## Forward-forward interest rate for periods up to one year

##### Published: Jul 2003

A company with a known future borrowing requirement can protect itself against any interest rate risk by borrowing the required amount today and then placing the borrowed sum on deposit until it is required. Because the act of placing the cash on deposit counteracts the cost of borrowing, this action means that the company is borrowing at one point in the future and repaying on a second date – hence, a forward-forward.

Because the initial cost of borrowing is mitigated by the time on deposit, the treasurer will want to calculate the effective cost of borrowing. This will be done by using two different interest rates – one for the full period of borrowing and one for the shorter period whilst the borrowed cash is on deposit. This is also the basis used by banks quoting forward-forward prices.

$$Forward-forward \:interest \:rate = \frac{1 \: + \: i_1\: \times \:\frac{d1}{year}}{1 \: + \: i_2\: \times \:\frac{d2}{year}} \: – \: 1 \: \times \: \frac{year}{d1 \: -\: d2}$$

where i1 is the interest rate for the longer period, d1 days, and i2 is the interest rate for the shorter period, d2 days.

Consider a company which knows that it has a borrowing requirement of €10m in one month for five months. It has a number of ways to protect itself against the interest rate risk. One solution is to borrow €10m today for six months. This €10m can then be placed on deposit for one month until it is required. Effectively, this means that the company will only start borrowing in one month for five months. In order to calculate the cost of borrowing, the treasurer will use the above formula. If the six month interest rate is 3.8%, the one month interest rate is 4.1% and interest is calculated on an actual/360 day basis, the forward-forward interest rate will be:

$$Forward-forward \:interest \:rate = \frac{1 \: + \: 0.038\: \times \:\frac{183}{360}}{1 \: + \: 0.041\: \times \:\frac{31}{360}} \: – \: 1 \: \times \: \frac{360}{183 \: -\: 31} = 0.03726 = 3.726\%$$
##### Conventional calculator

Using a conventional calculator, press the following buttons:

• 0.038 * 183 / 360 = + 1 = M + C
• 0.041 * 31 / 360 = + 1 = 1 / x * MR = 1 =
• MC * 360 = M + C 183 – 31 = 1 / x * MR = (this should give the result 0.03726)
##### Scientific calculator

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

• 1 + 0.038 * 183 / 360 / 1 + 0.041 * 31 / 360 1 * 360 / 183 / 31 = (this should give the result 0.03726)
##### HP12C

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

• 0.038 ENTER  183 x 360 ÷ 1 +
• 0.041 ENTER  31 x 360 ÷ 1 + 1 / x RCL x 1 360 x
• 183  ENTER  31 – 1 / x RCL x (this should give the result 0.03726)

This equation provides a check on forward-forward prices, which should be cheaper because of small differences between the borrowing and lending rates.