Question 7.2: Calculate the present value of a series of payments of $1000......
Calculate the present value of a series of payments of $1000 with the payments made for 5 years at a discount rate of 10%.
The regular payment A is 1000, the rate r is 0.1 and n = 5. The present value of the first payment received is 1000/1.1 = 909.91 at the end of year 1; at the end of year 2 it is 1000/(1.1)² = 826.45; and so on. At the end of year 5, its present value is 620.92. The net present value of the annuity is the sum of the present value of all the payments made over the 5 years, and it is given by the sum of the present values from Table 7.3. That is, the present value of the annuity is 909.91 + 826.44 + 751.31+ 683.01 + 620.92 = $3791.
We may derive a formula for the present value of a series of payments A made over a period of n years at a discount rate of r as follows: clearly, the present value is given by
\frac{A}{(1+r)}\quad+\ \ \ \frac{A}{(1+r)^{2}}\quad+\ \ldots+\ \ \frac{A}{(1+r)^{n}}This is a geometric series where the constant ratio is \frac{1}{1+r} and the present value of the annuity is given by its sum.
PRESENT VALUE OF ANNUITY
The present value of a series of payments of amount A (made at the
end of the compounding period) with interest rate r is given by
For the example above, we apply the formula and get
PV={\frac{100}{0.1}}\left[1\,-\,{\frac{1}{(1.1)^{5}}}\right]= 10000(0.3791)
= $3791
| TABLE 7.3 | |||
| Calculation of Present Value of Annuity | |||
| Year | Amount | Present Value (r = 0.1) | |
| 1 | 1000 | $909.91 | |
| 2 | 1000 | $826.44 | |
| 3 | 1000 | $751.31 | |
| 4 | 1000 | $683.01 | |
| 5 | 1000 | $620.92 | |
| Total | $3791 | ||