Copyright © 2012 by The McGraw-Hill Companies, Inc. All rights reserved. McGraw-Hill/Irwin Chapter Three Interest Rates and Security Valuation

3-2 Various Interest Rate Measures Coupon rate periodic cash flow a bond issuer contractually promises to pay a bond holder Required rate of return (r) rates used by individual market participants to calculate fair present values (PV) Expected rate of return or E(r) rates participants would earn by buying securities at current market prices (P) Realized rate of return ( r ) rate actually earned on investments Coupon rate periodic cash flow a bond issuer contractually promises to pay a bond holder Required rate of return (r) rates used by individual market participants to calculate fair present values (PV) Expected rate of return or E(r) rates participants would earn by buying securities at current market prices (P) Realized rate of return ( r ) rate actually earned on investments

3-3 Required Rate of Return The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate CF 1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon The fair present value (PV) of a security is determined using the required rate of return (r) as the discount rate CF 1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon

3-4 Expected Rate of Return The current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate CF 1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon The current market price (P) of a security is determined using the expected rate of return or E(r) as the discount rate CF 1 = cash flow in period t (t = 1, …, n) ~ = indicates the projected cash flow is uncertain n = number of periods in the investment horizon

3-5 Realized Rate of Return The realized rate of return ( r ) is the discount rate that just equates the actual purchase price ( ) to the present value of the realized cash flows (RCF t ) t (t = 1, …, n)

3-6 Bond Valuation The present value of a bond (V b ) can be written as: Par = the par or face value of the bond, usually $1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm)) The present value of a bond (V b ) can be written as: Par = the par or face value of the bond, usually $1,000 INT = the annual interest (or coupon) payment T = the number of years until the bond matures r = the annual interest rate (often called yield to maturity (ytm))

3-7 Bond Valuation A premium bond has a coupon rate (INT) greater than the required rate of return (r) and the fair present value of the bond (V b ) is greater than the face or par value (Par) Premium bond: If INT > r; then V b > Par Discount bond: if INT < r, then V b < Par Par bond: if INT = r, then V b = Par A premium bond has a coupon rate (INT) greater than the required rate of return (r) and the fair present value of the bond (V b ) is greater than the face or par value (Par) Premium bond: If INT > r; then V b > Par Discount bond: if INT < r, then V b < Par Par bond: if INT = r, then V b = Par

3-8 Equity Valuation The present value of a stock (P t ) assuming zero growth in dividends can be written as: D = dividend paid at end of every year P t = the stock’s price at the end of year t r s = the interest rate used to discount future cash flows The present value of a stock (P t ) assuming zero growth in dividends can be written as: D = dividend paid at end of every year P t = the stock’s price at the end of year t r s = the interest rate used to discount future cash flows

3-9 Equity Valuation The present value of a stock (P t ) assuming constant growth in dividends can be written as: D 0 = current value of dividends D t = value of dividends at time t = 1, 2, …, ∞ g = the constant dividend growth rate The present value of a stock (P t ) assuming constant growth in dividends can be written as: D 0 = current value of dividends D t = value of dividends at time t = 1, 2, …, ∞ g = the constant dividend growth rate

3-10 Equity Valuation The return on a stock with zero dividend growth, if purchased at current price P 0, can be written as: The return on a stock with constant dividend growth, if purchased at price P 0, can be written as: The return on a stock with zero dividend growth, if purchased at current price P 0, can be written as: The return on a stock with constant dividend growth, if purchased at price P 0, can be written as:

3-11 Relation between Interest Rates and Bond Values Interest Rate Bond Value Interest Rate Bond Value 12% 10% 8% ,0001,152.47

3-12 Impact of Maturity on Price Volatility (a) 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

3-13 Impact of Maturity on Price Volatility (b)

3-14 Impact of Coupon Rates on Price Volatility Bond Value Bond Value Interest Rate Low-Coupon Bond High-Coupon Bond

3-15 Impact of Coupon on Price Volatility (b)

3-16 Impact of r on Price Volatility Bond Price Interest Rate How does volatility change with interest rates? Price volatility isinversely relatedto the level of theinitial interestrate r

3-17 Duration Duration is the weighted-average time to maturity (measured in years) on a financial security Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes Duration captures the coupon and maturity effects on volatility. Duration is the weighted-average time to maturity (measured in years) on a financial security Duration measures the sensitivity (or elasticity) of a fixed-income security’s price to small interest rate changes Duration captures the coupon and maturity effects on volatility.

3-18 Duration Duration (Dur) for a fixed-income security that pays interest annually can be written as: P 0 = Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CF t = cash flow received at end of period t r = yield to maturity or required rate of return PV t = present value of cash flow received at end of period t Duration (Dur) for a fixed-income security that pays interest annually can be written as: P 0 = Current price of the security t = 1 to T, the period in which a cash flow is received T = the number of years to maturity CF t = cash flow received at end of period t r = yield to maturity or required rate of return PV t = present value of cash flow received at end of period t

3-19 9% Coupon, 4 year maturity annual payment bond with a 8% ytm Duration and Volatility Duration = years

3-20 Duration Duration (Dur) (measured in years) for a fixed- income security, in general, can be written as: m = the number of times per year interest is paid, the sum term is incremented in m units Duration (Dur) (measured in years) for a fixed- income security, in general, can be written as: m = the number of times per year interest is paid, the sum term is incremented in m units

3-21 Closed form duration equation: P 0 = Price INT= Periodic cash flow in dollars, normally the semiannual coupon on a bond or the periodic monthly payment on a loan. r = periodic interest rate = APR / m, where m = # compounding periods per year N = Number of compounding or payment periods (or the number of years * m) Dur = Duration = # Compounding or payment periods; Duration period is what you actually get from the formula

3-22 Duration Duration and coupon interest the higher the coupon payment, the lower the bond’s duration Duration and yield to maturity the higher the yield to maturity, the lower the bond’s duration Duration and maturity duration increases with maturity but at a decreasing rate Duration and coupon interest the higher the coupon payment, the lower the bond’s duration Duration and yield to maturity the higher the yield to maturity, the lower the bond’s duration Duration and maturity duration increases with maturity but at a decreasing rate

3-23 Duration and Modified Duration Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price given by

3-24 Duration and Modified Duration Modified duration (Dur Mod ) can be used to predict price changes for non-annual payment loans or securities: It is found as: where r period = APR/m Using modified duration to predict price changes: Modified duration (Dur Mod ) can be used to predict price changes for non-annual payment loans or securities: It is found as: where r period = APR/m Using modified duration to predict price changes:

3-25 Duration Based Prediction Errors

3-26 Convexity Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R Convexity incorporates the curvature of the price- yield curve into the estimated percentage price change of a bond given an interest rate change: Convexity (CX) measures the change in slope of the price-yield curve around interest rate level R Convexity incorporates the curvature of the price- yield curve into the estimated percentage price change of a bond given an interest rate change:

3-27 Practice Problem Using Modified Duration Predicted Price Change Using Modified Duration