The Time Value of Money Chapter 5 Brealey-Myers-Marcus 6th Edition Business Finance 620 General Overview Rules for Single Cash Flows Compounding and discounting Effective rates of interest Interest Rates and Inflation Rules for Streams of Cash Flows General streams of (uneven) cash flows Streams with special characteristics Perpetuities Annuities Amortization and amortizing loans Part 1 Topics Covered Future values (FV) and compounding Simple and compound interest FV of a single cash flow today FV of a stream of cash flows Effective annual rates of interest Present values (PV) and discounting PV of a single cash flow in the future PV of a stream of cash flows Interest rates and inflation FV with Simple Interest Simple interest is earned at an annual rate of 6% for 5 years on a principal balance of $100. Year Beginning Balance Interest Ending Balance 1 $ 100 $ 6 $ 106 2 106 6 112 3 112 6 118 4 118 6 124 5 124 6 130 FV with Compound Interest Compound Interest is earned at an annual rate of 6% for 5 years on a principal balance of $100. Year Beginning Balance Interest Ending Balance 1 $ 100.00 $ 6.00 $ 106.00 2 106.00 6.36 112.36 3 112.36 6.74 119.10 4 119.10 7.15 126.25 5 126.25 7.57 133.82 Ending balance comparison: With compound interest 133.82 With simple interest 130.00 Difference 3.82 Compound interest generates: interest of $30.00 on the original $100 investment interest of $3.82 in Years 2-5 on interest credited during Years 1-4 The $3.82 difference is called “interest on interest” and is the hallmark of interest compounding. Simple and Compound Interest FV of a Single Cash Flow FV of Single Cash Flow: In this formula: FV is future value PV is present value (not always today) R is the constant interest rate (%) K is the number of time periods separating FV and PV K can be any unit of time (e.g., years, months) FV of Single Cash Flow Application What is the FV of $1,500 if interest is compounded at a rate of 8% annually for seven years? 1,500 2,570.74 0 1 2 3 4 5 6 7 How much of the total interest earned represents interest earned on interest? FV of a Stream of Cash Flows The FV of each cash flow in the stream can be added to calculate the FV of the stream. The FV of a stream of multiple cash flows is the sum of the FV of each cash flow in the stream. The general formula for a stream of T cash flows (Ci) over T periods, compounded at a rate of R% per period: where Ci(1+R)T-i represents the FV of the ith cash flow in the stream. FV of a Stream of Cash Flows Application Consider two alternative 3-year savings plans: 1. You can deposit $5,500 in cash today 2. You can make 3 annual deposits: a. Deposit $800 in cash today b. Deposit $2,400 in cash one year from today c. Deposit $2,300 in cash two years from today If the interest rate is 7.50% annually, which savings option offers the better deal? FV of a Stream of Cash Flows Application Option Amount Years on Deposit Future Value Ending Balance 1 5,500 3 6,833 6,833 2 800 3 994 2,400 2 2,774 2,300 1 2,473 6,240 Annual Percentage Rates Interest rates normally are quoted on an annual basis. The quoted rate of interest is called the APR (Annual Percentage Rate). By definition, an APR is an annual rate. However, not all interest rates are paid annually (once a year). A 6% APR might be paid once every 6 months in two equal installments of 3% each. If interest is credited more frequently than once a year, does this affect the interest terms? Effective Annual Rates Suppose a bank CD pays interest at the rate of 6% APR. If the bank were to pay you interest just once a year (at yearend), your account would grow by 6% each year. Your ending CD account balance each year would always be 6% higher than your beginning balance. BUT, if the 6% APR your CD account earns is paid in 4 equal installment of 1.5% each, does your account still grow by 6% each year? Anatomy of Quarterly Compounding Quarter Beginning Balance Interest Rate Interest Income Ending Balance 1 1,000.00 0.015 15.00 1,015.00 2 1,015.00 0.015 15.23 1,030.23 3 1,030.23 0.015 15.45 1,045.68 4 1,045.68 0.015 15.69 1,061.37 Total 0.060 61.37 Suppose you purchase a $1,000 bank CD today. With $1,000 on account for one year at 6%, we would expect to earn $60 of interest – but instead, we earned $61.37. Anatomy of Quarterly Compounding Earned interest analysis: Interest earned on $1,000 investment $ 60.00 Interest earned on prior interest 1.37 Total interest earned $ 61.37 We earned interest in Quarters 2, 3 and 4 on interest credited in Quarters 1, 2 and 3. Dividing the total interest earned ($61.37) by the original investment ($1,000) yields a rate of 6.137%. This is the account’s Effective Annual Rate. The EAR Formula The general formula for the EAR: where: APR is the Annual Percentage Rate K is the number of times per year interest is paid (number of compounding periods per year) To illustrate, for an 8.4% APR, compounded monthly, what is the EAR? Compounding Frequency and EAR Compounding Frequency Interest Installments Per Year APR Periodic Interest Rate EAR Annual 1 6% 6.0% 6.000% Semiannual 2 6% 3.0% 6.090% Quarter 4 6% 1.5% 6.136% Month 12 6% 0.5% 6.168% Consider the following alternative interest terms: What conclusion should be drawn concerning the relationship between compounding frequency and EAR, given the APR? Present Value and Discounting Year 0 1 2 How much is $1 to be received in two years worth today if the annual interest rate (R) is 