The Time Value of Money

The Time Value of Money Chapter 8 October 3, 2012

Learning Objectives • The “time value of money” and its importance to you and business decisions • The future value and present value of a single amount. • The future value and present value of an annuity. • The present value of a series of uneven cash flows.

The Time Value of Money • Money grows in amount over time as it earns from investments. • However, money that is to be received at some time in the future is worth less than the same dollar amount to be received today. Why? • Similarly, a debt of a given amount to be paid in the future is less burdensome than that debt to be paid now. Why?

Some Examples • Bought Oakland house for $29,500 in 1969 $23,600 mortgage, $175 mo. pymt I bought my house in Los Altos in 1979 for $135,000 $40,000 30 yr mortgage, $300 mo In 2009, would still paying $300 mo! House sold for over $1.25 million in 2006 Current owner paying $5,500 per month I now own $935,000 home, no mortgage! Time value of money

Indians – Manhattan Island • In 1624, Indians got $24 for Manhattan island • People think they were “taken” • If invested at 8%, compounded annually, today they would have $223,166,200,000,000 (trillion) • If compounded semiannually, $396 trillion • If compounded quarterly, $534 trillion • You could buy Manhattan Island today for around $500 billion • They could pay off the nat’l debt/buy back US! • Time value of money!

16 year old saves for retirement! • Earns $2,000 per year for 6 years/stops • Reinvests at 10% per year • At 21 years old, she is worth $15,431 • At age 65, with no add’l investment, if she just lets it ride, she will be worth $1,022,535 • If she waits just one more year to get started, she would be worth only $929,578 • She loses $92,957! (final years earnings) • So start saving now! You’ll never miss it.

The Future Value of a Single Amount • Suppose that you have $100 today and plan to put it in a bank account that earns 8% (k) per year. • How much will you have after 1 year? • After one year: $100 + (.08 x $100) = $100 + $8 = $108Or • If k = 8%, then 1 + k = 1 + .08 or 1.08Then, $100 x (1.08)1 = $108

FV = PV (1 + k)n The Future Value of a Single Amount • Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year. • How much will you have after 1 year? 5? 15? • After one year: $100 x (1.08)1 = $100 x 1.08 = $108 • After five years: $100 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = $146.93 $100 x (1.08)5 = $100 x 1.4693 = $146.93 • After fifteen years: $100 x (1.08)15 = $100 x 3.1722* = $317.22 • Equation: *Table I, p. A-1 Appendix

The Future Value of a Single Amount Calculator solution: N = 15 I/Y = 8 PV = -$100 PMT = 0 Compute (CPT) FV = $317.22

1 (1 + k)n PV = FVn x 0 1 2 100 (1.10)1 PV = = Present Value of a Single Amount • Value today of an amount to be received or paid in the future. *Table II, p. A-2, Appendix Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today? $100 $100 x .9091* = $90.91 $

1 (1 + k)n PV = FVn x 0 1 2 3 4 5 6 7 8 100 (1+.10)8 = PV = Present Value of a Single Amount • Value today of an amount to be received or paid in the future. Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today? $100 $100 x .4665* = $46.65 *Table II, p. A-2, Appendix

100 (1+.10)8 = 46.65 PV = Using Formula: N I/YR PV PMT FV 100 8 10 ? Financial Calculator Solution - PV Previous Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today? Calculator Enter: N = 8 I/YR = 10 PMT = 0 FV = 100 CPT PV = ? - 46.65 0

Jan Feb Mar Dec $500 $500 $500 $500 $500 Annuities • An annuity is a series of equal cash flows spaced evenly over time. • For example, you pay your landlord an annuity since your rent is the same amount, paid on the same day of the month for the entire year.

0 1 2 3 $0 $100 $100 $100 Future Value of an Annuity You deposit $100 each year (end of year) into a savings account (saving up for an IPad). How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

0 1 2 3 $0 $100 $100 $100 Future Value of an Annuity $100(1.08)2 $100(1.08)1 $100(1.08)0 $100.00 $108.00 $116.64 $324.64 You deposit $100 each year (end of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

$100(1.08)2 $100(1.08)1 $100(1.08)0 $100.00 $108.00 $116.64 $324.64 ) (1+.08)3 - 1 .08 ( = 100 n (1+k) - 1 k FVA = PMTx( ) Future Value of an Annuity 0 1 2 3 $0 $100 $100 $100 How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? = 100(3.2464*) = $324.64 *Table III, p. A-3, Appendix

0 1 2 3 $0 $100 $100 $100 N I/YR PV PMT FV Future Value of an Annuity Calculator Solution Enter: N = 3 I/YR = 8 PV = 0 PMT = -100 CPT FV = ? 324.64 3 8 0 -100 ?

0 1 2 3 $0 $100 $100 $100 Present Value of an Annuity • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

0 1 2 3 $0 $100 $100 $100 Present Value of an Annuity • How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100/(1.08)1 $100 / (1.08)2 $100 / (1.08)3 $92.60 $85.73 $79.38 $257.71

0 1 2 3 $100/(1.08)1 $100 / (1.08)2 $100 / (1.08)3 $92.60 $0 $100 $100 $100 $85.73 $79.38 1 (1.08)3 $257.71 1 - ( ) = 100 1 (1+k)n 1 - .08 PVA = PMTx( ) k Present Value of an Annuity • How much would the following cash flows be worth to you today if you could earn 8% on your deposits? = 100(2.5771*) = $257.71 *Table IV, p. A-4, Appendix

