Time Value of Money. Lakehead University. Outline of the Lecture. Fall Future Value and Compounding. Present Value and Discounting

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1 Time Value of Money Lakehead University Fall 2004 Outline of the Lecture Future Value and Compounding Present Value and Discounting More on Present and Future Values 2

2 Future Value and Compounding Future value refers to the amount of money an investment would grow to over some length of time at a given rate of interest. To determine this value, it is important to know when interest is calculated. Is it once a year? Every six months? Each month? When many payments are involved, it is also important to know the timing of these payments. 3 Future Value and Compounding Investing for a Single Period Suppose $100 is invested in an account that pays 10% per year. This investment will then be worth after one year = ) 100 = = $110 4

3 Future Value and Compounding Investing for More than One Period Suppose $100 is invested in an account that pays 10% per year. After one year, this investment will be worth $110. If the interest payment is reinvested, this investment will be worth, after two years, = = = 1.1) = $ Future Value and Compounding Decomposing 1.1) gives us = ) 100 = 100 }{{} Capital + 20 }{{} Interest on capital Simple interest + 1 }{{} Interest on interest Compound interest 6

4 Future Value and Compounding More generally, $m invested at a period interest rate r will grow to ) t m = m }{{} Capital after t periods. + t r m }{{} Simple interest + Compound interest 7 Future Value and Compounding Compound interest can be significant over the long run. Take $100 invested for T years at 10% compounded annually: Ending Simple Compound T Amount Capital Interest Interest = = = = =

5 Future Value and Compounding Examples of Future Value Calculations 1. $2,250 invested for 30 years at 18% compounded annually gives 2, ) 30 = $322, $9,310 invested for 15 years at 6% compounded annually gives 9, ) 15 = $22, Future Value and Compounding One More Example 3. You are scheduled to receive $22,000 in two years. When you receive it, you will invest it for six more years at 6 percent per year. How much will you have in eight years? Answer: 22, ) 8 2 = 22, ) 6 = $31,

6 5.2 Present Value and Discounting Present value refers to the amount of money that has to be invested today to obtain a specific amount of money after a specific length of time at a given rate of interest. If, for example, we want to know how much to invest to obtain $1 after one year at 10% interest, we need to solve Present value 1.1 = $1 Present value = = $ Present Value and Discounting More generally, the amount of money that needs to be invested today to obtain $1 in t years at the annual rate of interest r is PV = 1 ) t. This amount is the present value, as of today, of $1 to be received in t years discounted at the annual rate r. 12

7 Present Value and Discounting Examples of Present Value Calculations 1. The present value of $15,000 to be received in 5 years, discounted at the annual rate 12%, is PV = 15, ) 5 = $8, The present value of $25,000 to be received in 10 years, discounted at the annual rate 8%, is PV = 25,000 = $11, ) More on Present and Future Values With a period rate of interest r and a number of periods t, we can define Future value factor = ) t Present value factor = 1 ) t 14

8 More on Present and Future Values Let PV 0 denote the present value, as of today date 0), of an investment that will grow to the future value FV t in t periods, the period interest rate being r. Then PV 0 ) t = FV t and, equivalently PV 0 = FV t ) t. This result is the basic present value equation. 15 More on Present and Future Values Determining the Discount Rate What must r be for PV 0 to grow to FV t in t periods? PV 0 ) t = FV t ) t = FV t PV 0 FVt = PV 0 ) 1/t r = ) 1/t FVt 1. PV 0 16

9 More on Present and Future Values Example of Discount Rate Determination You are offered an investment that requires you to put up $12,000 today in exchange for $40, years from now. What is the annual rate of return on the investment? Answer: In this example, PV 0 = 12,000, FV t = 40,000 and t = 12. Therefore, r = ) 40,000 1/12 1 = 10.55%. 12, More on Present and Future Values Finding the Number of Periods What must t be for PV 0 to grow to FV t at a rate r? Note: We will be using the following rules: lnab) = lna) + lnb) ln a b) = blna) a ln = lna) lnb). b) 18

