©2009, The McGraw-Hill Companies, All Rights Reserved 3-1 McGraw-Hill/Irwin Chapter Three Interest Rates and Security Valuation

©2009, The McGraw-Hill Companies, All Rights Reserved 3-2 McGraw-Hill/Irwin Various Interest Rate Measures Coupon rate –periodic cash flow a bond issuer contractually promises to pay a bond holder Required rate of return (rrr) –rates used by individual market participants to calculate fair present values (PV) Expected rate of return (Err) –rates participants would earn by buying securities at current market prices (P) Realized rate of return (rr) –rates actually earned on investments Coupon rate –periodic cash flow a bond issuer contractually promises to pay a bond holder Required rate of return (rrr) –rates used by individual market participants to calculate fair present values (PV) Expected rate of return (Err) –rates participants would earn by buying securities at current market prices (P) Realized rate of return (rr) –rates actually earned on investments

©2009, The McGraw-Hill Companies, All Rights Reserved 3-3 McGraw-Hill/Irwin Required Rate of Return The fair present value (PV) of a security is determined using the required rate of return (rrr) 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 (rrr) 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

©2009, The McGraw-Hill Companies, All Rights Reserved 3-4 McGraw-Hill/Irwin Expected Rate of Return The current market price (P) of a security is determined using the expected rate of return (Err) 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 (Err) 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

©2009, The McGraw-Hill Companies, All Rights Reserved 3-5 McGraw-Hill/Irwin Realized Rate of Return The realized rate of return (rr) 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)

©2009, The McGraw-Hill Companies, All Rights Reserved 3-6 McGraw-Hill/Irwin EXAMPLE A bond you purchased 2 years ago for $890 is now selling for $925. Coupon payment is $100 per year. You intend to hold the bond for 4 more years and project that you will be able to sell it at the end of year 4 for $960. You also project that the bond will continue paying $100 in interest per year. Given the risk associated with the bond required rate of return is 11.25%. What is its Fair Value? Expected rate of return? Realized rate of return?

©2009, The McGraw-Hill Companies, All Rights Reserved 3-7 McGraw-Hill/Irwin Bond Valuation The present value of a bond (V b ) can be written as: M = the par value of the bond INT = the annual interest (or coupon) payment T = the number of years until the bond matures i = the annual interest rate (often called yield to maturity (ytm)) The present value of a bond (V b ) can be written as: M = the par value of the bond INT = the annual interest (or coupon) payment T = the number of years until the bond matures i = the annual interest rate (often called yield to maturity (ytm))

©2009, The McGraw-Hill Companies, All Rights Reserved 3-8 McGraw-Hill/Irwin EXAMPLE A bond with a face value of $1000 pays 10%coupon interest rate. Coupon payments are made semiannually. The bond matures in 12 years. If the required rate of return (rrr) is 8%, what is the market value of the bond? What if rrr is 10% What if rrr is 12%

©2009, The McGraw-Hill Companies, All Rights Reserved 3-9 McGraw-Hill/Irwin Bond Valuation A premium bond has a coupon rate (INT) greater then the required rate of return (rrr) and the fair present value of the bond (V b ) is greater than the face value (M) Discount bond: if INT < rrr, then V b < M Par bond: if INT = rrr, then V b = M A premium bond has a coupon rate (INT) greater then the required rate of return (rrr) and the fair present value of the bond (V b ) is greater than the face value (M) Discount bond: if INT < rrr, then V b < M Par bond: if INT = rrr, then V b = M

©2009, The McGraw-Hill Companies, All Rights Reserved 3-10 McGraw-Hill/Irwin Yield to Maturity It is the yield (return) the bondholder will earn on the bond if he or she buys it at its current market price, receives all coupon payments and holds the bond until maturity.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-11 McGraw-Hill/Irwin EXAMPLE You purchase bond with $1000 face value that pay 11% coupon interest per year. Coupons are paid semiannually. The bond matures in 15 years. If the current market prce of the bond is $931.76

©2009, The McGraw-Hill Companies, All Rights Reserved 3-12 McGraw-Hill/Irwin Equity Valuation Finding the PVs of infinite CFs discounted at an appropriate interest rate. Even if the equity holder does not want to hold it forever, he or she can sell it to someone. D t = Dividend paid out at he end of year t P t = Price of the common stock at he end of year t P 0 = Current price İ s = Interest rate used to discount the cash flows

©2009, The McGraw-Hill Companies, All Rights Reserved 3-13 McGraw-Hill/Irwin EXAMPLE Suppose you owned a stock for 2 years. You originally bought the stock two years ago for $15 and just sold it for $35. The stock paid an annual dividend of $1 on the last day of each of the past two years. What is the raealized rate of return (rr) on the stock?

