# Copyright © 2000 Addison Wesley Longman Slide #3-1 Chapter Three UNDERSTANDING INTEREST RATES Part II Principles of Financial Markets.

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Copyright © 2000 Addison Wesley Longman Slide #3-1 Chapter Three UNDERSTANDING INTEREST RATES Part II Principles of Financial Markets

Copyright © 2000 Addison Wesley Longman Slide #3-2 Present Value Four Types of Credit Instruments 1. Simple Loan 2. Fixed Payment Loan 3. Coupon Bond 4. Discount Bond Concept of Present Value Simple loan of \$1 at 10% interest Year123 n \$1.10 \$1.21 \$1.33 \$1x(1+i) n PV of future \$1 = \$1 (1+i) n

Copyright © 2000 Addison Wesley Longman Slide #3-3 Yield to Maturity: Loans Yield to maturity = interest rate that equates today's value with present value of all future payments 1. Simple Loan (i = 10%) \$100 = \$110/(1+i)  i = \$110 - \$100 = \$10 =.10 = 10% \$100 \$100  YTM= 約定利率

Copyright © 2000 Addison Wesley Longman Slide #3-4 2. Fixed Payment Loan (i = 12%) \$1000 = \$126 + \$126 + \$126 +... + \$126 (1+i) (1+i) 2 (1+i) 3 (1+i) 25 LV = FP + FP + FP +... + FP (1+i) (1+i) 2 (1+i) 3 (1+i) N Yield to Maturity: Loans  YTM= 約定利率

Copyright © 2000 Addison Wesley Longman Slide #3-7 Yield to Maturity: Bonds 3. Coupon Bond (Coupon rate = 10% = C/F) P = \$100 + \$100 + \$100 +... + \$100 + \$1000 (1+i) (1+i) 2 (1+i) 3 (1+i) 10 (1+i) 10 P = C + C + C +... + C + F (1+i) (1+i) 2 (1+i) 3 (1+i) N (1+i) N Consol: Fixed coupon payments of \$C forever P = C i = C i P  YTM  coupon interest rate

Copyright © 2000 Addison Wesley Longman Slide #3-8 4. One-year Discount Bond (P = \$900, F = \$1000) \$900 = \$1000  (1+i) i = \$1000 - \$900 =.111 = 11.1% \$900 i = F - P P Yield to Maturity: Bonds

Copyright © 2000 Addison Wesley Longman Slide #3-9 Relationship Between Price and Yield to Maturity Three Interesting Facts in Table 1 1. When bond is at par, yield equals coupon rate 2. Price and yield are negatively related 3. Yield greater than coupon rate when bond price is below par value

Copyright © 2000 Addison Wesley Longman Slide #3-10 Current Yield i c = C P Two Characteristics 1. Is better approximation to yield to maturity, nearer price is to par and longer is maturity of bond 2. Change in current yield always signals change in same direction as yield to maturity To approximate coupon bond 的 YTM

Copyright © 2000 Addison Wesley Longman Slide #3-11 Yield on a Discount Basis One-year bill, P = \$900, F = \$1000 Two Characteristics 1.Understates yield to maturity; longer the maturity, greater is understatement 2.Change in discount yield always signals change in same direction as yield to maturity To approximate discount bond 的 YTM

Copyright © 2000 Addison Wesley Longman Slide #3-13 Distinction Between Real and Nominal Interest Rates Real interest rate Interest rate that is adjusted for expected changes in the price level i r = i - π e 1. Real interest rate more accurately reflects true cost of borrowing 2. When real rate is low, greater incentives to borrow and less to lend if i = 5% and π e = 0% then: i r = 5% - 0% = 5% if i = 10% and π e = 20% then i r = 10% - 20% = - 10% ∴有 index bond 其利率與本金皆隨物 價水準調整

Copyright © 2000 Addison Wesley Longman Slide #3-15 Distinction Between Interest Rates and Returns Rate of Return

Copyright © 2000 Addison Wesley Longman Slide #3-17 Maturity and the Volatility of Bond Returns Key Findings from Table 2 1. Only bond whose return = yield is one with maturity = holding period 2. For bonds with maturity > holding period, i  P  implying capital loss 3. Longer is maturity, greater is price change associated with interest rate change 4. Longer is maturity, more return changes with change in interest rate 5. Bond with high initial interest rate can still have negative return if i  Initial YTM

Copyright © 2000 Addison Wesley Longman Slide #3-18 Maturity and the Volatility of Bond Returns Conclusion from Table 2 Analysis 1. Prices and returns more volatile for long- term bonds because have higher interest- rate risk 2. No interest-rate risk for any bond whose maturity equals holding period

Copyright © 2000 Addison Wesley Longman Slide #3-19 Reinvestment Risk 1. Occurs if hold series of short bonds over long holding period 2. i at which reinvest uncertain 3. Gain from i , lose when i 

Copyright © 2000 Addison Wesley Longman Slide #3-21 Calculating Duration, i = 20% 10-yr 10% Coupon Bond

Copyright © 2000 Addison Wesley Longman Slide #3-22 Formula for Duration Key facts about duration Everything else equal, 1. When the maturity of a bond lengthens, the duration rises as well. 2. When interest rates rise, the duration of a coupon bond falls. =effective maturity for n 個 zero coupon bond

Copyright © 2000 Addison Wesley Longman Slide #3-23 3. The higher is the coupon rate on the bond, the shorter is the duration of the bond. 4. Duration is additive: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each. Formula for Duration

Copyright © 2000 Addison Wesley Longman Slide #3-24 Duration and Interest-Rate Risk %ΔP  - DUR x Δi/(1+i) i  10% to 11%: Table 4 -10% coupon bond %ΔP = -6.76 x.01/(1+.10) = -.0615 = -6.15%. Actual decline = 6.23% 20% coupon bond, DUR = 5.98 years %ΔP = - 5.98 x.01/(1+.10) = -.0540 = -5.40%

Copyright © 2000 Addison Wesley Longman Slide #3-25 The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates. Therefore, the greater is the duration of a security, the greater is its interest-rate risk. Duration and Interest-Rate Risk