Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-1 Chapter Three Interest Rates and Security Valuation

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-2 Chapter Outline 1.Bond Valuation Review 2.Interest Rate Risk and Factors Affecting Interest Rate Risk 3.Duration 1.Bond Valuation Review 2.Interest Rate Risk and Factors Affecting Interest Rate Risk 3.Duration

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin Bond Valuation Formula V b = INT/m + INT/m INT/m __ (1 + i d /m) 1 (1 + i d /m) 2 (1 + i d /m) Nm + M_ _ _ (1 + i d /m) Nm Where: V b = Present value of the bond M = Par or face value of the bond INT = Annual interest (or coupon) payment per year on the bond; i d = Interest rate used to discount cash flows on the bond

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-4 Bond Valuation Example V b = 1,000(.1) (PVIFA 8%/2, 12(2) ) + 1,000(PVIF 8%/2, 12(2) ) 2 Where: V b = $1, (solution) M = $1,000 INT = $100 per year (10% of $1,000) N = 12 years i d = 8% (rrr) PVIF = Present value interest factor of a lump sum payment PVIFA = present value interest factor of an annuity stream V b = 1,000(.1) (PVIFA 8%/2, 12(2) ) + 1,000(PVIF 8%/2, 12(2) ) 2 Where: V b = $1, (solution) M = $1,000 INT = $100 per year (10% of $1,000) N = 12 years i d = 8% (rrr) PVIF = Present value interest factor of a lump sum payment PVIFA = present value interest factor of an annuity stream

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-5 Premium, Discount, and Par Bond Premium bond— when the coupon rate, INT, is greater than the required rate of return, rrr, the fair present value of the bond (V b ) is greater than its face value (M) Discount bond— when INT<rrr, then V b <M; bond in which the present value of the bond is less than its face value Par bond— when INT=rrr, then V b =M; bond in which the present value of the bond is equal to its face value Premium bond— when the coupon rate, INT, is greater than the required rate of return, rrr, the fair present value of the bond (V b ) is greater than its face value (M) Discount bond— when INT<rrr, then V b <M; bond in which the present value of the bond is less than its face value Par bond— when INT=rrr, then V b =M; bond in which the present value of the bond is equal to its face value

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin Interest Rate Risk There is a negative relation between interest rate changes and present value changes As interest rate increases, security price decrease at a decreasing rate The higher the interest rate level, the less sensitive of bond price to the change of interest rate, that is the lower the interest rate risk There is a negative relation between interest rate changes and present value changes As interest rate increases, security price decrease at a decreasing rate The higher the interest rate level, the less sensitive of bond price to the change of interest rate, that is the lower the interest rate risk

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-7 Impact of Interest Rate Changes on Security Values Interest Rate Bond Value Interest Rate Bond Value 12% 10% 8% ,0001,152.47

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-8 Impact of Interest Rate Changes on Security Values 12% 10%8% ,000 1, Bond Value Interest Rate

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-9 Balance sheet of an FI before and after an Interest Rate Increase (a) Balance Sheet before the Interest Rate Increase Assets Bond (8% required rate of return) $1, Liabilities and Equity Bond (10% required rate of return) $1,000 Equity $ (b) Balance Sheet after 2% increase in the Interest Rate Increase Assets $1,000 Bond (10% required rate of return) Liabilities and Equity Bond (12% required rate of return) Equity $ $125.50

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-10 Factors Affecting Interest Rate Risk Time Remaining to Maturity –The shorter the time to maturity, the closer the price is to the face value of the security –The longer time to maturity, the larger the price change of the securities for a given interest rate change –which increases at a decreasing rate Coupon Rate –The higher the coupon rate, the smaller the price change for a given change in interest rates Time Remaining to Maturity –The shorter the time to maturity, the closer the price is to the face value of the security –The longer time to maturity, the larger the price change of the securities for a given interest rate change –which increases at a decreasing rate Coupon Rate –The higher the coupon rate, the smaller the price change for a given change in interest rates

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-11 Summary of Factors that Affect Security Prices and Price Volatility when Interest Rates Change Interest Rate –negative relation between interest rate changes and present value changes –increasing interest rates correspond to security price decrease (at a decreasing rate) Time Remaining to Maturity –shorter the time to maturity, the closer the price is to the face value of the security –longer time to maturity corresponds to larger price change for a given interest rate change (at a decreasing rate) Coupon Rate –the higher the coupon rate, the smaller the price change for a given change in interest rates (and for a given maturity) Interest Rate –negative relation between interest rate changes and present value changes –increasing interest rates correspond to security price decrease (at a decreasing rate) Time Remaining to Maturity –shorter the time to maturity, the closer the price is to the face value of the security –longer time to maturity corresponds to larger price change for a given interest rate change (at a decreasing rate) Coupon Rate –the higher the coupon rate, the smaller the price change for a given change in interest rates (and for a given maturity)

