Effective Rate of Discount

• Recall the effective rate of interest is equal to the amount of interest if it was just one installment at the end of the period
• The effective rate of discount is when the interest is paid at the beginning of the period
  • Useful for loans and treasury bills
• “A loan of $1200 is made for one year at an effective rate of discount of 5%”
  • The borrower pays $1200 \times 0.05 = 60$ (known as the discount amount) at the beginning to be able to use the money
  • The borrower pays back the $1200 loan after the year is over
• If $k$ is borrowed at a discount rate of $d$, the discount amount is $kd$ to receive the use of $k$
  • This means the borrower will only have $(k-kd) = k(1-d)$ to use from the loan amount after paying the interest upfront
• Effective rate of discount is therefore:
  • $A(n)$ is the amount at the end of the $n$th period
  • $D$ is the discount amount within the interval, which is just equal to $A(n)-A(n-1)$

Interest Vs Discount Model

• Interest model
  • Payment for loan is made at the end of the interest period
  • The borrower can make full use of the loan amount
  • The effective rate is calculated based on the initial balance
• Discount model
  • Payment for loan is upfront
  • Borrower can make use of $k(1-d)$ of the loan
  • Effective rate calculated from final balance

Relationship Between Effective Rates

and

• We can use this relationship to derive the following:

and

and

Compound Discount Function

• Theorem: $d_{n} = d$ for all $n \ge 1$

Simple Discount Function

• Theorem: $d_n$ is increasing until $t = \frac{1}{d}$

Growth of Simple and Compound Discount Functions

• $(1-d)^t < 1 - dt, t \in (0, 1)$
• $(1-d)^t = 1-dt, t \in \{0, 1\}$
• $(1-d)^t > 1-dt, t > 1$