0 1 2 3 $0 $100 $100 $100 N I/YR PV PMT FV Present Value of an Annuity Calculator Solution PV=? Enter: N = 3 I/YR = 8 PMT = 100 FV = 0 CPT PV = ? -257.71 3 8 ? 100 0

Annuity Due • An annuity is a series of equal cash payments spaced evenly over time. • Ordinary Annuity: The cash payments occur at the END of each time period. • Annuity Due: The cash payments occur at the BEGINNING of each time period. • Lotto is an example of an annuity due

0 1 2 3 $100 $100 $100 FVA=? Future Value of an Annuity Due You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

0 1 2 3 $100 $100 $100 Future Value of an Annuity Due $100(1.08)3 $100(1.08)2 $100(1.08)1 $108 $116.64 $125.97 $350.61 You deposit $100 each year (beginning of year) into a savings account. How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

0 1 2 3 (1+k)n - 1 k FVA= PMTx( ) $108 $100(1.08)3 $100(1.08)2 (1+k) $100(1.08)1 $100 $100 $100 $116.64 $125.97 $350.61 ) ( (1+.08)3 - 1 .08 = 100 (1.08) Future Value of an Annuity Due How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually? =100(3.2464)(1.08)=$350.61

Calculator solution to annuity due • Same as regular annuity, except • Multiply your answer by (1 + k) to account for the additional year of compounding or discounting • Future value of an annuity due: n = 3, i/y = 8%, pmt = -100, PV = 0 CPT FV = 324.64 (1.08) = 350.61

0 1 2 3 $100 $100 $100 Present Value of an Annuity Due • How much would the following cash flows be worth to you today if you could earn 8% on your deposits? PV=?

How much would the following cash flows be worth to you today if you could earn 8% on your deposits? 0 1 2 3 $100 $100 $100 Present Value of an Annuity Due $100/(1.08)1 $100 / (1.08)2 $100/(1.08)0 $100.00 $92.60 $85.73 $278.33

0 1 2 3 $100 $100 $100 1 (1.08)3 1 - ( ) (1.08) = 100 1 (1+k)n 1 - .08 (1+k) PVA = PMTx() k Present Value of an Annuity Due • How much would the following cash flows be worth to you today if you could earn 8% on your deposits? $100/(1.08)1 $100 / (1.08)2 $100/(1.08)0 $100.00 $92.60 $85.73 $278.33 = 100(2.5771)(1.08) = 278.33

Calculator solution to annuity due • Same as regular annuity, except • Multiply your answer by (1 + k) to account for the additional year of compounding or discounting • Present value of an annuity due: N = 3, i/y = 8%, PMT = 100, FV = 0, CPT PV = -257.71 (1.08) = -278.33

Amortized Loans • A loan that is paid off in equal amounts that include principal as well as interest. • Solving for loan payments (PMT). • Note: The amount of the loan is the present value (PV)

0 1 2 3 4 5 $5,000 $? $? $? $? $? N I/YR PV PMT FV Amortized Loans • You borrow $5,000 from your parents to purchase a used car. You agree to make payments at the end of each year for the next 5 years. If the interest rate on this loan is 6%, how much is your annual payment? ENTER: N = 5 I/YR = 6 PV = 5,000 FV = 0 CPT PMT = ? –1,186.98 5 6 5,000 ? 0

Compounding more than once per Year • If m = number of compounds, then N = n x m and K = k / m • Annual i.e. N = 4 K = 12% • Semi-annual N = 4 x 2 = 8 • K = 12% / 2 = 6% • Quarterly N = 4 x 4 = 16 • K = 12% / 4 = 3% • Monthly N = 4 x 12 = 48 • K = 12% / 12 = 1%

1 - ) ( = PMT $20,000 1 (1.0075)48 .0075 1 (1+k)n 1 - PVA = PMTx( ) k Amortized Loans • You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? $20,000 = PMT(40.184782) PMT = 497.70 Note: Tables no longer work

N I/YR PV PMT FV Amortized Loans • You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment? ENTER: N = 48 I/YR = .75 PV = 20,000 FV = 0 CPT PMT = ? – 497.70 Note: N = 4 * 12 = 48 I/YR = 9/12 = .75 48 .75 20,000 ? 0

PMT k PVP = Perpetuities • A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity.

PMT k PVP = Perpetuities • A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity (i.e., retirement payments) If k = 8%: PVP = $5/.08 = $62.50 Proof: $62.50 x .08 = $5.00 Example:A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%?

0 1 2 $200 $230 FV= PV(1+ k)n 1.15 = (1+ k)2 Solving for k Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? 230 = 200(1+ k)2 1.15 = (1+ k)2 1.0724 = 1+ k k = .0724 = 7.24%

N I/YR PV PMT FV 2 ? -200 230 Solving for k - Calculator Solution Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment? Enter known values: N = 2 I/YR = ? PV = -200 PMT = 0 FV = 230 Solve for: I/YR = ? 7.24 0

N = 1.9995, or 2 years N I/YR PV PMT FV Solving for N Example: A $200 investment has grown to $230. If the ANNUAL return on this investment is 7.24%, how long would it take? • Enter known values: • N = ? • I/YR = 7.24 • PV = -200 • PMT = 0 • FV = 230 ? 7.24 -200 0 230

Compounding more than Once per Year • $500 invested at 9% annual interest for 2 years. Compute FV. Compounding Frequency $500(1.09)2 = $594.05 Annual $500(1.045)4 = $596.26 Semi-annual $500(1.0225)8 = $597.42 Quarterly $500(1.0075)24 = $598.21 Monthly $500(1.000246575)730 = $598.60 Daily

The Time Value of Money