Present Value of Single Cash Flow for Multiple Periods with Multiple Interest Rates
Present Value with a Single Cash Flow:
Present value (PV) is the current value of a single future cash flow discounted at the appropriate discount rate.
Time value of money for a single cash flow is a cash inflow or outflow that investors or lenders received from or paid to once respectively for specific periods
There are often four parts to equation (time value of money for a single cash flow): the present value (PV), the future value (FV), the discount rate (r), and the number of periods of the investment (t). If three of these (FV, r and t) are given, so we can find the present value (PV).
Multiple-Period Investment:
Multiple-period investment may be investment for more than one year. Term multiple-period can refer to more than one day or one month investment, so it isn’t always more than one year investment.
If period investments are based on one day, one month or one year etc., so discount rates are based on one day, one month or one year respectively.
Definition of Multiple Interest Rates :
We can calculate present value with different interest rates or growth rates for different periods.
Formula of Present Value with multiple discount rates for Multiple-Period Investment:
FV = PV(1 + r1 ) (1 + r2) (1 + r3) . . . (1 + rt)
So PV=FV/(1+r1)(1+r2)(1+r3)…(1+rt)
where
PV: Present Value or principal is worth today
FV: Future value is worth in the future
r1,r2,r3…,rt : Interest rate, rate of return, or discount rate per period (but not always one year ) for period 1, 2, 3,…,t respectively
Question
Suppose you want get $26,618 after three years, and investment at bank will be paid 9% during the first year, 10% during the second year and 11% during the third year.
What is the present value of investment ?
Solution
PV=$26,618/(1+9%)x(1+10%)x(1+11%)=$20,000
You invests $20,000 today, so you will get $26,618 after three years from bank.