1. Time Value of Money – Part Two (Ch. 3) 04/12/06

2. Valuing a series of cash flows We use the same approach and intuition as we applied in the previous chapter to value a series of future cash flows. We use the same approach and intuition as we applied in the previous chapter to value a series of future cash flows. However, remember that cash flows at different points in time cannot be compared or aggregated unless they are either discounted or compounded to the same point in time. However, remember that cash flows at different points in time cannot be compared or aggregated unless they are either discounted or compounded to the same point in time. A timeline may be useful in visualizing the cash flow streams. A timeline may be useful in visualizing the cash flow streams.

3. Cash flow types There are four types of multiple cash flow streams that we will consider- There are four types of multiple cash flow streams that we will consider-  mixed (or different) periodic cash flows,  annuities,  perpetuities, and  growing perpetuities

4. Mixed periodic cash flows To find the present value or future value of a different future cash flows, we must treat each cash flow separately, find the present or future value of each cash flow and then add them up. To find the present value or future value of a different future cash flows, we must treat each cash flow separately, find the present or future value of each cash flow and then add them up.

5. Annuities An annuity is a stream of equal cash flows that occurs at regular intervals for a fixed period of time. Defining PMT to be the periodic cash flow, An annuity is a stream of equal cash flows that occurs at regular intervals for a fixed period of time. Defining PMT to be the periodic cash flow, PMTPMTPMTPMT | | | | | | | | 0 1 2 3 4 0 1 2 3 4 For ordinary annuities, cash flows occur at the end of each period whereas for annuities due cash flows occur at the beginning of the period. For ordinary annuities, cash flows occur at the end of each period whereas for annuities due cash flows occur at the beginning of the period.

6. Present value of an ordinary annuity The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [PMT = periodic cash flow; r = Discount Rate; n = Number of years] The present value of an annuity can be calculated by taking each cash flow and discounting it back to the present, and adding up the present values. Alternatively, there is a short cut that can be used in the calculation [PMT = periodic cash flow; r = Discount Rate; n = Number of years] The term in the square brackets is also referred to as the Present Value Interest Factor of an Annuity (PVIFA) The term in the square brackets is also referred to as the Present Value Interest Factor of an Annuity (PVIFA)

7. Annuity, given present value The reverse of this problem, is when the present value is known and the periodic cash flow is to be estimated - PMT(PV, r, n). The reverse of this problem, is when the present value is known and the periodic cash flow is to be estimated - PMT(PV, r, n).

8. Future value of an ordinary annuity The future value of an annuity can also be calculated by taking the future value of each cash flow and adding up the future values. Alternatively, a similar short cut can be used in the calculation. The future value of an annuity can also be calculated by taking the future value of each cash flow and adding up the future values. Alternatively, a similar short cut can be used in the calculation. The term in the square brackets is also referred to as the Future Value Interest Factor of an Annuity (FVIFA) The term in the square brackets is also referred to as the Future Value Interest Factor of an Annuity (FVIFA)

9. Annuity, given future value If you are given the future value, you can calculate the periodic cash flows required to obtain that future value: If you are given the future value, you can calculate the periodic cash flows required to obtain that future value:

10. From ordinary annuities to annuities due To find the future value or present value of annuities due, you To find the future value or present value of annuities due, you –Calculate the present value or future value of the ordinary annuity –Multiply this value by (1+r)

11. Waiting time We can rearrange our FVA equation to solve for the number of periods it will take us to accumulate a certain amount of money at a given interest rate and a given periodic cash flow: We can rearrange our FVA equation to solve for the number of periods it will take us to accumulate a certain amount of money at a given interest rate and a given periodic cash flow:

12. How long will our money last? We can rearrange our PVA equation to solve for the number of periods that a certain amount of money today will last at a given interest rate if we take out periodic cash flows: We can rearrange our PVA equation to solve for the number of periods that a certain amount of money today will last at a given interest rate if we take out periodic cash flows:

13. Interest rate Unlike with our simple PV/FV equation, where we could solve for the interest rate, to solve for the interest rate associated with annuities we can use the equation and trial and error or solve using a financial calculator or spreadsheet. Unlike with our simple PV/FV equation, where we could solve for the interest rate, to solve for the interest rate associated with annuities we can use the equation and trial and error or solve using a financial calculator or spreadsheet.

14. Perpetuity A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is- A perpetuity is a constant cash flow at regular intervals forever. The present value of a perpetuity is-

15. Growing perpetuities A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is - A growing perpetuity is a cash flow that is expected to grow at a constant rate forever. The present value of a growing perpetuity is -where – CF 1 is the expected cash flow next period, –g is the constant growth rate and –r is the discount rate.

16. Loans Typical consumer loans make equal payments every period. A portion of this payment goes towards interest and a portion goes towards reducing the principal of the loan. This is referred to as an amortized loan. Typical consumer loans make equal payments every period. A portion of this payment goes towards interest and a portion goes towards reducing the principal of the loan. This is referred to as an amortized loan. Two other types of acceptable loan structures are: Two other types of acceptable loan structures are: –pure discount loans, where the principal and any accumulated interest is paid at loan maturity, and –interest-only loans, where each year only interest is paid. At maturity, the principal and the last year’s interest is paid.

17. Amortization schedule An amortization schedule provides a tabulation of each amortized loan payment’s application to interest expense and principal reduction. An amortization schedule provides a tabulation of each amortized loan payment’s application to interest expense and principal reduction.

18. Amortization schedule How to build an amortization schedule: How to build an amortization schedule: –Step One – Determine the payment for each period given number of periods (n), interest rate (r), and the present value (PV). –Step Two – Determine each periods required interest payment Outstanding Balance x Interest rate = Interest expense Outstanding Balance x Interest rate = Interest expense –Step Three – Determine principal reduction Payment – Interest expense = Principal Reduction Payment – Interest expense = Principal Reduction –Step Four – Determine remaining principal Beginning of Period Principal – Principal reduction = End of Period Principal Beginning of Period Principal – Principal reduction = End of Period Principal –Step Five – Repeat Steps Two through Four, n-1 times