# What is the time value of money

## Finance Chapter 4: The Time Value of Money

Transcript

1 Chapter 4: The time value of money by summer semester 2010

2 Fundamentals of investment theory Every investment project can be viewed abstractly as a temporal distribution of cash flows. Time of cash in / out flow An investment that begins with a payout and only generates deposits in the following periods is called a normal investment (there is only one change in sign in the cash flow profile). The amounts must not be added together directly: giving up capital today is not free. The price of money is the interest rate.

3 Fundamentals of investment theory Interest rate: Percentage amount of a loaned principal. Usually notated in decimal notation. Interest factor: 1 + interest rate Simple interest Interest that is only earned on the original investment amount. Compound Interest Interest earned on interest.

4 Fundamentals of investment theory Future value Amount to which an investment, including interest, grows after a certain period of time (usually the end of the company associated with the investment). Present Value Value that a future payment has today. With the help of the interest calculation, final values can be converted into present values.

5 Future Values Example of simple interest and taking compound interest into account Interest earned on a sum of 100 invested for 5 years at 6% interest. Time cash flow value of the investment with simple interest rate of the investment when taking compound interest into account, 36 119,,, 822558

6 End values (future values) The end value or future value of $ 100 is calculated taking into account compound interest according to: FV $ 100 (1 + r)) t See example: FV $ 100 (1 +.06) 5 $

7 End values at compound interest 7000 interest rates: FV of $% 5% 10% 15% End values after 30 years 0% 100 5% 432,% 1744,% 6621, Number of Years

8 The Sale of Manhattan Island Peter Minuit bought Manhattan Island in 1626 for $ 24 from the Indians living there. To answer if the business was worth it, one needs to calculate the final Ed value of $ 24 in 2003. An annual interest rate of 8% is assumed. 377 FV $ 24 (1 +.08) $ Problems: Find the right interest rate and consider future cash flows.

9 Present Values Present value: The current value of a future amount of money. Discount rate: Interest rate used to calculate the present value of future payments. Discount factor: The present value of $ 1 future payment PV value after t (1 + - r) periods t

10 Present Values Example You bought a new computer today. The payment terms stipulate that you have to pay 3000 in 2 years. How much money do you have to invest today to make the payment due in two years at an interest rate of 8%? PV (1.08) $ 2,572

11 Present Values Discounting factors The discounting factor (DF) multiplied by the future value of a cash flow results in its present value. It depends on the length of time and the interest rate. DF 1 (1 + r) t

12 The Time Value of Money The present value formula has many uses. If you have the details except for one, you can solve the equation for the remaining variables. PVFV 1 (1 + r) t

13 Present Value of Multiple Cash Flows Example Your car dealer gives you a choice of paying $ 15,500 in cash today or making three payments of $ 8,000 now and $ 4,000 each at the end of the next two years. If your money is 8% what will you prefer? 8,000 PV 0 Payments today PV PV 1 2 4,000 () 1 4,000 () Total PV 2 3,, $ 15, () 8,000

14 Present value of several cash payments Present values can be added to evaluate the value of several cash payments. Of course, cash values must be based on the same point in time! PV C 1 + C (1 + r) 1 (1 + r) 2

15 Special Cases of the Cash Flow Profile: Perpetuities & Annuities Perpetuity: A stream of equal, equidistant cash payments that never ends. Annuity: A steady stream of cash payments for a limited number of periods.

16 Perpetuities & Annuities Assumption: Today you deposit 100 into a call money account and receive 3% interest p.a. As long as you do not withdraw the capital, you will receive 3. bank insolvency every year, excluding changes in interest rates. At an interest rate of 3% p.a., the 100 can then be interpreted as the present value of a perpetual annuity of 3. Therefore, the present value of a perpetual annuity results as: PV C r C Cash payment r Interest rate

17 Perpetuities & Annuities Example Perpetual Annuity In order to achieve an endowment that makes $ 100 per year forever, how much money has to be put aside today if the interest rate is 10%? PV 100, $ 1,000,000