©2009, The McGraw-Hill Companies, All Rights Reserved 3-14 McGraw-Hill/Irwin EXAMPLE You are considering to purchase the stock that you expect to own for the next 3 years. The current market price is $32 and you expect to sell it for $45 in three years’ time. You also expect the stock to pay an annual dividend of $1.50 on the last day of each of the next 3 years. Calculate the expected rate of return (Err)?

©2009, The McGraw-Hill Companies, All Rights Reserved 3-15 McGraw-Hill/Irwin Equity Valuation Zero-growth dividends: 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 i s = the interest rate used to discount future cash flows Zero-growth dividends: 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 i s = the interest rate used to discount future cash flows

©2009, The McGraw-Hill Companies, All Rights Reserved 3-16 McGraw-Hill/Irwin Equity Valuation The return on a stock with zero dividend growth, if purchased at price P 0, can be written as: If the fair market price is applied to this formula, the return we solve for is required rate of return If the current market price is applied, the return is expected rate of return. In effcient market rrr=Err The return on a stock with zero dividend growth, if purchased at price P 0, can be written as: If the fair market price is applied to this formula, the return we solve for is required rate of return If the current market price is applied, the return is expected rate of return. In effcient market rrr=Err

©2009, The McGraw-Hill Companies, All Rights Reserved 3-17 McGraw-Hill/Irwin EXAMPLE Suppose the company pays a onstant dividend of $5 per year. What is the current market price of the stock if the expected rate of return is 12%

©2009, The McGraw-Hill Companies, All Rights Reserved 3-18 McGraw-Hill/Irwin Equity Valuation Constant growth dividends: 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 Constant growth dividends: 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

©2009, The McGraw-Hill Companies, All Rights Reserved 3-19 McGraw-Hill/Irwin Equity Valuation The return on a stock with constant dividend growth, if purchased at price P 0, can be written as:

©2009, The McGraw-Hill Companies, All Rights Reserved 3-20 McGraw-Hill/Irwin EXAMPLE You are evaluating a stock paid a dividend at the end of last year of $3.5. Dividend has grown at a constant rate of 2% per year over the last 20 years, this constant growth is expected to continue. The required rate of return is 10%. Calculate the fair value of the stock

©2009, The McGraw-Hill Companies, All Rights Reserved 3-21 McGraw-Hill/Irwin EXAMPLE A stock you are evaluating paid a dividend at the nd of the last year of $4.80. Dividends have grown at a constant rate of 1.75 over the last 15 years. And this growth rate is expected to continue into the future. The stock is currently selling for $52 per share. What is the expected rate of return?

©2009, The McGraw-Hill Companies, All Rights Reserved 3-22 McGraw-Hill/Irwin Equity Valuation Supernormal or Non-constant Growth in Dividends: Firms ofteh experience periods of supernormal or non-constnt dividend growth. Dividends during the periods of supernormal must be evaluated individually.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-23 McGraw-Hill/Irwin Equity Valuation Step 1: Find the PV of dividends during the supernormal period. Step 2: Find the price of the stock at the end of the supernormal growth using constant growth model. Step 3: Add the two components of the stock price together.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-24 McGraw-Hill/Irwin EXAMPLE A stock is expected to experience supernormal growth in dividends of 10% over the next 5 years. Following this period, dividends are expected to grow at a constant rate of 4%. The stock is paid a dividend of $4 last year and the required rate of return is 15%. Calculate the fair value of the stock.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-25 McGraw-Hill/Irwin Relation between Interest Rates and Bond Values Interest Rate Bond Value Interest Rate Bond Value 12% 10% 8% ,0001,152.47

©2009, The McGraw-Hill Companies, All Rights Reserved 3-26 McGraw-Hill/Irwin Impact of Maturity on 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

©2009, The McGraw-Hill Companies, All Rights Reserved 3-27 McGraw-Hill/Irwin Impact of Coupon Rates on Interest Rate Sensitivity Bond Value Bond Value Interest Rate Low-Coupon Bond High-Coupon Bond

©2009, The McGraw-Hill Companies, All Rights Reserved 3-28 McGraw-Hill/Irwin 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 (bond) price to small interest rate changes 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 (bond) price to small interest rate changes

©2009, The McGraw-Hill Companies, All Rights Reserved 3-29 McGraw-Hill/Irwin Duration Macauley’s Duration For investors and financial managers duration is a tool that can be used to estimate the change in the value of a portfolio of securities or even firm value for a given change in interest rates.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-30 McGraw-Hill/Irwin EXAMPLE Consider a bond 1 year remaining to maturity, $1000 face value, 8% coupon rate (paid semiannually) and an interest rate of 10%. Calculate duration?

