Time Value of Money Future Value Present Value Annuities Fixed-Rate Mortgages
Future Value • Single Period • Multiple compounding periods within a year • Longer Horizons • Examples
Future Value Example: Deposit $10,000 in bank Interest rate = 6% How much money will you have at the end of the year?
Future Value • PV=$10,000 • i=.06 • FV=10,000(1.06) = $10,600
Future Value -- With Compounding • What if the bank offers to “compound” your interest monthly? • How much money will you have?
Future Value • What happened here? • Mathematically: • Intuitively -- we are earning interest on interest with monthly compounding
Periodic Compounding • To find the value of the bank account at the end of the year for any number of periods: • divide the interest rate by the number of periods to get the periodic rate • Raise the quantity 1+ periodic rate to the power equal to the number of periods
Observations • For fixed annual nominal interest rate, future wealth increases with increasing number of compounding periods • Same “nominal” annual rate • Getting money sooner is always better than getting it later • There is not much difference between monthly and continuous compounding
Effective Annual Yield • For fixed annual nominal rate: • Different Compounding Periods Different Terminal Wealth Values • Effective Annual Yield is a measure that captures the different wealth • . For One Year:
Example • For semiannual compounding: • For daily compounding:
Effective Annual v. Nominal Yield • When discussing interest bearing investments, people usually speak in terms of the “nominal” or “simple” annual interest rate. • The nominal rate does not completely describe the investments return -- you must know the compounding period too. • The compounding period is standard (and so presumed to be known) for certain kinds of investments: • U.S. and Japan Bonds: Semiannual • Residential Mortgages; monthly • Eurobonds: Annual • The nominal rate is sufficient to rank investments when comparing within a single type with the same compounding • When comparing mortgages and bonds, you must use effective yields
More examples • Calculate the EAY for a15% Treasury bond and a 15% mortgage: • Bond: .1556 • Mortgage:.16075 • The mortgage has an EAY more than 50 basis points higher than the bond. • Note that we didn’t need to know the maturity of these to calculate EAY
One More Time... • Calculate the EAY for a 5% Treasury bond and a 5% mortgage: • bond: .0506 • mortgage: .0512 • For 5% rates, the difference between the two is 6 basis points • The difference in EAY depends a lot on the level of the interest rate
Longer Investment Horizons • What if we plan to leave the investment in the bank for more than one year? • For annual compounding: • For periodic compounding:
Calculator Tips • Some calculators let you set a “mode” feature for a certain number of compounding periods per year (e.g., 2 or 12 or 365) • The calculator then takes care of dividing the interest rate by p and/or multiplying the years by p. • Some calculators do both, some only one. As a result, using this feature is one of the most common sources of errors. • HP calculators divide the interest rate by p but do not multiply the years by p • I strongly recommend keeping your calculator in the 1 payment per period mode until you are comfortable with these calculations
Present Value • Present Values and Future Values are related by the same formula we have been using -- we just solve for PV not FV. • For annual compounding and a single year • For periodic compounding:
Present Value • Present values can be viewed as: • The amount you would have to put in the bank today (I.e., invest today) in order to generate a future wealth equal to FV, for a specific interest rate, or, • The price you would be willing to pay for the right to receive a cash flow of FV at some point in the future, when your opportunity cost of capital is equal to the specific interest rate
Annuities • The formula for the future value of an annuity is: • Notice that the formula simply reflects the repeated application of the initial FV formula: • The FV of each of the periodic deposits is calculated and added together to get the overall future value.
Annuities • Calculate the future value of $2,000 deposited at the end of each year into a bank account earning 6% compounded annually. • N=5 • i=6 • PMT=2000 • FV=??? ($11,274.19)
Present Value of Annuities • As with single cash flows, it is straightforward to switch to calculating the present value of an annuity. First since: • As with future values, the present value of an annuity is simply the sum of the present values of each individual payment.
Annuities • We can obtain a condensed mathematical expression for the present value of an annuity: • Note that if n is very large: • The second factor adjusts (P/i) for the fact that the payment is only received for a specific period of time.
Extensions • Annuities and lump sum payments can be readily combined by combining the use of the PV ,PMT, and FV keys • What is the future value of your bank account if you deposit $5,000 now and deposit $2,000 per year for five years - interest rate is 6% annual? • N=5 • i=6 • PMT=$2,000 • PV=5,000 • FV=???($17,965.31)
Yields • Consider a general form of the present value/future value equation: or
Yields • When calculating a yield, we are asking to find a value of i that makes this equation hold as an equality -- given known values for the payments, the price (or present value) and the lump sum payment at the end (FV). • Mathematically, we are solving for the roots of an nth order polynomial. • With a financial calculator, we are solving for i.