18 Perpetuities & Annuities Example continued If the first payment of the perpetual annuity is not to begin for three years from now, how much money do you have to put aside today? PV 1,000,000 $ 751,315 () 3

19 Perpetuities & Annuities The present value of an annuity is calculated as follows: Time PV of the perpetual annuity PV of a delayed perpetual annuity PV of the annuity [PV C 1 1 rr (1+ r) t] The r PV PV PV C r C r ( 1+ r) 3 CC rr (1 + r) The present value of an annuity is calculated as follows: C Cash payment Interest rate t Number of years in which the cash payment is due 3

20 Perpetuities & Annuities Present Value Annuity Factor (PVAF): The present value of one dollar per year for each of t years. [] PVAF 1 r 1 r (1+ r) t

21 Perpetuities & Annuities Example of an annuity: You buy a new car. You are required to make three equal payments of $ 4,000 per year each year. What is the price / present value of the car at an interest rate of 10% today? PV 4, PV $ 9, [] () 3

22 Perpetuities & Annuities Examples of uses of the formula: Value of payments Implicit interest rates on an annuity Calculation of periodic payments Mortgage payment Annual income from an investment The final value of annual payments FV [C PVAF] (1+ r) t

23 Perpetuities & Annuities Example: The Ending Value of Annual Payments You plan to save $ 4,000 each year for 20 years before you retire. At 10% interest, what will the value of your account be when you retire in 20 years? 1 FV 4.000 1 []] () (1 + .10) 20 FV $ 229, (1 10) 20

24 Annuities and perpetual annuities In Germany, the following terms had become established: Annuity * Present value factor for annuities Annuity recovery factor * Annuity present value * End value factor End value end value * Repayment factor annuity ATTENTION: So far we have considered excess cash flows. In the case of direct / advance payment consequences, each payment must be multiplied by 1 + r (annuity due!)

25 Inflation Inflation rate at which prices rise in an economy / currency area. Nominal interest rate Rate at which the invested money grows nominally. Real interest rate Rate at which the purchasing power of the money invested increases. There is a connection between nominal and real interest rates! This was formulated by I. Fisher to explain the Fisher effect:

26 Inflation 1 + real interest rate 1 + nominal interest rate 1 + inflation rate Approximation: 1 + R Real interest rate 1 + N Nominal interest rate (expected) tt) iflti Inflationsratet Dr. Tobias Effertz

27 Inflation Example i If the interest rate on one-year government bonds is 5.0% and the inflation rate is 2.2%, what is the real interest rate? 1 + real interest rate real interest rate real it interest t rate. 027 or 27% 2.7% Approximation i or 28% 2.8% Dr. Tobias Effertz

28 Effective Annual Interest Rate: Annual interest rate taking compound interest into account g of the period-related individual interest Example: 1% per month 1.01 ^ 12 Effective annual interest rate 12.68% Annual Percentage Rate Annual interest rate without taking compound interest effects into account. (US specialty) Example 1% per month 12%

29 Effective interest rate Credit institutions in particular have the additional option of controlling the effective interest rate on a receivable / loan through the so-called premium or discount (also damnum). In order to protect creditors / investors / borrowers, credit institutions are obliged to indicate or disclose the effective interest rate in order to avoid a (possible) misorientation of the investor / borrower. 492 I, 5 BGB

30 Effective interest rate The effective interest rate can also be changed by paying interest more frequently:

31 Example of effective annual interest rate The Lüneburger-Krösus-Bank grants a loan of 5000, which is to be repaid in 24 monthly installments of 217.82 each. What is the monthly interest rate and the effective two-year interest rate?

32 Example of effective annual interest rate 2 The Lüneburger-Krösus-Bank grants a loan of 5000, which is to be repaid in 24 monthly installments of 217.82 each. The effective annual interest rate is specified as 4.4%. What is the monthly interest rate and the effective two-year interest rate? 1 1,,,, The monthly interest is 0.3594%, the effective two-year interest is 8.9936. The final value is 5449.68.

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