Discount Rate

Study guide: Exam FM-2

The effective annual discount rate $d$ is the interest paid at the beginning of a period on a loan of 1, rather than at the end. If you borrow 1 today and repay 1 at year-end, the interest charge of $d$ is deducted upfront so you receive only $1-d$ now.

$d = \frac{i}{1+i}$

$= 1 - v$

The key relationships among $d$ , the effective rate $i$ , and the discount factor $v = (1+i)^{-1}$ are:

$d = iv$

$i = \frac{d}{1-d}$

Because interest is collected at the start rather than the end, $d < i$ for any positive interest rate.
The nominal discount rate convertible $m$ -thly, $d^{(m)}$ , satisfies $\left(1 - \frac{d^{(m)}}{m}\right)^m = 1 - d = v$ , and as $m \to \infty$ , $d^{(m)} \to \delta$ .

Finding the Discount Rate from an Effective Rate The effective annual interest rate is $i = 8\%$ . Find the effective annual discount rate $d$ and verify the relationship $d = iv$ .

Answer $d = \frac{i}{1+i} = \frac{0.08}{1.08} \approx 0.07407 = 7.407\%$ Check: $v = 1/1.08 \approx 0.92593$ , so $iv = 0.08 \times 0.92593 \approx 0.07407$ .