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# Chapter 4 Time Value of Money (cont.)

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1. Present value of multiple cash flows Nominal interest rate and real interest rate Effective interest rate Chapter 4 Time Value of Money(cont.)

2. Multiple Cash Flows • Usually an investment involve multiple/a stream of (negative/positive) cash flows instead of just one payment and one initial investment. One term deposit Several term deposits that end at the same time Several withdrawals out of one deposit

3. FV of Multiple Cash Flows • The future value of several cash flows paid (or several cash flows received) at a certain point of time can be calculated by adding up the future values of each of the cash flows. • N specifies how many periods away from now is the FV that we want to calculated. • Ct denotes the actual cash flow that is paid/received at the end of the tth period.

4. FV of Multiple Cash Flows Example: If you make one term deposit of \$300 now and another 2 of \$200 at the end of each of the following two years, and all the deposit expires at the end of the 4th year from now. Interest rate is 8%. How much will your bank account balance be? (draw a time line and assign values to variables in the formula)

5. FV of Multiple Cash Flows Example: (cont.)

6. PV of Multiple Cash Flows • The present value of several cash flows paid (or several cash flows received) in future can be calculated by adding up the present values of each of the cash flows. • Ct denotes the actual cash flow that is paid/received at the end of the tth period.

7. PV of Multiple Cash Flows Example: If you need to make 3 payments at different point of time: one of \$250 now, a second payment of \$300 at the end of next year (the first year) and a third one of \$500 at the end of the year after next (the second year) . Interest rate is 8%. How much money should you have in your bank account now so that you would be able to make all the three payments at the specified time? (draw a time line and assign values to variables in the formula)

8. PV of Multiple Cash Flows Example: (cont.)

9. Multiple Cash Flows • Using financial calculators: • Calculate the FV/PV of each cash flows independently then sum the results together • Make sure the correct t (i.e. N) is used for each cash flow • When there are several cash flows paid and also several cash flows received, the formula to be used are the same: • Make sure the correct sign is given to each cash flow

10. Perpetuities & Annuities Perpetuity: A stream of level cash payments that never ends. Annuity: Equally spaced level stream of cash flows for a limited period of time.

11. Perpetuities Assume: • Deposit \$100 • Annual interest rate is 8% and it never changes • Interests are withdrawn at the end of every year but never the principal Cash flows: • Pay \$100 now • Receive \$8 at the end of every year forever

12. Perpetuities PV of Perpetuity: the value of all future cash flows from a perpetuity in terms of a one time payment now Formula: for a perpetuity whose cash flows occur at the end of every period starting from now. C = cash payment r = interest rate / discount rate

13. Perpetuities Example - Perpetuity In order to create an endowment, which pays \$100,000 per year, forever, how much money must be set aside today if the rate of interest is 10%?

14. Perpetuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?

15. Annuities • Annuity can be viewed as the difference between two perpetuities

16. Annuities PV of Annuity: the value of all future cash flows from an annuity in terms of a one time payment now Formula: for an annuity whose cash flows occur at the end of every period starting from now and lasting for t periods. C = cash payment every period r = interest rate t = number of periods cash payment is received

17. Annuities PV Annuity Factor (PVAF) - The present value of \$1 a year for each of t years. [Table A.3 on page 704 ] • Find the appropriate PVAF according to the right t and r

18. Annuities Example - Annuity To purchase a car, you are scheduled to make 3 annual installments of \$4,000 per year starting one year from now. Given a rate of annual interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?

19. Annuities • Example – Annuity (cont.)

20. (Ordinary) Annuity and Annuity Due

21. Annuity Due Calculation • Adjust your financial calculator • Switch from “End” to “Begin • The inputs are the same as an ordinary annuity • Example: start paying the installments right now

22. Switch From “End” to “Begin” • HP Press {shift} (i.e. the yellow button) and then press {BEG/END} • TI • Press {2nd}, then {BGN} • Press {2nd}, then {SET} • Press {2nd}, then {QUIT} • To switch back from “Begin” to “End”, just repeat the procedure

23. Annuity Due Calculation (cont.) • PV of and annuity due equals the multiple of the PV of the ordinary annuity and (1+r) • Both annuities have the same annual payment and number of periods • Example: start paying the installments right now • Calculate the PV of corresponding ordinary annuity • Multiply by (1+r)

24. Annuities Applications • Present Value of payments • Implied interest rate for an annuity • Calculation of periodic payments • Mortgage payment • Annual income from an investment payout • Future Value of annuity

25. Present Value of payments • Example: In 1992, a nurse in a Reno casino won the biggest jack pot - \$9.3 million. That sum was paid in 20 annual installments of \$465,000. What is the PV? r=10% (draw a time line and assign values to variables in the annuity formula)

26. Present Value of payments

27. Home Mortgages • Example:Suppose you are buying a house that costs \$125,000, and you want to put down 20% (\$25,000) in cash. Assume that the mortgage loan lasts 30 years, i.e. 360 months. What will be your monthly payment for each option, if the monthly interest rate is 1%? (draw a time line and assign values to variables in the annuity formula)

28. Home Mortgages

29. Future Value of Annuity Example - Future Value of annual payments You plan to save \$4,000 every year for 20 years starting from the end of this year, and then retire. Given a 10% rate of interest, what will be the balance of your retirement account in 20 years?

30. Future Value of Annuity

31. Inflation Inflation: Rate at which prices as a whole are increasing. • Consumer price index, CPI Real Interest Rate: Rate at which the purchasing power of the return of an investment increases. • Real value of money Nominal Interest Rate: Rate at which money invested grows. • Nominal value of money • The quoted interest rate

32. Inflation • Exact formula • Approximation formula

33. Inflation • Let r= real interest rate, i=inflation rate, and R= nominal interest rate.

34. Inflation Example 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?

35. Effective Interest Rates • Effective Annual Interest Rate - Interest rate that is annualized using compound interest. • Give the actual annual interests • Annual Percentage Rate - Interest rate that is annualized using simple interest. • Only a way to quote interest rates • Imposed by legal requirements

36. Effective Interest Rates Example Given APR of 12% and monthly compounding, what is the Effective Annual Rate(EAR)? • First, calculate month interest rate • Then, calculate the annual rate after compounding

37. Amortizing Loan • Mortgage Amortization (page 88) • Periodic Payment = Amortization + Periodic Interest • Periodic Interest = interest rate * prior period loan balance Example: pay off 100,000 mortgage loan in 360 months at interest rate of 1% per month

38. Amortizing Loan Summary: • Each periodic payment include amortization and interests due. • As the loan approaches maturity, the amortizations paid increase every period. • As the loan approaches maturity, the loan balances and interests due decrease every period. • The last amortization is just enough to payoff the last part of principal.

39. Problem 25 on page 108 (4/e 24 on page 105) Annuity Values You want to buy a new car, but you can make an initial payment of only \$2,000 and can afford monthly payments of at most \$400. • If the APR on auto loans is 12% and you finance the purchase over 48 months, what is the max price you can pay for the car? • How much can you afford if you finance the purchase over 60 months?

40. Problem 28 on Page 109 (Problem 27 on Page 105) • Rate on a Loan If you take out an \$8,000 car loan that calls for 48 monthly payments of \$240 each, what is the APR of the loan? What is the EAR?

41. Problem 37 on Page 109 (Problem 36 on Page 106) Amortizing Loan You take out a 30-year \$100,000 mortgage loan with an APR of 6% and monthly payments. In 12 years you decide to sell your house and pay off the mortgage. What is principal balance on the loan