10 Finding the Number of Periods PV 0 ) t = FV t ln PV 0 ) t) = lnfv t ) lnpv 0 ) + ln ) t) = lnfv t ) lnpv 0 ) + t ln) = lnfv t ) t = lnfv t) lnpv 0 ) ln) t = lnfv t/pv 0 ) ln) 19 How Long to Double Your Money? Knowing r, how many periods is needed for PV 0 to double? t = lnfv t/pv 0 ) ln) = ln2pv 0/PV 0 ) ln) = ln2) ln) 20

11 The Rule of 72 Note that when r is small, ln) r slightly below r); ln2) = slightly below 0.72). A good approximation of the time it takes to double an investment is 0.72 = 72 r 100r. If r = 8%, PV 0 will double in approximately 72/8 = 9 years. 21 The Rule of 72 r ln) ln2) ln1+r) r 2% % % % % % %

12 The Rule of 72 The rule of 72 holds exactly at around 7.85%. The rule of 72 will overestimate the time it takes to double an investment when r < 7.85%; underestimate the time it takes to double an investment when r > 7.85%; 23 The Rule of 72 When r is small, the error will be insignificant. The error is significant when using large numbers. Take r = 72%, for instance. According to the rule of 72, an investment doubles in approximately one year at this rate. This makes no sense: it takes r = 100% to double an investment in one year. 24

13 Finding the Number of Periods: An Example You are trying to save to buy a new $120,000 Ferrari. You have $40,000 today that can be invested at 8% compounded annually. How long will it take before you have enough money to buy the car? Answer: t = ln120,000/40,000) ln1.08) = ln3) ln1.08) = years. 25 Future and Present Values of Multiple Cash Flows Future Value with Multiple Cash Flows Suppose $100 is invested today and another $100 is invested in one year at an annual rate of 8%. How much will this investment be worth in two years? $100 $108 $ $208 $ Time 26

14 Future Value with Multiple Cash Flows The same example, put differently: $ $ $100 $ $ $ Time That is, FV = $ ) 2 + $ = $ Future Value with Multiple Cash Flows Suppose now that the two payments are made at the end of each period. This gives us $100 $108 $100 $208 Time and thus, in this case FV = $ $100 = $

15 Future Value with Multiple Cash Flows More generally, let d t payment made in period t; r period interets rate; T the total number of periods. Then FV = ) T d 0 + ) T 1 d )d T 1 + d T = T t=0 ) T t d t. 29 An example with T = d 0 ) ) ) ) ) 4 d 0 d 1 ) ) ) ) 3 d 1 d 2 ) ) ) 2 d 2 d 3 ) )d 3 d 4 4 t=1 ) 4 t d t 30

16 Present Value with Multiple Cash Flows What is the present value of $100 to be received one year from now and another $100 to be received in two years, the annual rate of interest being 8%? $ /1.08 $100 $ /1.08 1/1.08 $100 $ Time That is, PV = $ $ ) 2 = $ Future Value with Multiple Cash Flows More generally, let d t payment made in period t; r period interets rate; T the total number of periods. Then PV = d 0 + d 1 + d 2 ) d T ) T = T t=0 d t ) t. 32

17 A Note on Cash Flow Timing Unless specified otherwise, cash flows are assumed to take place at the end of each period. A cash flow in year 2, for instance, means a cash flow to be received two years from now, and thus at the end of the second year. 33 A Note on Cash Flow Timing If you are told that a three-year investment has a first-year cash flow fo $100, a second-year cash flow of $200 and a third-year cash flow of $300, then the timing of cash flows is as follows: $100 $200 $300 34

18 6.2 Valuing Level Cash Flows A ordinary annuity is a series of constant, or level, cash flows that occur at the end of each period for some fixed number of periods. An annuity due is a series of constant, or level, cash flows that occur at the beginning of each period for some fixed number of periods. A perpetuity is an annuity in which the cash flows continue forever. 35 A Note on How to Value Level Cash Flows Let Then S = T q t = q + q 2 + q q T 1 + q T. t=1 qs = q T t=1 and thus, if q 0 and q 1, q t = q 2 + q 3 + q q T + q T +1, S qs = q q T +1 S = q qt +1 q = q q T ). q 36