9%? $1 The 9% interest rate used to discount the $1 payment to the present (today) is called the discount rate. Present Value = ? Compounding builds interest into a stream of cash flows, while discounting removes the interest from the stream. PV of a Single Cash Flow The present value (PV) of a future cash flow is given by the formula: where: FV is the future value after K time periods K is the number of time (compounding) periods K can be any time period (e.g., year, month) R is the interest rate per time period (the discount rate) PV of Single Cash Flow Application You just bought a new Dell PC for $1,200. Dell’s payment terms are “2 years same as cash.” If you can earn interest at the rate of 6% per year on your funds, how much should you set aside today in order to make the payment due in two years? $1,200 1 0 2 PV How much should you set aside today if the interest rate were compounded monthly? Why is your initial deposit different? PV of a Stream of Cash Flows The PV of each cash flow in the stream can be added to calculate the PV of the stream. The PV of a stream of multiple cash flows is the sum of the PV of each cash flow in the stream. The general formula for a stream of T cash flows (Ci) over T periods, compounded at a rate of R% per period: where Ci/(1+R)i represents the PV of the ith cash flow in the stream. PV of a Stream of Cash Flows Application Consider two alternative 3-year car financing plans: 1. You can buy the car today for $25,700 in cash. 2. You can make 3 annual installment payments: a. $10,700 in cash today b. $7,500 in cash one year from today c. $7,500 in cash two years from today If the interest (discount) rate is 8.35% a year, which financing option offers better terms? PV of a Stream of Cash Flows Application The PV of financing Option 1 is $25,700. The PV of financing Option 2 is $24,011: The car costs less to finance over two years; that is, $ 24,011 is less than $25,700. Interest Rates and Inflation Over time, inflation reduces the purchasing power of money. Quoted interest rates (rates at which money grows) are called nominal rates. Nominal rates can be adjusted to isolate two component effects: Price increases (inflation) Increases in the purchasing power of money The portion of the nominal rate representing increases in the purchasing power of money is called the real rate of interest. Real Interest Rates The formula for the real rate of interest is: where RINF is the annual rate of inflation. For many practical applications, the real rate can be approximated by the expression: Real Interest Rates Application A bank CD earns interest at the rate of 8.5% annually (this is the nominal rate). If the rate of inflation is 2.7% per year, what is the CD’s real rate of interest? Note that 5.6% is the actual rate at which the purchasing power of CD interest grows (thus, the label, “real” rate of interest). Part 2 Topics Covered Cash flows with special characteristics Perpetuities and annuities Perpetuity applications: PV of a constant perpetuity PV of a growing perpetuity Annuity applications: PV of an annuity (including amortizing loans) FV of an annuity PV of a growing annuity FV of a growing annuity Streams of Cash Flows with Special Characteristics When a stream of cash flows exhibits certain characteristics, both the stream’s PV and FV have analytical solutions – can be found with reasonably simple formulas. Two important patterns of cash flows: level, equally spaced cash flows that never end level, equally spaced cash flows that begin and end (fixed time period) Definitions Perpetuity A stream of level, equally spaced cash payments that never tends Perpetuities can be constant or growing Used in the valuation of common stock Annuity A stream of level, equally spaced cash payments between two points in time Extremely versatile analytical tool Used for bond valuation, mortgage loan analysis, and personal investment planning A Classification Scheme Streams of Equally-Spaced Cash Flows Trend Duration Measure Description Constant Perpetual PV PV of constant perpetuity Finite PV PV of ordinary annuity FV FV of ordinary annuity Growing Perpetual PV PV of growing perpetuity Finite PV PV of growing annuity FV FV of growing annuity PV of a Constant Perpetuity The formula: where C is the level cash flow and R is the periodic rate of interest (discount rate). In this formula: The first instance of C is one time period (year) from the date of the PV (usually today). The time interval can be any length, provided it remains fixed. PV of a Constant Perpetuity Application How much must be set aside today to fund a perpetual trust account paying $6,000 a year and earning 7.5% annual interest? If the account is funded today with $______, and continues to earn 7.5% interest, it can pay out $6,000 at the end of each year, and never run out of money! PV of a Growing Perpetuity The formula: where C is the initial cash flow in the stream, R is the periodic interest (discount) rate, and G is the constant periodic growth rate of C. In this formula: The first payment of C is one period (year) from the date of the PV (usually today). The time interval between payments can be any length, provided it remains fixed PV of a Growing Perpetuity Application How much should you set aside today to get successive annual payments growing at 5% per year indefinitely? The first payment, due one year from today, is $6,000. The account earns 7.5% annual