1 Chapter 7 The Time Value of Money

2 Time Value A. Process of expressing 1. The present value of $1 invested now in future terms. (Compounding) Compounding – Process by which interest is paid on interest that was previously earned. 2. The future value of $1 invested in terms of the present Future Value of a dollar – Amount to which a single payment will grow at some rate of interest. B. Payments are either: 1. a single payment 2. A series of equal payments (an annuity)

3 Time Value C.Time value of money problems may be solved by using: 1. Interest tables 2. Financial calculators 3. Software

4 Variables for Time Value of Money Problems  PV = present value  FV = future value  PMT = annual payment  N = number of time periods  I = interest rate per period

5 Future Value  Future value of $1 takes a single payment in the present into the future.  General equation for the future value of $1: P 0 (1 + i ) n = P n FV = PV (1 + i) n

6 Future Value Illustrated  PV = -100  I = 5  N = 20  PMT = 0  FV = ?  =

7 Greater Terminal Values  Higher interest rates  Longer time periods  Result in greater terminal values

8 Greater Terminal Values

9 Present Value  Present value of $1 brings a single payment in the future back to the present. Present Value – Current value of a dollar to be received in the future. Discounting – Process of determining present value.

10 Present Value  General equation for the present value of $1: P 0 = P n (1+ i ) n PV = FV [ 1 / (1 + i) n ] 1. PV = Present Value 2. FV = Future Value 3. i = Interest Rate per Period 4. n = Number of Compounding Periods

11 Present Value Illustrated  FV = 100  I = 6  N = 5  PMT = 0  PV = ?  =

12 Lower Present Values  Higher interest rates  Longer time periods  Result in lower present values

13 Lower Present Values

14 Financial Calculators and Excel  Express the cash inputs (PV, FV, and PMT) as cash inflows and cash outflows  At least one of the cash variables must be –an inflow (+) –an outflow (-)

15 Simple Interest  Simple Interest – No Compounding –Good to have if you withdraw interest each period. –SI = Principal (PV) x rate (i) x time (n)

16 Future Value of a Single Amount Example  You buy a stock for $10 and expect the price to increase 9 percent annually. After 10 years, what is the anticipated price of the stock?

17 Future Value of a Single Amount Example  The unknown: FV  The givens: –PV = 10 –PMT = 0 –N = 10 –I = 9 The answer: $23.67

18 FV Interpretation  A $10 stock will be worth $23.67 after 10 years if its price grows 9% annually.

19 Present Value of a Single Amount Example  What is the cost of a stock that was sold for $23.67, held for 10 years and whose value appreciated 9 percent annually?

20 Present Value of a Single Amount Example  The unknown: PV  The givens: –FV = –PMT = 0 –N = 10 –I = 9 The answer: $10

21 Interpretation  $23.67 received after ten years is worth $10 today if the rate of return is 9 percent.

22 Interpretation of Future and Present Values These two problems are mirror images:  In the first case, the $10 is compounded into its future value ($23.67).  In the second case, the future value ($23.67) is discounted back to its present value ($10).

23 Rate of Return Example  A stock was purchased for $10 and sold for $23.67 after 10 years. What was the return?

24 Future Determination of the Interest Rate (can use Present too)  The unknown: I  The givens: –PV = 10 –PMT = 0 –N = 0 –FV = The answer: 9%

25 Interpretation  The yield on a $10 investment that was sold after 10 years for $23.67 is 9%.

26 Non-annual Compounding  More than one interest payment a year  State Interest rates are always annual interest rates.  More frequent compounding

27 Non-annual Compounding  Multiply number of years by frequency of compounding  Divide interest rate by frequency of compounding

28 Periods less than One Year  Same variables as in all time value problems except N < 1.  Calculate by dividing the number of days by 365.

29 Illustration for Return on Investment  What is the return on an investment that costs $98,543 and pays $100,000 after 45 days?

30 Determination of Return  The unknown: I  The givens: –PV = -98,543 –N = (45/365) –FV = 100,000 –PMT = 0 The answer: 12.64%

31 Interpretation  $98,543 invested for 45 days grows to $100,000 at percent.