19 A Note on How to Value Level Cash Flows If q = 1, then S = T t=1 qt = T. What happens when T is arbitrarily large? lim T q q T ) = q q 1 q if 0 q < 1, if q > A Note on How to Value Level Cash Flows Suppose that we have S = 1 + ) ) 1 3. Let q = 1+r 1, where r > 0 is a discount rate. Then S = q + q 2 + q 3 = q q 3 ). q 38

20 A Note on How to Value Level Cash Flows Replace q with 1+r 1 in the last equation. This gives ) ) 1/) 1 3 S = 1/) ) ) = 1 = 1 ) ) 1 3 r 39 A Note on How to Value Level Cash Flows So if we have S = then T t=1 1 ) t = 1 + S = 1 r ) ) ) 1 T. ) 1 T, 40

21 Present Value of Annuity Cash Flows Consider an ordinary annuity that pays $C each period for T periods, the first payment being made one period from now. What is the present value of this annuity if the period rate of interest is r? 41 PV = C + C ) 2 + C ) C ) T 1 = C + = C 1 r = C r ) ) ) T 1 ) ) T ) ) ) T 42

22 Present Value of Annuity Cash Flows The term ) 1 T 1+r r is often referred to as the present value interest factor for annuities and abbreviated PVIFAr,T ). Note that PVIFAr,T ) = 1 1+r r ) T = Present Value factor r 43 Present Value of Annuity Cash Flows Consider now an annuity due involving T payments of $C, the period interest rate being r. Then PV = C + C + C ) C ) T 1 1 = )C + ) ) ) ) T = )C 1 r 1 ) ) T = ) C PVIFAr,T ). 44

23 Present Value of Annuity Cash Flows The present value of an annuity due is equal to times its ordinary counterpart. Using the above equations, we can answer questions similar to those in chapter 5, such as finding the fixed payment that will repay a loan, or finding the number of periods necessary to repay a loan, etc.. 45 Present Value of Annuity Cash Flows Example 1 An investment offers $2,250 per year for 15 years, with the first payment occurring one year from now. If the required return is 10 percent, what is the value of the investment? Answer: PV = 2, ) ) = $17,

24 Present Value of Annuity Cash Flows Example 2 Betty s Bank offers a $25,000, seven-year loan at 11 percent annual interest payable in equal annual amounts. What will the annual payment be? Answer: C = PV 1 r 1 1+r ) T ) = , ) 7 ) = $5, Present Value of Annuity Cash Flows Example 3 How long does it take to repay a $25,000 loan with fixed annual payments of $4,000 at an 11% annual interest rate? We will be using the ln trick to solve this problem. 48

25 Example 3 Answer: PV = C r 1 ) ) T rpv C ) 1 T = ) 1 T = rpv C ) 1 T ln T = = ln rpv ) C ln rpv ) C ln ) 1 1+r 49 Example 3 Answer: T = = ln rpv ) C ln ) 1 1+r ) ln ,000 4,000 ln ) = years, and thus it takes 12 years to repay such a loan, the last payment being less than $4,

26 Present Value of Annuity Cash Flows Example 4 What must the annual rate of interest be in order to fully repay a $25,000 loan in 10 years with fixed annual payments of $4,000? Answer: PV = C ) ) 1 T r 25,000 = 4,000 r Can t solve this equation analytically. ) ) Example 4 The solution can be found by trial-and-error or by using a computer. In Excel: RATENPER,PMT,PV) = RATE10,-4000,25000) = 9.61%. 52

27 Future Value of Annuity Cash Flows Consider an ordinary annuity that pays $C each period for T periods, the first payment being made one period from now. What is the future value of this annuity if the period rate of interest is r? 53 Future Value of Annuity Cash Flows Answer: FV = ) T 1 C + ) T 2 C )C + C ) = C ) T 1 + ) T ) + 1 = C 1 + ) ) T 2 + ) T 1) ) T ) = C ) ) T ) = C r ) T ) 1 = C r 54

28 Future Value of Annuity Cash Flows The term ) T 1 r is often referred to as the future value interest factor for annuities and abbreviated FVIFAr,T ). Note that FVIFAr,T ) = )T 1 r = 1+r) T ) 1 T 1+r r = 1+r) T PVIFAr,T ). 55 Perpetuities A perpetuity is an annuity with perpetual cash flows. The future value of a perpetuity is always infinite. The present value of a perpetuity paying $C forever at the end of each period, the period interest rate being r > 0, is 1 ) T ) lim C 1+r = C T r r, since lim T 1 1+r) T = 0. 56