interest. Your withdrawals could increase by 5% each year, and you would never run out of money! PV of a Constant Annuity The formula: where: C is the constant periodic payment, R is the periodic interest (discount) rate, and K is the number of periods the payment C is made. In this formula: The first payment of C is one period from the date of the PV (usually today). The interval between payments can be any length, provided it remains fixed. PV of a Constant Annuity Application You decide to buy a new car for five annual installment payments of $6,000 each. If you could borrow money at 7.5% annual interest, what is the PV of the 5 payments? If you purchased the car today for the PV in the previous question (rather than making 5 annual payments), would you still be paying the same price for the car? FV of a Constant Annuity The formula: where: C is the constant periodic payment, R is the periodic interest (discount) rate, and K is the number of periods the payment C is made. In this formula: The first payment of C is one period from the date of the PV (usually today). The interval between payments can be any length, provided it remains fixed. FV of a Constant Annuity Application If you save $6,000 annually for the next 5 years, what will be your account balance at the end of that period? Assumptions: Your account currently has a zero balance. Your first $6,000 deposit is one year from today. You will not withdraw funds along the way. The account will earn 7.5% interest annually for the entire 5-year period. How would your answer change if you had a starting account balance of $3,000? PV of a Growing Annuity The formula: where: C is the constant periodic payment, R is the periodic interest (discount) rate, G is the constant growth rate of C, and K is the number of periods the payment C is made. In this formula: The first payment of C is one period from the date of the PV (usually today). The interval between payments can be any length, provided it remains fixed. PV of a Growing Annuity Application You want to purchase an annuity to provide 5 years of annual payments with the following terms: The first payment, due one year from today, is $6,000. Second and subsequent payments increase by 5% per year. The annuity earns 7.5% APR, compounded annually. How much do you need to deposit today to fund this annuity? If the five payments were to remain constant ($6,000 each), how much would you need to deposit today? Why is the amount of the deposit different? FV of a Growing Annuity The formula: where: C is the constant periodic payment, R is the periodic interest (discount) rate, G is the constant growth rate of C, and T is the number of periods the payment C is made. In this formula: The first payment of C is one period from the date of the PV (usually today). The interval between payments can be any length, provided it remains fixed. FV of a Growing Annuity Application You want to purchase an annuity to generate 5 years of annual payments with the following terms: The first payment, due one year from today, is $6,000. Second and subsequent payments increase by 5% per year. The annuity earns 7.5% APR, compounded annually. How much will the annuity be worth at the end of five years? If the five payments were to remain constant ($6,000 each), how much would you have in five years? Why is the ending balance different? Amortizing Loans Loans in which each payment includes some mix of interest and repayment of principal are called amortizing loans. Loan payment amounts are constant, but the mix of interest and principal varies among payments. One of the best known examples of this type of loan is the mortgage loan. Each mortgage payment includes interest as well repayment of principal. After the final payment, the principal balance has been repaid in full, at the stated rate of interest. Amortizing Loans Application You purchase a house for $187,500, paying 15% down, and financing the balance at 6% APR, compounded monthly, over 20 years. What is your monthly mortgage payment? First, note that: The mortgage is an amortizing loan. Amortizing loans have constant (level) payments. Thus, the stream of monthly mortgage payments represents an annuity. Amortizing Loans Application Solving the following expression for monthly mortgage payment (PMT) yields $1,141.81: where: $159,375 is the amount of the price financed. 240 is the number of monthly payment periods. 0.5% is the monthly (periodic) rate of interest (in this case, the APR divided by 12). Amortizing Loans Application Month Beginning Balance Mortgage Payment Interest Payment Principal Repayment Ending Balance 1 159,375.00 1,141.81 796.88 344.94 159,030.07 2 159.030.07 1,141.81 795.15 346.66 158,683.41 3 158,683.41 1,141.81 793.42 348.39 158,335.01 4 158,335.01 1,141.81 791.68 350.13 157,984.88 5 157,984.88 1,141.81 789.92 351.89 157,632.99 6 157,632.99 1,141.81 788.16 353.65 157,279.35 First 6 months of the mortgage loan’s amortization schedule Interest Principal Amortizing Loans Application Amortizing Loans Application If you make all the mortgage payments, how much interest will you pay over the life of the loan? If you sell the property after making the 70th payment, how much will you owe the lender (what is the loan’s “pay-off amount”)? Starting with the 49th payment, suppose you pay an additional $300 each month, making your total payment $1,441.81. How will this affect the loan’s total interest cost?