## Bond Pricing – The Time Value of Money

##### Published: Jun 2002

The key to bond pricing is the concept of the time value of money. In Treasury Today April 2001, we showed how to calculate the total proceeds of a long term investment. We showed that if we invested £100 for 10 years at an annual rate of 8%, we would receive £215.89 at the end of the ten year investment period. This assumes that the interest earned annually would be reinvested in the same vehicle as the original investment. The initial £100 is the present value of the investment and the £215.89 is the future value of the investment.

When making an investment an investment such as this, the investor will know the present value of the sum to be invested, the time period for which the investment will be made and the interest rate that will be applied. The unknown is the future value.

In contrast, the potential investor in a bond will know the future value of the bond, which is the payment due on maturity, and will want calculate its present value, given the time until the bond’s maturity and the current market interest rate.

The following formula links present value (PV), future value (FV), the interest rate (i) and the length of time to maturity in years (n):

FV = PV x 1 + in

This is the formula that we used to calculate the total proceeds of our £100 ten year investment. This formula can be rearranged so that we can calculate the present value of the future proceeds:

$$PV = \frac{FV} { 1 \: + \: in}$$

So, to calculate the present value of £200 in ten years time, assuming an interest rate of 8%:

$$PV = \frac{200} {1 \: + \: 0.08^{10}} = 92.64$$

In other words, an investment of £92.64 now will generate total proceeds of £200 in ten years time, assuming interest is earned annually at a rate of 8% and the interest is reinvested at that rate.

In order to calculate this:

##### Conventional calculator

Using a conventional calculator, press the following buttons:

• 1 + 0.08 = M + C
• MR * MR * MR * MR * MR * MR * MR * MR * MR * MR =
• 1 / * 200 =this will give the result  92.64

##### 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 + 0.08  x ^ 10 = 1 / x * 200 = this will give the result  92.64

##### HP12C

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

• 0.08 ENTER
• 1 +
• ENTER  10yx
• 1 / x 200 x this will give the result  92.64

The HP12C also has a time value of money function, which shortcuts the computation process for more complex calculations:

• 10 n  8 i 200  FV PV this will give the result 92.64

For short term investments (those under a year), the formula link between PV and FV is as follows:

$$PV = \frac{FV}{ 1 \: + \: i \:* \:\frac {number\: of \:days\: until\: payment}{number\: of\: days\: in \:the\: year}}$$