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-12 Impact of Maturity on Security Values 12 Years to Maturity 16 Years to Maturity Required Rate of Return Fair Price* Price Change Percentage Price Change 8% $1, $ % 10% $1, $ % 12% $ Fair Price* Price Change Percentage Price Change $1, $ % $1, $ % $ *The bond pays 10% coupon interest compounded semiannually and has a face value of $1,000

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-13 Impact of a Bond’s Maturity on its Interest Rate Sensitivity Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Absolute Value of Percent Change in a Bond’s Price for a Given Change in Interest Rates Time to Maturity

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-14 Impact of Coupon Rate on the Relation between Security Values and Its rrr 10 Percent Coupon Bond 12 Percent Coupon Bond Required Rate of Return Fair Price* Price Change Percentage Price Change 8% $1, $ % 10% $1, $ % 12% $ Fair Price* Price Change Percentage Price Change $1, $ % $1, $ % $1, *The bond has 12 years remaining to maturity and has a face value of $1,000

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-15 Impact of a Bond’s Coupon Rate on Its Interest Rate Sensitivity Bond Value Low-Coupon Bond High-Coupon Bond Interest Rate

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin Macauley’s Duration: A Measure of Interest Rate Sensitivity The weighted-average time to maturity on an investment N N  CF t  t  PV t  t t = 1 (1 + R) t t = 1 D = N = N  CF t  PV t t = 1 (1 + R) t t = 1 The weighted-average time to maturity on an investment N N  CF t  t  PV t  t t = 1 (1 + R) t t = 1 D = N = N  CF t  PV t t = 1 (1 + R) t t = 1

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-17 FV=1000, PMT=40, I/Y=5, N=2 CPT PV= Macauley’s Duration (p.76) PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin /(1+.05)= /(1+.05) 2 = Macauley’s Duration PV 1 = PV 0.5 =38.1 PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin /(1+.05)= /(1+.05) 2 = Macauley’s Duration PV 1 = PV 0.5 =38.1 PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin /(1+.05)= /981.41=3.88% 1040/(1+.05) 2 = /981.41=96.12% Macauley’s Duration PV 1 = PV 0.5 =38.1 PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin /(1+.05)= /981.41=3.88% 1040/(1+.05) 2 = /981.41=96.12% So 3.88% of the initial investment will be paid back in 0.5 year, 96.12% of the initial investment will be paid back in 1 year. Macauley’s Duration PV 1 = PV 0.5 =38.1 PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-22 D = (38.1/981.41)×(0.5)+(943.31/981.41) ×(1) =.0388×(0.5)+.9612×(1)=.9806 years Macauley’s Duration PV 1 = PV 0.5 =38.1 PV= CF 0.5 = 40 CF 1 = 1040

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-23 Features of the Duration Measure Duration and Coupon Interest –the higher the coupon payment, the lower its duration Duration and Maturity –The longer the maturity, the higher the duration Duration and Yield to Maturity –The higher the yield to maturity, the lower the duration Duration and Coupon Interest –the higher the coupon payment, the lower its duration Duration and Maturity –The longer the maturity, the higher the duration Duration and Yield to Maturity –The higher the yield to maturity, the lower the duration

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-24 Example of Duration Calculation (10% Semiannual Coupon & 8% YTM) 1 CF t CF t x t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity 1 CF t CF t x t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity , , , (48.08/ ) = (46.23/1,067.34) = (44.45/1,067.34) = (42.74/1,067.34) = (41.10/1,067.34) = (39.52/1,067.34) = (38.00/1,067.34) = (767.22/1,067.34) = D = 3, , = 3.42 years

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin CF t CF t × t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity 1 CF t CF t × t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity , …… D = 3, = 3.6 years …… (28.84/932.68)=0.01 …….. 4(752.62/932.68)= …….. 3, ,356.5 Base case: D = 3.42 years Coupon rate changes from 10% to 6%

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin CF t CF t X t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity 1 CF t CF t X t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-maturity , …… D = 3, = 3.39 years …… (47.62/1000)=0.02 …….. 4(710.68/1000)= …….. 2, , Base case: D = 3.42 years YTM change from 8% to 10%

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin CF t CF t X t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-Maturity 1 CF t CF t X t Weighted t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Time-to-Maturity …… D = = 2.67 years …… (48.08/ )=0.02 …….. 4(829.82/ )= …….. 2, , Base case: D = 3.42 years Time to maturity changes from 4 years to 3 years

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-28 Economic Meaning of Duration Measure of a bond’s interest rate sensitivity (elasticity)

Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-29 Errors in Duration Estimation Bond Value Yield For large interest rate increases, duration overestimates the fall in security prices; for large interest rate decreases, duration underestimates the rise in security.