The correct way to calculate the price of treasury bills

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

21 December 2008

Kenya is slowly becoming a mathematical society. I say this because major institutions have taken to publishing formulae in the general press. Last month there was one from the Energy Regulatory Commission (ERC) proposing a method of calculating the maximum retail price of petroleum. Then last week we saw another one from the Central Bank of Kenya (CBK) showing how to calculate the price of a Treasury Bill (T-Bill).

The problem with T-Bills is that they are sold at a discount. This means that the interest earn is not paid separately at the end of the investment period. It is deducted at the time of purchase. An investor may put in, say, Sh98,000 and get a T-Bill of Sh100,000 which matures after three months.

In such a case, what is the percentage interest earned per year? It is quite straightforward to see that the Sh98,000 has “given birth” to an additional Sh2,000. Thus the earning rate for three months is 2.04 percent. This works out to 8.16 percent per annum.

The reason we want to get the rate per annum is that other investment avenues quote their returns in percent per year, thus we are able to make a quick comparison before investing.

When buying T-Bills, however, investors are required to quote the price they will pay for every sh100. So the question they are faced with is this: if you want to earn X percent per annum, how much should you pay per Sh100?

Suppose X is 10 percent and the investor wished to put money in a 91-day T-Bill (approximately three months). Here are the steps to follow

The starting point is to convert the annual rate to a 91-day rate. This is done by simply dividing the rate by 365 (number of days in a year) and then multiplying the result by 91. In the case of 10 percent, the answer is 2.49 percent in 91 days.

It is tempting to subtract the 2.49 from 100 to get 97.51 and then quote sh97.51 per sh100 of face value, but that would be wrong. The reason is that if you increase sh97.51 by 2.49 percent, the result is NOT sh100 (try it and see!)

Now, increasing a number by 2.49 percent is equivalent to multiplying it by 1.0249. Thus to get the correct price we divide sh100 by 1.0249. The answer is sh97.57.

Finally, we calculate the withholding tax on the interest earned at the rate of 15 percent. In this example, the gross earning is sh100 minus sh97.57, which equals sh2.43. The tax on this is, sh0.3645 per sh100.

This tax is added to the purchase price to get the amount paid to the CBK for the T-Bill. The result is sh97.9345 per sh100. Thus to buy a 91-day, sh100,000 T-Bill at 10 percent per annum, the investors would pay sh97,934.50. At the end of the period, she gets sh100,000 back making a profit of sh2,065.50