## Interpolating interest rates

##### Published: Apr 2002

Interest rates are usually quoted for standard periods – one month, two months, three months and six months. In order to calculate an interest rate for an interim period, you have to interpolate a rate from the two nearest given rates. The interpolation assumes that the interest rate increases or decreases uniformly from one date to the next – in other words, the relationship is a straight line.

### To interpolate the interest rate

$$Interest\: rate\: r = r_a \: + \: r_b – r_a\: \times \: {\frac{d \: – \: \alpha}{b \: – \: \alpha}}$$

Where:

• r is the interest rate applicable for d days
• rais the interest rate known for a days

And:

• rb is the interest rate known for b days.

So, if we wanted to apply an interest rate for 68 days, we would use the quoted rates for two months (61 days) and three months (92 days).

Assuming that the two months rate was 6.4% and the three months rate was 6.5%, we would calculate the interest rate for 68 days as follows:

• $$R = 6.4 \: + \: 6.5 \:-\:6.4 \times \: \frac{ 68 \: – \: 61}{92 \: – \: 61}$$
• $$= 6.4 \: + \: 6.5 \: – \: 6.4 \: x \: \frac{7}{31}$$
• = 6.4 + 0.023
• = 6.423
##### Conventional calculator

Using a conventional calculator, press the following buttons:

• 7 ÷ 31 = M + C
• 6.5 – 6.4 = x MR + 6.4 = this should give the result 6.423
##### Scientific calculator

Using the scientific calculator on a Windows computer (Start, Programs, Accessories, Calculator, View, Scientific), you would need to press the following keys:

• 6.4 + 6.5 – 6.4 * 7 / 31 = this should give the result 6.423
##### HP12C

Using an HP12C (or a similar calculator using Reverse Polish Notation):

• 6.5 ENTER 6.4 – 7 x 31 6.4 + this should give the result 6.423

The calculation can also be seen graphically: