Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-1 Chapter Three Interest Rates and Security Valuation

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-2 Various Interest Rate Measures Coupon rate: interest rate on a bond used to calculate the annual cash flows the issuer promises to pay to bond holder Required Rate of Return: interest rate an investor should receive on a security given it’s risk (used to calculate the fair present value on a security) Expected rate of return: interest rate an investor would receive on a security if the security is bought at it’s current market price, receives all expected payments and sells at the end of the investment horizon Realized Rate of Return: actual interest rate earned on an investment (ex post measure of the interest rate) Coupon rate: interest rate on a bond used to calculate the annual cash flows the issuer promises to pay to bond holder Required Rate of Return: interest rate an investor should receive on a security given it’s risk (used to calculate the fair present value on a security) Expected rate of return: interest rate an investor would receive on a security if the security is bought at it’s current market price, receives all expected payments and sells at the end of the investment horizon Realized Rate of Return: actual interest rate earned on an investment (ex post measure of the interest rate)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-3 Required Rate of Return ~ ~ ~ ~ FPV = CF 1 + CF 2 + CF 3 + … + CF n (1 + rrr) 1 (1 + rrr) 2 (1 + rrr) 3 (1 + rrr) n Where: rrr = Required rate of return CF 1 = Cash flow projected in period t (t = 1, …, n) ~ = Indicates that projected cash flow is uncertain (due to default and other risks) n = Number of periods in the investment horizon ~ ~ ~ ~ FPV = CF 1 + CF 2 + CF 3 + … + CF n (1 + rrr) 1 (1 + rrr) 2 (1 + rrr) 3 (1 + rrr) n Where: rrr = Required rate of return CF 1 = Cash flow projected in period t (t = 1, …, n) ~ = Indicates that projected cash flow is uncertain (due to default and other risks) n = Number of periods in the investment horizon

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-4 Expected Rate of Return ~ ~ ~ ~ P = CF 1 + CF 2 + CF 3 + … + CF n (1 + Err) 1 (1 + Err) 2 (1 + Err) 3 (1 + Err) n Where: Err = Expected rate of return CF 1 = Cash flow projected in period t (t = 1, …, n) ~ = Indicates that projected cash flow is uncertain (due to default and other risks) n = Number of periods in the investment horizon ~ ~ ~ ~ P = CF 1 + CF 2 + CF 3 + … + CF n (1 + Err) 1 (1 + Err) 2 (1 + Err) 3 (1 + Err) n Where: Err = Expected rate of return CF 1 = Cash flow projected in period t (t = 1, …, n) ~ = Indicates that projected cash flow is uncertain (due to default and other risks) n = Number of periods in the investment horizon

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-5 Realized Rate of Return The actual interest rate earned on an investment in a financial security P = RCF 1 + RCF 2 + … + RCF n (1 + rr) 1 (1 + rr) 2 (1 + rr) n Where: RCF = Realized cash flow in period t (t = 1, …, n) rr = Realized rate of return on a security The actual interest rate earned on an investment in a financial security P = RCF 1 + RCF 2 + … + RCF n (1 + rr) 1 (1 + rr) 2 (1 + rr) n Where: RCF = Realized cash flow in period t (t = 1, …, n) rr = Realized rate of return on a security

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-6 Bond Valuation The valuation of a bond instrument employs time value of money concepts –Reflects present value of all cash flows promised or projected, discounted at the required rate of return (rrr) –Expected rate of return (Err) is the interest rate that equates the current market price to the present value of all promised cash flows received over the life of the bond –Realized rate of return (rr) on a bond is the actual return earned on a bond investment that has already taken place The valuation of a bond instrument employs time value of money concepts –Reflects present value of all cash flows promised or projected, discounted at the required rate of return (rrr) –Expected rate of return (Err) is the interest rate that equates the current market price to the present value of all promised cash flows received over the life of the bond –Realized rate of return (rr) on a bond is the actual return earned on a bond investment that has already taken place

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-7 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; equals the par value of the bond times the (percentage) coupon rate N = Number years until the bond matures m = Number of times per year interest is paid i d = Interest rate used to discount cash flows on the bond

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-8 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 m = 2 (semiannual) i d = 8% (rrr) 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 m = 2 (semiannual) i d = 8% (rrr)

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-9 Description of a Premium, Discount, and Par Bond Premium bond— when the coupon rate, INT, is greater then 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 Par bond— when INT=rrr, then V b =M Premium bond— when the coupon rate, INT, is greater then 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 Par bond— when INT=rrr, then V b =M

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-10 Yield to Maturity The return or yield the bond holder will earn on the bond if he or she buys it at its current market price, receives all coupon and principal payments as promised, and holds the bond until maturity V b = INT (PVIFA ytm/m, Nm ) + M(PVIF ytm/m,Nm ) m The return or yield the bond holder will earn on the bond if he or she buys it at its current market price, receives all coupon and principal payments as promised, and holds the bond until maturity V b = INT (PVIFA ytm/m, Nm ) + M(PVIF ytm/m,Nm ) m

Copyright © 2004 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 © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-12 Impact of Interest Rate Changes on Security Values Interest Rate Bond Value Interest Rate Bond Value 12% 10% 8% ,0001,524.47

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-13 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 © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-14 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 © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-15 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 © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-16 Impact of a Bond’s Coupon Rate on Its Interest Rate Sensitivity Interest Rate Interest Rate Bond Value Low-Coupon Bond High-Coupon Bond

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-17 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 © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-18 Example of Duration Calculation 1 CF t CF t  _ t Percent of Initial t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Investment Recovered Totals 1, , CF t CF t  _ t Percent of Initial t CF t (1 + 4%) 2t (1 + 4%) 2t (1 + 4%) 2t Investment Recovered Totals 1, , , , /1, = /1, = /1, = /1, = /1, = /1, = /1, = ,068.88/1, = 2.88 D = 3, , = 3.42 years

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-19 Features of the Duration Measure Duration and Coupon Interest –the higher the coupon payment, the lower is a bond’s duration Duration and Yield to Maturity –duration increases as yield to maturity increases Duration and Maturity –Duration increases with the maturity of a bond but at a decreasing rate Duration and Coupon Interest –the higher the coupon payment, the lower is a bond’s duration Duration and Yield to Maturity –duration increases as yield to maturity increases Duration and Maturity –Duration increases with the maturity of a bond but at a decreasing rate

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-20 Discrepancy Between Maturity and Duration on a Coupon Bond

Copyright © 2004 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill /Irwin 3-21 Economic Meaning of Duration Measure of the average life of a bond Measure of a bond’s interest rate sensitivity (elasticity) Measure of the average life of a bond Measure of a bond’s interest rate sensitivity (elasticity)