Effective Discount Rate

Study guide: Exam FM-2

The effective discount rate d measures interest paid at the beginning of the period rather than the end: it is the amount of discount during the period divided by the amount at the end of the period. On a loan of 1 repayable at the end of one period, the borrower pays d upfront and has the use of only 1-d:

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

= iv

= 1 - v

• i is the effective Interest Rate and v is the Discount Factor.
• The relationship between i and d is:

1 - d = v

= \frac{1}{1+i}

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

• Under compound discount, 1 today accumulates to \frac{1}{1-d} after one period and to (1-d)^{-t} after t periods.

Interest Paid in Advance on a Loan A lender makes a one-year loan with 5{,}000 repayable at the end of the year, at an effective annual discount rate of 6\%. How much does the borrower receive today, and what effective annual interest rate is the borrower paying?

Answer The discount of 0.06 \times 5000 = 300 is taken upfront, so the borrower receives 5000 - 300 = 4700. The interest of 300 is earned on the 4700 actually lent:

\begin{align*}
i &= \frac{300}{4700} \\
  &= 0.063830 = 6.383\%
\end{align*}

The same rate follows from i = \frac{d}{1-d} = \frac{0.06}{0.94} = 0.063830. Collecting interest in advance makes a 6% discount rate cost the borrower 6.383% effective.

Bank Discount on a Treasury Bill A 91-day T-bill with face value 10{,}000 is purchased at a bank discount rate of 4% (simple discount, actual/360 basis). Find the purchase price, the effective annual interest rate, and the effective annual discount rate.

Answer Discount = 10000 \times 0.04 \times (91/360) = 101.11. Price = 10000 - 101.11 = 9898.89.

\begin{align*}
i &= \left(\frac{10000}{9898.89}\right)^{365/91} - 1 \\
  &= (1.0102143)^{365/91} - 1 \\
  &= 0.041604 = 4.160\%
\end{align*}

\begin{align*}
d &= \frac{i}{1+i} \\
  &= \frac{0.041604}{1.041604} \\
  &= 0.039942 = 3.994\%
\end{align*}

The quoted 4% is a simple bank-discount rate, not the effective discount rate d: the effective annual discount rate is 3.994%.