (YTM) - The IRR on an interest bearing instrument

Term Structure What Determines the Shape of the TS? 1 - Unbiased Expectations Theory 2 - Liquidity Premium Theory 3 - Market Segmentation Hypothesis Term Structure & Capital Budgeting CF should be discounted using Term Structure info Since the spot rate incorporates all forward rates, then you should use the spot rate that equals the term of your project. If you believe in other theories take advantage of the arbitrage.

Yield To Maturity All interest bearing instruments are priced to fit the term structure This is accomplished by modifying the asset price The modified price creates a New Yield, which fits the Term Structure The new yield is called the Yield To Maturity (YTM)

Yield to Maturity Example A $1000 treasury bond expires in 5 years. It pays a coupon rate of 10.5%. If the market price of this bond is 107.88, what is the YTM?

Yield to Maturity Example A $1000 treasury bond expires in 5 years. It pays a coupon rate of 10.5%. If the market price of this bond is 107.88, what is the YTM? C0 C1 C2 C3 C4 C5 -1078.80 105 105 105 105 1105 Calculate IRR = 8.5%

Web Resources Web Links www.fintools.com www.loanpricing.com Click to access web sites Internet connection required www.fintools.com www.loanpricing.com www.smartmoney.com www.stockcharts.com/charts/yieldcurve.html