2 Present Value: Discounting the Future Present Value (PV) - the value today of a payment that is promised to be made in the future
3 Discounting the Future First need to Understand Future Value (FV) Let i = 0.10 (10 percent)In one year: $100 +(.10 x $100) = $100 x (1+.10) = $110In two years: $110+(.10 x $110) = $110 x (1+.10) = $121OR $100 x (1+.10)2 = $121In three years: $121+(.10 x $121)= $121x(1+.10) = $133OR $100 x (1+.10)3 = $133In n years: FV = $100 x (1+ .10)n
4 Future ValueFuture value in n years of an investment of PV today at interest rate i(i is measured as a decimal, 10% = .10)FVn = PV x (1+i)nCalculate one plus the interest rate (measured as a decimal) raised to the nth power and multiply it by the amount invested (present value).
5 Future ValueNote:When computing future value, both n and i must be measured in same time units - if i is annual, then n must be in years.The future value of $100 in 18 months at 5% annual interest rate is:FV = 100 *(1+.05)1.5
6 Future ValueThe future value of $100 in one month at a 5% annual interest rate is:FV = $100 *(1+.05)1/12 = $(1+.05)1/12 converts the annual interest rates to a monthly rate.(1+.05)1/12 = , which converted to percentage is % or 0.41%(rounded)Note: 0.05/12 =
7 Basis Point Faction of a percentage point is called basis point. A basis point is one-one hundredth of a percentage pointOne basis point (bp) = 0.01 percent.On the previous slide: 0.41% is 41 basis points.
8 Discounting the Future Present Value (PV) Reverses the FV Calculation FVn = PV x (1+i)nPV = FVn /(1+i)n
10 Present Value Example 1: Present Value of $100 received in 5 years discounted at an interest rate of 8%.PV = $100 / (1.08)5 = $68.05Example2:PV of $20,000 received 20 years from now discounted at 8% is:PV = $20,000 / (1+0.08)20 = $20,000/ = $4,291Discounted at 9%:PV = $20,000 / (1+0.09)20 = $20,000/ = $3,568
11 Present Value Example3: PV of $20,000 received 19 years from now discounted at 8% is:PV = $20,000 / (1+0.08)19 = $20,000/ = $4,634In general, present value is higher:1. The higher the future value of the payment (CF).2. The shorter the time period until payment (n).3. The lower the interest rate. (i)
12 Mishkin Discusses Four Types of Credit Market Instruments Simple LoanFixed Payment Loan (I will just mention this)Coupon Bond- Special case: consol bondDiscount Bond
13 We will focus on Bonds and look at three types of bonds Coupon Bonds: which make periodic interest payments and repay the principal at maturity.U.S. Treasury Bonds and most corporate bonds are coupon bonds.Discount or Zero-coupon bonds: which promise a single future payment, such as a U.S. Treasury Bill.Consols: which make periodic interest payments forever, never repaying the principal that was borrowed. (There aren’t many examples of these.)
14 Yield to Maturity -YTMThe interest rate that equates the present value (PV) of cash flow payments (CF) received from a debt instrument with its value todayGiven values for PV, CF and n, solve for i.
17 Bonds: Our Objectives Bond price is a present value calculation. YTM is the interest rate.Supply and Demand determine the price of bonds.- We will also discuss loanable funds andmoney supply/money demand.Why bonds are risky.
19 Present Value of Coupon Payments Coupon Bond PricePresent Value of Coupon PaymentsPresent Value of Principal PaymentPresent Value of Coupon Bond (PB) =Present value of Yearly Coupon Payments (C)+ Present Value of the Face Value (FV),where: i = interest rate and n = time to maturity
20 Example: Price of a n-year Coupon Bond Coupon Payment =$100, Face value = $1,000,and n = time to maturityGiven values for i and n, we can determine the bond price PBDefinition: Coupon Rate = Coupon Payment / Face Value
