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## Chapter Three

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**Chapter Three**Interest Rates and Security Valuation McGraw-Hill/Irwin**Various Interest Rate Measures**• Coupon rate • Required Rate of Return • Expected rate of return • Realized Rate of Return McGraw-Hill/Irwin**Required Rate of Return**~ ~ ~ ~ FPV = CF1 + CF2 + CF3 + … + CFn (1 + rrr)1 (1 + rrr)2 (1 + rrr)3 (1 + rrr)n Where: rrr = Required rate of return CF1 = 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 McGraw-Hill/Irwin**Expected Rate of Return**~ ~ ~ ~ P = CF1 + CF2 + CF3 + … + CFn (1 + Err)1 (1 + Err)2 (1 + Err)3 (1 + Err)n Where: Err = Expected rate of return CF1 = 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 McGraw-Hill/Irwin**Realized Rate of Return**The actual interest rate earned on an investment in a financial security P = RCF1 + RCF2 + … + RCFn (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 McGraw-Hill/Irwin**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 McGraw-Hill/Irwin**Bond Valuation Formula**Vb = INT/2 + INT/2 + . . . + INT/2 __ (1 + id/2)1 (1 + id/2)2 (1 + id/2)2N + M_ _ _ (1 + id/2)2N Where: Vb = 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 id = Interest rate used to discount cash flows on the bond McGraw-Hill/Irwin**Bond Valuation Example**Vb = 1,000(.1) (PVIFA8%/2, 12(2)) + 1,000(PVIF8%/2, 12(2)) 2 Where: Vb = $1,152.47 (solution) M = $1,000 INT = $100 per year (10% of $1,000) N = 12 years id = 8% (rrr) PVIF = Present value interest factor of a lump sum payment PVIFA = present value interest factor of an annuity stream McGraw-Hill/Irwin**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 (Vb) is greater than its face value (M) • Discount bond—when INT<rrr, then Vb <M • Par bond—when INT=rrr, then Vb =M McGraw-Hill/Irwin**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 Vb = INT (PVIFAytm/m, Nm) + M(PVIFytm/m,Nm) m McGraw-Hill/Irwin**Summary of Factors that Affect Security Prices and Price**Volatility when Interest Rates Change • Interest Rate • Time Remaining to Maturity • Coupon Rate McGraw-Hill/Irwin**Impact of Interest Rate Changes on Security Values**Interest Rate Bond Value 12% 10% 8% 874.50 1,000 1,152.47 McGraw-Hill/Irwin**Balance sheet of an FI before and after an Interest Rate**Increase (a) Balance Sheet before the Interest Rate Increase Assets Liabilities and Equity Bond (8% required rate of return) Bond (10% required rate of return) $1,152.47 $1,000 $152.47 Equity (b) Balance Sheet after 2% increase in the Interest Rate Increase Assets Liabilities and Equity Bond (10% required rate of return) Bond (12% required rate of return) $874.50 $1,000 Equity $125.50 McGraw-Hill/Irwin**Impact of Maturity on Security Values**12 Years to Maturity 16 Years to Maturity Required Rate of Return Percentage Price Change Percentage Price Change Fair Price* Price Change Price Change Fair Price* 8% $1,152.47 $1,178.74 -$178.74 -15.16% -$152.47 -13.23% $1,000.00 10% $1,000.00 -$140.84 -14.08% -$125.50 -12.55% 12% $874.50 $859.16 *The bond pays 10% coupon interest compounded semiannually and has a face value of $1,000 McGraw-Hill/Irwin**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 Time to Maturity McGraw-Hill/Irwin**Impact of a Bond’s Coupon Rate on Its Interest Rate**Sensitivity Interest Rate High-Coupon Bond Low-Coupon Bond Bond Value McGraw-Hill/Irwin**Duration: A Measure of Interest Rate Sensitivity**The weighted-average time to maturity on an investment N N CFt tPVt t t = 1(1 + R)tt = 1 D = N = N CFt PVt t = 1 (1 + R)t t = 1 McGraw-Hill/Irwin**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 McGraw-Hill/Irwin**Discrepancy Between Maturity and Duration on a Coupon Bond**McGraw-Hill/Irwin**Economic Meaning of Duration**• Measure of the average life of a bond • Measure of a bond’s interest rate sensitivity (elasticity) McGraw-Hill/Irwin