Discount Function

The discount function $v(t)$ is the Present Value at time 0 of 1 unit due at time $t$ . It is the reciprocal of the Accumulation Function $a(t)$ .

$v(t) = \frac{1}{a(t)}$

Under a constant effective rate $i$ , $v(t) = (1+i)^{-t}$ , with one-period factor $v = (1+i)^{-1}$ .
Under a constant Force of Interest $\delta$ , $v(t) = e^{-\delta t}$ .
Multiplying a future cash flow by $v(t)$ gives its present value — the basic building block of annuity and bond valuation.

Present Value of a Future Payment At an annual effective rate of $i = 5\%$ , what is the present value of $ $1{,}000$ due in 8 years?

Answer $v(8) = (1.05)^{-8} \approx 0.67684$ $PV = 1{,}000 \times 0.67684 = \$676.84$