21 Price of a 10-year Coupon Bond If n = 10, i = 0.10, C = $100 and FV = $1000.What’s the Coupon Rate?
22 Price of a 10-year Coupon Bond If n = 10, i = 0.12, C = $100 and FV = $1000.What’s the Coupon Rate?
23 Coupon Bond - YTM Suppose n = 10, PB = $950, C = $100 and FV = $1000. What’s the Coupon Rate?What’s the YTM?YTM = or 10.85%
25 Using the Approximation Formula Previous example:n = 10, PB = $950, C = $100 and FV = $1000.
26 For our 10-year bond selling at $950: Coupon rate = 10%YTM = 10.85%Current Yield = 10.52%
27 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 relatedThe yield to maturity is greater than the coupon rate when the bond price is below its face value
28 Consol – Special Case Coupon Bond Infinite maturityNo face value.Fixed coupon payment of C forever.Pconsol = C/(1+i) + C/(1+i)2 + C/(1+i)3 + … + C/(1+i)tAs t goes to infinity this collapses to:PConsol = C / i => i = C / P$2,000 = $100/.05
29 Discount or Zero Coupon Bond Definition: A discount bond is sold at some price P, and pays a larger amount (FV) after t years. There is no periodic interest payment.Let P = price of the bond, i= interest rate, n = years to maturity, and FV = Face Value (the value at maturity):
30 Zero Coupon Bonds - Price Examples: Assume i=4%Price of a One-Year Treasury Bill with FV = $1,000:Price of a Six-Month Treasury Bill with FV = $1,000:Price of a 20-Year zero coupon bond at 8% and FV = $20,000:
32 Zero Coupon Bonds - YTM i = 1.0657 -1=> i =6.57% For a discount bond with FV = $15,000 and P = $4,200, and n = 20, the interest rate (or yield to maturity) would be:i = => i =6.57%Note: This is the formula for compound annual rate of growth
33 Zero Coupon Bonds - YTM i = 1.06368 – 1 = .06368 or 6.368% For a discount bond with FV = $10,000 and P = $6,491, and n = 7, the interest rate (or yield to maturity) would be:i = – 1 = or 6.368%
34 From a Coupon Bond to Zero Coupon Bonds (called Strips) Create n+1 discount bonds
36 Current YieldTwo Characteristics of Current YieldIs a better approximation of yield to maturity, nearer price is to par (face value) and longer is maturity of bond2. Change in current yield always signals change in same direction as yield to maturity
45 Key Conclusions From Table 2 The return equals the yield to maturity (YTM) only if the holding period equals the time to maturityA rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the holding period is less than the time to maturityThe more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest-rate change
46 Key Conclusions From Table 2 The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rateEven if a bond has a substantial initial interest rate, its return can be negative if interest rates rise
47 Interest-Rate Risk Change in bond price due to change in interest rate Prices and returns for long-term bonds are more volatile than those for shorter-term bondsThere is no interest-rate risk for a bond whose time to maturity matches the holding period
48 Reinvestment (interest rate) Risk If investor’s holding period exceeds the term to maturity proceeds from sale of bond are reinvested at newinterest rate the investor is exposed to reinvestment riskThe investor benefits from rising interest rates, and suffers from falling interest rates
49 Real and Nominal Interest Rates Nominal interest rate (i) makes no allowance for inflationReal interest rate (r) is adjusted for changes in price level so it more accurately reflects the cost of borrowingEx ante real interest rate is adjusted for expected changes in the price level (πe)Ex post real interest rate is adjusted for actual changes in the price level (π)
50 Real and Nominal Interest Rates Fisher Equation:i = r + πeFrom this we get -rex ante = i - πerex post = i - π
53 Real and Nominal Interest Rates Real Interest Rate:Interest rate that is adjusted for expected changes in the price levelr = i - πe1. Real interest rate more accurately reflects true cost of borrowingWhen real rate is low, greater incentives to borrow and less to lend.if i = 5% and πe = 3% then:r = 5% - 3% = 2%if i = 8% and πe = 10% thenr = 8% - 10% = -2%