Time Value of Money (TVM)

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1 BUSI Financial Management Time Value of Money 1 Time Value of Money (TVM) Present value and future value how much is \$1 now worth in the future? how much is \$1 in the future worth now? Business planning compare current to future cash flows is investment in a particular project worthwhile? is one project a better investment than another? Personal financial planning should you consume today or invest and consume later? how will your money grow over time? how much do you need to save today to retire later? 2 Professor Gordon Phillips 1

2 \$1 Today versus \$1 Later Interest rates: the supply and demand of money must be equal the value of being patient; the cost of impatience depends on horizon (maturity) reflects expected inflation and a real return What about \$ 1 for certain in five year s time versus \$1 on average in five year s time? Discount rates reflect both the riskless time value of money as well as the riskiness of the cash flows 3 What techniques will we review? What is the value today of \$C T at time T? What is the value at time T of \$C 0 today? Compounding Effective annual interest rate (EAIR) vs. Stated annual interest rate (SAIR) Continuous vs. discrete compounding Multiple cash flow streams perpetuity growing perpetuity annuity 4 Professor Gordon Phillips 2

3 Time Value of Money: Notation C 0 C T PV FV \$ invested today \$ received in future Define some terms C T = Future value at time T = FV; or cash flow at T C 0 = Current value = PV; or cash flow at O r = Discount or interest rate T = Number of future time periods 5 Present Value and Discounting Discounting: if we receive \$1 in two years, what is it worth to us today (r = 9%)? Year C \$0.842 \$1 Present value (C0) = \$1 / = \$0.842 The interest rate (9%) is also called the discount rate 6 Professor Gordon Phillips 3

4 Present Value - another example How much would an investor have to set aside today in order to have \$20,000 five years from now if the current rate is 15%? PV \$20, \$ 9, = \$20,000 5 (1.15) 7 Future Value Compounding: how much will \$1 invested today at 9% be worth in two years? The Time Line Year C \$1 \$ Future value (C2) = \$1 x = \$ Professor Gordon Phillips 4

5 Compounding Compounding: interest on interest Benjamin Franklin: Money makes money, and the money that money makes makes more money Which one would you prefer? 1. simple interest: no compounding FV T = PV 0 (1 + r*t) 2. annual compound interest FV T = PV 0 ( 1 + r ) T 3. Semi-annual compound interest (bonds and banks) FV T = PV 0 ( 1 + r/2 ) 2T, where r is annualized,t in years 4. continuous compound interest: compound every second FV T = PV 0 *e rt (to be shown later) 9 Money Makes Money INTEREST RATE EXAMPLE re turn 10% contribution 100 years Simple Inte re st Compound Inte re st \$ \$ \$ \$ \$ \$ \$1, Continuous Compound Inte rest 100 \$ \$ \$ \$ \$ \$ \$ \$2, Professor Gordon Phillips 5

6 Compounding Simple Interest FV of \$ Compound Interest 3. Continuous Compound Interest Years 11 Effective Annual Rates (EAIR) The formula for computing the return when there are multiple compounding periods is: (1+SAIR/m) m = (1+EAIR) m is the number of times interest is paid annually SAIR represents the stated annual interest rate EAIR represents the effective annual interest rate If the SAIR is 12%, how much will I receive at the end of the year if interest is paid monthly vs. semi-annually vs. yearly? 12 Professor Gordon Phillips 6

7 EAIR: Example If you invest \$50 for 1 year at 12% (the SAIR) compounded semi-annually - what is the EAIR on that investment? FV = \$50 (1 + ) = \$50 (1.06) = \$ The EAIR is the annual rate that would give us the same end-of-investment wealth after 3 years: \$ 50 (1 + EAIR) = \$ EAIR: Solution FV = \$ 50 (1 + EAIR) = \$56.18 ( 1+ EAIR) = EAIR =.1236 \$56.18 \$50 So, investing at12.36% compounded annually (i.e. the EAIR) is the same as investing at the SAIR of 12% compounded semiannually. 14 Professor Gordon Phillips 7

8 Continuous Compounding The general formula for the future value of an investment compounded continuously over many periods can be written as: FV = C 0 e rt Where C 0 is cash flow at date 0, r is the stated annual interest rate, T is the number of periods over which the cash is invested, and e is a special number (= ). e x is a key on your calculator, and the inverse function is the natural logarithm (ln). 15 Present Values with Continuous Compounding Two types of compounding - be careful! Discrete Compounding annual or semi-annual most common: Continuous Compounding: very close for small r but not for large r: watch out for those small discrepancies that make your bankers rich! While no bank offers continuously compounded rates of return, continuous compounding is often used in finance because it facilitates calculations. PV PV = FV ( 1+ ) r T FV = e r * T 16 Professor Gordon Phillips 8

9 How long does it take to double your money? If we deposit \$5,000 today in an account paying 10% annually, how long does it take to grow to \$10,000? T FV = C (1 r) \$ 10,000 = \$5,000 (1.10) 0 + T T \$10,000 ( 1.10) = = 2 \$5,000 T ln( 1.10 ) = ln 2 T ln(1.10 ) = ln 2 T = ln 2 ln(1.10) = = 7.27 years Double your money... using continuous compounding Continuous compounding: FV/PV = e rt Double your money FV/PV = 2 e rt = 2 rt = ln (2) (if y=e x, then x = ln(y)) rt =.69 if r=.10, T = 6.9 years (compare to 7.3 on last slide) if r=.20, T = 3.5 years The rate or time required to double your money can be quickly figured out using the Rule of 72 (an approximate rule based on annual compounding). 18 Professor Gordon Phillips 9