29 Perpetuities The present value of a perpetuity paying $C forever at the beginning of each period, the period interest rate being r > 0, is 1 ) T ) 1+r )C lim C) =. T r r 57 Relationship between Annuities and Perpetuities Consider the following perpetuities: Perpetuity P 1: Pays $C forever, the first payment being made one period from now. Perpetuity P 2: Pays $C forever, the first payment being made at time T + 1 i.e. at the end of period T, which is the beginning of period T + 1). P1 : P2 : T 1 T T +1 T +2 T C C C C C C C C C C C C C 58

30 Relationship between Annuities and Perpetuities Note that P P2 = AC,T ), where AC,T ) denotes the ordinary) annuity that pays $C for T periods. Hence, the present value of P P2 must be equal to the present value of AC,T ). 59 Relationship between Annuities and Perpetuities Let r denote the period interest rate. Then PVP1) = C r and PVP2) = C/r ) T, and thus PVP P2) = C r C/r ) T = C r 1 ) ) T = PVAC,T )). 60

31 Growing Annuities Consider an annuity in which the payment grows at the rate g from one period to the other. That is, the cash flows from this annuity are as follows: T 1 T... C 1+g)C 1+g) 2 C 1+g) T 2 C 1+g) T 1 C 61 The present value of this annuity is PV = = = = C 1 + g)c + ) g)2 C ) g)t 1 C ) T C 1 + g 1 + g + ) 1 + g 2 + ) 1 + g ) ) C 1 + g)/) 1 + g T 1 + g 1 + g)/) ) ) C 1 + g g T )/1 + g) 1 ) ) 1 + g T = C ) ) g T 1 + g) = C ) ) 1 + g T r g 62

32 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, r g where T is the number of payments and r is the period discount rate. What is the present value of a growing perpetuity? 63 Growing Annuities If g < r, then 1 + g If g > r, then 1 + g Therefore, lim T C r g ) 1 + g T < 1 and lim = 0. T ) 1 + g T > 1 and lim =. T ) ) 1 + g T = C r g if g < r, if g r. 64

33 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? 65 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 = = ) ) ) ) ) ) 8 ) ) ) 8 = $108,

34 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? 67 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? 68

35 Effective Annual Rates and Compounding With a 10% interest rate compounded semiannually, the EAR is EAR = 1.05) 2 1 = = 10.25%. Note that 0.25% = 5% 5% is the interest on interest charged during the year. 69 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 = m Compare the following rates: Bank A: 15% compounded daily Bank B: 15.5% compounded quarterly Bank C: 16% compounded annually 70

36 Effective Annual Rates and Compounding Bank A: Bank B: Bank C: EAR A = EAR B = EAR C = ) = 16.18% ) 4 1 = 16.42% ) 1 1 = 16.00% Quoting a Rate What is the quoted rate, compounded monthly, that provides an effective return of 15%? 72

37 Quoting a Rate Answer: 0.15 = 1.15 = ) Quoted rate ) Quoted rate ) 1/12 = ) 1/12 1 = ) ) 1/12 1 Quoted rate 12 Quoted rate 12 = Quoted rate = 14.06%. 73 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. 74

38 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 ) 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 = ) 1/12 1 = 1.13%. 75 Mortgages Example 1 continued) Find the monthly payment T = = 300): PV = C r 300, 000 = 1 ) ) T ) ) C C = , ) 300 C = $3,

39 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? 77 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 % = 18%. 78

40 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? 79 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. 80

41 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. 81 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, = $ If the repayment L) takes place after t periods, the present value of the loan is L PV = ) t. 82

42 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 $80 $80 $80 $80 $80 $1, Interest-Only Loans The present value of the above loan, at a discount rate r, is PV = 80 ) ) ,000 r ) 5. Note that PV > $1,000 if r < 8%, = $1,000 if r = 8%, < $1,000 if r > 8%. 84

43 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? 85