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Chapter 4 Understanding Interest Rates Measuring Interest Rates • Present Value: • A dollar paid to you one year from now is less valuable than a dollar paid to you today • Why? – A dollar deposited today can earn interest and become $1 x (1+i) one year from today. 4-2 © 2013 Pearson Education, Inc. All rights reserved. Discounting the Future L et i = .10 In one year $100 X (1+ 0.10) = $110 In tw o years $110 X (1 + 0.10) = $121 or 100 X (1 + 0.10) 2 In three years $121 X (1 + 0.10) = $133 or 100 X (1 + 0.10) In n years $100 X (1 + i ) 4-3 © 2013 Pearson Education, Inc. All rights reserved. n 3 Simple Present Value P V = to d ay's (p resen t) valu e C F = fu tu re cash flo w (p aym en t) i = th e in terest rate PV = CF (1 + i ) 4-4 © 2013 Pearson Education, Inc. All rights reserved. n Time Line •Cannot directly compare payments scheduled in different points in the time line Year PV 4-5 $100 $100 $100 $100 0 1 2 n 100 100/(1+i) 100/(1+i)2 100/(1+i)n © 2013 Pearson Education, Inc. All rights reserved. Four Types of Credit Market Instruments • • • • 4-6 Simple Loan Fixed Payment Loan Coupon Bond Discount Bond © 2013 Pearson Education, Inc. All rights reserved. Yield to Maturity • The interest rate that equates the present value of cash flow payments received from a debt instrument with its value today 4-7 © 2013 Pearson Education, Inc. All rights reserved. Simple Loan P V = am o u n t b o rro w ed = $ 1 0 0 C F = cash flo w in o n e year = $ 1 1 0 n = n u m b er o f years = 1 $110 $100 = (1 + i ) 1 (1 + i ) $ 1 0 0 = $ 1 1 0 (1 + i ) = $110 $100 i = 0 .1 0 = 1 0 % F o r sim p le lo an s, th e sim p le in terest ra te eq u als th e yield to m atu rity 4-8 © 2013 Pearson Education, Inc. All rights reserved. Fixed Payment Loan T he sam e cash flow paym ent every period throughout the life of the loan L V = loan value FP = fixed yearly paym ent n = num ber of years until m aturity LV = FP 1+ i 4-9 FP (1 + i ) © 2013 Pearson Education, Inc. All rights reserved. 2 FP (1 + i ) 3 ...+ FP (1 + i ) n Coupon Bond U sing the sam e strategy used for the fix ed-paym ent loan: P = price of coupon bond C = yearly coupon paym ent F = face value of the bond n = years to m aturity date P = C 1+ i 4-10 C (1+ i ) 2 © 2013 Pearson Education, Inc. All rights reserved. C (1+ i ) 3 . . . + C (1+ i ) n F (1 + i ) n Table 1 Yields to Maturity on a 10%Coupon-Rate Bond Maturing in Ten Years (Face Value = $1,000) • When the coupon bond is priced at its face value, the yield to maturity equals the coupon rate • The price of a coupon bond and the yield to maturity are negatively related • The yield to maturity is greater than the coupon rate when the bond price is below its face value 4-11 © 2013 Pearson Education, Inc. All rights reserved. Consol or Perpetuity • A bond with no maturity date that does not repay principal but pays fixed coupon payments forever P C / ic Pc price of the consol C yearly interest payment i c yield to maturity of the consol can rewrite above equation as this : ic C / Pc For coupon bonds, this equation gives the current yield, an easy to calculate approximation to the yield to maturity 4-12 © 2013 Pearson Education, Inc. All rights reserved. Discount Bond For an y one year discount bo nd i = F-P P F = Face value of the discoun t bo nd P = current price of the discoun t bo nd T h e yield to m aturity equals the increase in price over the year divided by the initial price. A s w ith a co upon bond , th e yield to m aturity is negativ ely related to the current b ond p rice. 4-13 © 2013 Pearson Education, Inc. All rights reserved. The Distinction Between Interest Rates and Returns • Rate of Return: T h e p ay m en ts to the ow ner plus the change in value ex pressed as a fraction o f the purchase price RET = C Pt + Pt 1 - Pt Pt R E T = retu rn fro m holding the bond from tim e t to tim e t + 1 Pt = price of bond at tim e t Pt 1 = price of the b on d at tim e t + 1 C = coupon p aym ent C Pt Pt 1 - Pt Pt 4-14 © 2013 Pearson Education, Inc. All rights reserved. = current yield = ic = rate of capital gain = g The Distinction Between Interest Rates and Returns (cont’d) • The return equals the yield to maturity only if the holding period equals the time to maturity • A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if time to maturity is longer than the holding period • The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change 4-15 © 2013 Pearson Education, Inc. All rights reserved. The Distinction Between Interest Rates and Returns (cont’d) • The more distant a bond’s maturity, the lower the rate of return the occurs as a result of an increase in the interest rate • Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise 4-16 © 2013 Pearson Education, Inc. All rights reserved. Table 2 One-Year Returns on DifferentMaturity 10%-Coupon-Rate Bonds When Interest Rates Rise from 10% to 20% 4-17 © 2013 Pearson Education, Inc. All rights reserved. Interest-Rate Risk • Prices and returns for long-term bonds are more volatile than those for shorter-term bonds • There is no interest-rate risk for any bond whose time to maturity matches the holding period 4-18 © 2013 Pearson Education, Inc. All rights reserved. The Distinction Between Real and Nominal Interest Rates • Nominal interest rate makes no allowance for inflation • Real interest rate is adjusted for changes in price level so it more accurately reflects the cost of borrowing • Ex ante real interest rate is adjusted for expected changes in the price level • Ex post real interest rate is adjusted for actual changes in the price level 4-19 © 2013 Pearson Education, Inc. All rights reserved. Fisher Equation i ir e i = nom inal interest rate ir = real interest rate e = expected inflation rate W hen the real interest rate is low , there are greater incentives to borrow a nd few er incentives to lend. T he real inter est rate is a better indicator of the in centives to borrow and lend. 4-20 © 2013 Pearson Education, Inc. All rights reserved. Figure 1 Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2011 Sources: Nominal rates from www.federalreserve.gov/releases/H15 and inflation from ftp://ftp.bis.gov/special.requests/cpi/cpia.txt. The real rate is constructed using the procedure outlined in Frederic S. Mishkin, “The Real Interest Rate: An Empirical Investigation,” Carnegie-Rochester Conference Series on Public Policy 15 (1981): 151–200. This procedure involves estimating expected inflation as a function of past interest rates, inflation, and time trends and then subtracting the expected inflation measure from the nominal interest rate. 4-21 © 2013 Pearson Education, Inc. All rights reserved.