Yields • Example: Calculate the yield of a $10,000, five year maturity, Treasury bond that has a coupon of 7% and is priced at 9,950. • N=5*2=10 • PMT=.07/2*(10,000)=350 (inflow) • FV=10,000 (inflow) • PV=-9,950 (outflow) • I=???(3.56% per period)
Yields • . Lets look at this example closely--starting with the inputs: • N-- the number of periods. Because this is a treasury bond and “everyone knows” it pays interest semiannually, we must set p=2 for two periods per year and a total of 10 semiannual periods. • PMT=$350. A U.S. bond with a nominal interest rate of 7% pays .07/2 = .035 times the face value of the bond every six months • FV=10,000. When people describe a bond, they will typically give you a “face” value or the principal value. In this case it is $10,000. That means the Treasury promises to pay you interest at the coupon rate on this face amount and pay you this amount at maturity in a lump sum. • PV= -9,950. The price is what you must pay out now in order to receive the PMT and FV. Note that it is critical that you enter the negative sign to denote that this is an outflow.
Discount Securities • After we enter the necessary inputs, we solve for the interest rate that makes them equal. The calculator tells us that it is 3.56% • The number the calculator solves for is i/p, not i, the annual nominal interest rate. You need to multiply this number by p to get i. Here we find i=7.12%. • Does this make sense? We paid less than the face amount of the debt (i.e., bought the security at a discount). Had we paid “par” for the security (i.e., $10,000 price for $10,000 of principal), we would have earned a yield of 7%. We earn a slightly higher yield here because we bought the bond at a discount.
Premium Securities • Example: Calculate the yield of a $10,000, five year maturity, Treasury bond that has a coupon of 7% and is priced at $11,000 • N=5*2=10 • PMT=.07/2*(10,000)=350 • FV=10,000 • PV=-11,000 • I=???(2.37% per period, 4.74% per year)
Bond Values Viewed as a Function of the Discount Rate • The bond indenture describes the promised future cash flows associated with a bond. • Maturity, coupon payment, timing of payments. • Treating these cash flows as “certain”, we can calculate the present value for many different discount rates.
Treasury Bond Cash Flows 8% coupon rate; 5 year maturity Return of Principal $1,000=FV $40 Coupon Payments=PMT N=10 Semiannual periods $ Price =PV
8% Treasury Bond Prices for Different Discount Rates Change Price Discount Rate
Interest Rate Risk • The price or market value of a bond varies with interest rates • If you purchase $1000 face value of an 8% 10-year bond when interest rates are 8%, you initially have an asset with a market value of $1000. • If interest rates change after your purchase, the value of your asset changes: • Rates rise to 10%, your asset value declines to 922.78 • Rates fall to 6%, your asset value increases to $1085.30
Interest Rate Risk • Uncertainty about the future value of an asset because of uncertainty about the level of interest rates is one way of viewing interest rate risk • How significant is this risk? • How much do interest rates change on a monthly basis? • Over the last 45 years, the average month to month change (in 10-year Treasury rates) has been about +/-20 bp • The standard deviation has been about 30 bp • The max change was -1.76% and +1.61%
Interest Rate RiskRisk to Market Value • Quantify the risk to market value • Based on the average change in interest rates, the market value of a $1,000 8% 10-year bond can change • +/- 20 bp. +/-1.3% in value • +/- 90 bp.+ + /- 6% in value • +/- 200 bp. +/- 13% in value
Interest Rate Risk • What we have seen is that uncertainty about future interest rates translates into uncertainty about future values of a financial asset • The market value of a fixed income security varies with interest rates • i prices • i prices
Duration • Macaulay (1938) developed duration as a measure of the remaining life of bonds and other fixed income securities. • Time to maturity of a bond is an unambiguous measure of life only for zero coupon bonds because it ignores the amount and timing of all cash flows except the last. • Hicks & Samuelson (1939) developed duration as a direct measure of the price sensitivity of bonds to changes in yield-to-maturity.
Final Maturity Final maturity is the length of time until all principal is repaid. The last cash flow from the example portfolio occurs at the end of the fifth year Final Maturity=5 Years Final maturity does not measure interest rate sensitivity. Recall the example with the four 10 year investments. Weighted Average Maturity. Weights = the % of principal received in a given time period Weighted average maturity is a slightly better predictor of price sensitivity to rate changes recall the mortgage and bond But WAM is not very accurate as shown by the three bonds with the same maturity Measures of Maturity
Duration • Duration is an alternative weighted average measure of maturity: • Weights are the % of the total present value of the security received in a time period. • All cash flows are considered in the calculation • Both principal and interest
Numerical Example • Consider an annuity with cash flows: • C1=C2=100 • Price of the annuity is 177.88 • Implies an IRR of 8.18%