©2009, The McGraw-Hill Companies, All Rights Reserved 3-31 McGraw-Hill/Irwin Duration Duration (D) for a fixed-income security that pays interest annually can be written as: 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 (D) for a fixed-income security that pays interest annually can be written as: 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

©2009, The McGraw-Hill Companies, All Rights Reserved 3-32 McGraw-Hill/Irwin Duration Duration (D) (measured in years) for a fixed- income security in general can be written as: m = the number of times per year interest is paid Duration (D) (measured in years) for a fixed- income security in general can be written as: m = the number of times per year interest is paid

©2009, The McGraw-Hill Companies, All Rights Reserved 3-33 McGraw-Hill/Irwin EXAMPLE Suppose that you have a bond that offers a coupon rate of 10% paid semiannually. Face value is $1000, it matures in 4 years, its current yield to maturity is 8%. Calculate duration.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-34 McGraw-Hill/Irwin Duration The Duration of Zero-coupon Bond: Zero- coupon bonds sell at discount from face value on issue and pay their face value on maturity. P = 1000/(1+R/2) T

©2009, The McGraw-Hill Companies, All Rights Reserved 3-35 McGraw-Hill/Irwin EXAMPLE Suppose that you have a zero-coupon bond with a face value of $1000, a maturity of 4 years, current yield to maturity of 8% compounded semiannually.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-36 McGraw-Hill/Irwin 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 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 at a decreasing rate

©2009, The McGraw-Hill Companies, All Rights Reserved 3-37 McGraw-Hill/Irwin Duration and Modified Duration Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price is found by rearranging the duration formula: MD = modified duration = D/(1 + R) Given an interest rate change, the estimated percentage change in a (annual coupon paying) bond’s price is found by rearranging the duration formula: MD = modified duration = D/(1 + R)

©2009, The McGraw-Hill Companies, All Rights Reserved 3-38 McGraw-Hill/Irwin EXAMPLE Four-year bond 10% coupon paid semiannually. YTM is 8%. Duration (D)= 3.42 years. At 8%, Bond Value =$ Suppose that YTM increase by 10 basis points (1/10 of 1%) from 8% to 8.10%. Bond Value =$ Calculate the sensitivity of bond price to changes in interest rate.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-39 McGraw-Hill/Irwin EXAMPLE Four-year bond 6% coupon paid semiannually. YTM is 8%. Duration (D)= 3.6 years. At 8%, Bond Value =$ Suppose that YTM increase by 10 basis points (1/10 of 1%) from 8% to 8.10%. Bond Value =$ Calculate the sensitivity of bond price to changes in interest rate.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-40 McGraw-Hill/Irwin Duration The higher the coupon rate, the shorter the duration and the smaller the % decrease in bond’s price.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-41 McGraw-Hill/Irwin Duration Duration accurately measures the price sensitivity of financial securities only for small changes in interest rates. For large interest rate increases, duration overpredicts the fall in price. For large interest rate decreases, duration underpredicts the increase in price. Bond price and interest rate relationship exibiting a property called convexity.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-42 McGraw-Hill/Irwin Figure 3-7

©2009, The McGraw-Hill Companies, All Rights Reserved 3-43 McGraw-Hill/Irwin 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:

©2009, The McGraw-Hill Companies, All Rights Reserved 3-44 McGraw-Hill/Irwin EXAMPLE 4 year bond, $1000 face value with 10% coupon paid semiannually and a 8% yield. D= 3.42 years Current price = $ at 8% yield. What will be the bond value if yield increase from 8% to 10% according to the duration model? What will be the exact change in price after yield increase to 10%

©2009, The McGraw-Hill Companies, All Rights Reserved 3-45 McGraw-Hill/Irwin EXAMPLE What will be the bond value if yield decrease from 8% to 6% according to the duration model? What will be the exact change in price after yield decrease to 6%

©2009, The McGraw-Hill Companies, All Rights Reserved 3-46 McGraw-Hill/Irwin Convexity Imp. Question for investors and FI is whether the error in the duration equation is big enough to be concerned about. It depends on the size of interest rate change and size of their portfolios. High convexity means that for equally large changes of int. rates, (up and down 2%), the capital gain effect of a rate decrease is more than the capital loss effect of a rate increase.

©2009, The McGraw-Hill Companies, All Rights Reserved 3-47 McGraw-Hill/Irwin 1) Convexity is desirable. 2) Convexity diminishes the error in duration. 3) All-fixed income securities are convex.