Time Value of Money Part III September 2003 Outline of the Lecture Growing Annuities The Effect of Compounding Loan Type and Loan Amortization 2
Growing Annuities The present value of an annuity in which the payments grow at the constant rate g, the first payment being C, is PV = C ) ) 1 + g T 1, r g 1 + r where T is the number of payments and r is the period discount rate. What is the present value of a growing perpetuity? 3 Growing Annuities If g < r, then 1 + g 1 + r If g > r, then 1 + g 1 + r Therefore, lim T C 1 r g ) 1 + g T < 1 and lim = 0. T 1 + r ) 1 + g T > 1 and lim =. T 1 + r ) ) 1 + g T 1 + r = C r g if g < r, if g r. 4
Growing Annuities Example Problem 77. Consider a firm that is expected to generate a net cash flow of $10,000 at the end of the first year. The cash flows will increase by 3 percent a year for seven years and then the firm will be sold for $120,000. The relevant discount rate for the firm is 11 percent. What is the present value of the firm? 5 Answer: The total cash flows generated by this firm are the 8 cash flows from its operations and the terminal value of $120,000, which will materialize eight years from now. The present value of the firm is then numbers in 000 s) PV = = 10 1.11 + 1.03 10 1.11) 2 + 1.03)2 10 1.11) 3 +... + 1.03)7 10 1.11) 8 + 120 1.11) 8 ) ) 10 1.03 8 1 0.11 0.03 1.11 + 120 1.11) 8 = $108, 360. 6
The Effect of Compounding Interest rates can be quoted in many different ways. How rates are quoted may come from tradition or regulation. Very often, rates are quoted in a misleading manner. What s under a quoted rate? 7 Effective Annual Rates and Compounding Suppose a rate is quoted at 10% compounded semiannually. This means that 5% is charged every six months. 10%, the quoted rate, is the interest charged on the principal during the year, it does not include the interest on interest compound interest). The rate that takes into account compound interest is called the effective annual rate EAR). What is the EAR in the above example? 8
Effective Annual Rates and Compounding With a 10% interest rate compounded semiannually, the EAR is EAR = 1.05) 2 1 = 1.1025 1 = 10.25%. Note that 0.25% = 5% 5% is the interest on interest charged during the year. 9 Effective Annual Rates and Compounding More generally, the EAR of a quoted annual rate compounded m times during the year is ) Quoted Rate m EAR = 1 + 1. m Compare the following rates: Bank A: 15% compounded daily Bank B: 15.5% compounded quarterly Bank C: 16% compounded annually 10
Effective Annual Rates and Compounding Bank A: Bank B: Bank C: EAR A = EAR B = EAR C = 1 + 0.15 ) 365 1 = 16.18%. 365 1 + 0.155 ) 4 1 = 16.42%. 4 1 + 0.16 ) 1 1 = 16.00%. 1 11 Quoting a Rate What is the quoted rate, compounded monthly, that provides an effective return of 15%? 12
Quoting a Rate Answer: 0.15 = 1.15 = 1 + 1 + ) Quoted rate 12 1 12 ) Quoted rate 12 12 1.15) 1/12 = 1 + 1.15) 1/12 1 = ) 12 1.15) 1/12 1 Quoted rate 12 Quoted rate 12 = Quoted rate = 14.06%. 13 Mortgages Regulations for Canadian institutions require that mortgage rates be quoted with semiannual compounding. Payments, however, are made each month. How to calculate monthly payments from a quoted mortgage rate? i) When quoting a rate, a financial institution is thinking EAR, so the first step is to find the EAR implied by the quoted rate. ii) Calculate the monthly rate prodiving the EAR in i). iii) Using the annuity formula, find the monthly payment. 14
Mortgages Example 1 Find the monthly payment on a $300,000 mortgage quoted at 14 percent and amortized over 25 years. EAR = 1 + ) Quoted rate m 1 = m 1 + 0.14 ) 2 1 = 14.49%. 2 Find the monthly rate that gives an EAR of 14.49%: 1 + Monthly Rate) 12 1 = 14.49% Monthly Rate = 1.1449) 1/12 1 = 1.13%. Mortgages 15 Example 1 continued) Find the monthly payment T = 25 12 = 300): PV = C r 300, 000 = 1 1 ) ) T 1 + r ) ) C 1 300 1 0.0113 1.0113 C = 0.0113 300,000 1 1 1.0113 ) 300 C = $3, 510.61. 16
Mortgages Example 2 An entrepreneur is considering the purchase of an office in a new high-rise complex. The office is worth $1,000,000 and a bank is offering a mortgage for the whole amount at 8 percent APR. If the entrepreneur s budget allows payments of $7,000 a month, how long will it take to pay off the purchase? 17 EARs and APRs Cost of borrowing disclosure regulations in Canada require that lenders disclose an annual percentage rate APR) in a prominent and unambiguous manner. By law, the APR is the interest rate per period multiplied by the number of periods in a year. This is indeed the quoted rate mentioned earlier. For example, the APR on a loan at 1.5% monthly interest rate is 12 1.5% = 18%. 18
Continuous Compounding What is the EAR when the quoted rate is compounded every nanosecond? Take, for example, a 12% APR: Compounded EAR Annually 12.00% Quarterly 12.55% Monthly 12.68% Weekly 12.73% Daily 12.75% Continuously? 19 Continuous Compounding The more often a quoted rate is compounded, the greater the EAR. Continuous compounding thus yields the maximum EAR from a given APR. Given a quoted rate q, lim m 1 + q m) m 1 = e q 1. Note that e q 1 is the highest EAR that can be obtained with an APR of q. 20
Loan Types and Loan Amortization We will see three types of loan in this section: Pure Discount Loans: Usually short-term loans, such as T-bills. Interest-Only Loans: Usually long-term loans, such as government and corporate bonds. Amortized Loans: Majority of individual loans. 21 Pure-Discount Loans In a pure discount loan, the borrower receives money today and makes one lump-sum payment at some time in the future. Consider, for example, a T-bill that promises to pay $1,000 in one year. When the interest rate is 3.48%, the value of this T-bill is PV = 1,000 1.0348 = $966.37. If the repayment L) takes place after t periods, the present value of the loan is L PV = 1 + r) t. 22
Interest-Only Loans With this type of loan, the borrower pays interest each period and repays the principal at some point in the future. Take, for example a 5-year loan of $1,000 at an 8% annual interest rate. Each year the borrower pays $80 in interest and the principal $1,000) is repaid after 5 years. Cash flows to the lender are then Interest Principal 0 1 2 3 4 5 $80 $80 $80 $80 $80 $1,000 23 Interest-Only Loans The present value of the above loan, at a discount rate r, is PV = 80 ) ) 1 5 1 + 1,000 r 1 + r 1 + r) 5. Note that PV > $1,000 if r < 8%, = $1,000 if r = 8%, < $1,000 if r > 8%. 24
Amortized Loans An amortized loan is such that interest and principal are repaid each period. This type of loan can be such that a constant amount of the principal is repaid each period, or can be such that a constant payment is made each period. How long would it take to repay a $5,000 loan with an APR of 10% compounded monthly if $500 in principal has to be repaid each month? 25