10 What Rate Is Enough? Assume the total cost of a college education will be \$50,000 when your child enters college in 12 years. You have \$5,000 to invest today. What annual rate of interest must you earn on your investment to cover the cost of your child s education? About 21.15%. T 12 = C0 (1 + r \$ 50,000 = \$5,000 (1 + r) FV ) (1 + r) 12 = \$50,000 \$5,000 = 10 ( 1+ r) = r = = = Multiple Cash Flows Now, a more realistic and complex setting Most investments generate multiple cash flows over time Special cases: Annuities: same cash flow repeated for a fixed number of periods Perpetuities: same cash flow repeated periodically, forever Growing annuities and perpetuities Objective: understand Discounted Cash Flow (DCF) and Net Present Value (NPV) so as to: value and price securities and companies evaluate projects: capital budgeting 20 Professor Gordon Phillips 10

11 Present and Future Value: Multiple Cash Flows Net Present Value and Investment Projects Net Present Value is a measure of the difference between the market value of an investment and its cost. If the NPV is positive, this indicates a profitable investment (if the NPV < 0, the investment should not be made. Estimating Net Present Value Discounted cash flow (DCF) valuation consists of finding the market value of assets or their benefits by taking the present value of future cash flows, i.e., by estimating what the future cash flows would trade for in today's dollars. 21 Net Present Value (NPV) Some rules for computing NPV: use a simple time line to lay out all the cash flows only do simple addition or subtraction of cash flows if they are from the same time period use an appropriate discount rate: later in the course The general formula for calculating NPV: NPV = -C 0 + C 1 /(1+r) + C 2 /(1+r) C T /(1+r) T C 1 C C CT -C 0 22 Professor Gordon Phillips 11

12 Simple NPV Example Assume r = 10% -100 NPV = /(1+.10) + 35/(1+.10) /(1+.10) 3 = Annuities Series of equal period payments T C 1 = C 2 = C 3 = C 4 = C T For simplicity, we will assume a single rate of interest which allows us to easily use the interest tables. 24 Professor Gordon Phillips 12

13 PV of an Annuity 1 1 PV of annuity: T T C ( 1 + r) PV = = C t t= 1 (1 + r) r NOTE: first payment is received in one year s time Table A.2 in RWJ Appendix shows annuity factor values generalization: RWJ give the formula for a growing annuity (equation 4.15 on page 92) How much are 5 equal annual payments of \$8,190 (starting one year from today) worth today if the annual interest rate is 10%? Answer = \$31, Annuity Practice Problem Installment purchases: you are considering purchasing a house that costs \$70,000 you intend to put 20% down and finance the remainder if the 30 year mortgage rate is 7%, what will be your monthly mortgage payment? Note: we implicitly assume monthly compounding with end of period payments 26 Professor Gordon Phillips 13

14 Amortization Table Initial Data LOAN DATA TABLE DATA House Price \$700, Loan amount: \$560, Table starts at date: 1/1/03 Annual interest rate: 7.00% or at payment number: 1 Term in years: 30 Payments per year: 12 PVIFA First payment due: 1/1/03 PERIODIC PAYMENT Entered payment: The table uses the calculated periodic payment amount, Calculated payment: \$3, unless you enter a value for "Entered payment." After-Tax payment \$2, CALCULATIONS Use payment of: \$3, Beginning balance at payment 1: \$560, st payment in table: 1 Cumulative interest prior to payment 1: \$0.00 Table Payment Beginning Ending Cumulative No. Date Balance Interest Principal Balance Interest 1 1/1/03 560, , , , /1/03 559, , , , /1/03 559, , , , /1/03 558, , , , /1/03 558, , , , /1/03 557, , , , /1/03 557, , , , /1/03 556, , , , /1/03 556, , , , /1/03 555, , , , /1/03 555, , , , /1/03 554, , , , FV of an Ordinary Annuity Future Value T 1 ( 1 + r) FV = C r FV annuity factors in Table A4 of RWJ Note that you can just calculate the PV of the annuity, and then find the FV by compounding the PV forwards (in other words, you don t need to learn yet another formula) You wish to take a year off work in 5 years If you save \$8,190 each year for 5 years, how much will you have saved by the end of the time period if the interest rate is 10%? (assume you begin saving at the end of the year). Hint: see page 30 (take FV of the PV calculated you get \$50K) 28 Professor Gordon Phillips 14

15 Perpetuities A perpetuity is a stream of infinite cash flows with the first cash flow beginning one period from now C C C etc C PV of a perpetuity: PV = r The value of \$10 every year forever if the interest rate is 5% is \$200 (= \$10 /.05) quite simple! 29 Valuing Growing Perpetuities A cash flow stream growing at a steady rate cash flow each year; the first one (in one year s time) is \$60, but cash flow grows at 2% a year if the discount rate is 7%, how much would you be willing to pay? (Answer = \$1200) Present value of growing perpetuity: C PV =, for r > g r g Useful when valuing companies (or equity). 30 Professor Gordon Phillips 15

16 Summary Present values and Future values allow us to easily switch back and forth between cash flows separated in time calculations are mechanical: look for shortcuts, use formulas / tables Simplifications: if possible reduce PV problems to Perpetuity: an everlasting stream of constant cash flows Growing perpetuity: stream of cash flows that grows at a constant rate forever Annuity: a stream of constant cash flows that lasts for a fixed number of periods Growing annuity: a stream of cash flows that grows at a constant rate for a fixed number of periods Important question for the future: where do the cash flows come from? what are their risks? 31 Professor Gordon Phillips 16

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