Over the last three months, we have been examining the following equation, which is developed as a result of some simplifying assumptions and is the one that is usually used in the bond markets:

\( Price = \frac{ 100 \:}{1\: + \: \frac{i}{n}w} \frac{c}{n} \: \times \: \frac{1 \: -\: \frac{1}{1\: + \: \frac{i}{n}t}}{1 \: -\: \frac{1}{1\: + \: \frac{i}{n}}} \: + \: \frac{1}{1\: + \: \frac{i}{n}t\: – \: 1} \)Where i = yield, C is the coupon expressed as an annual rate which is paid n times a year, w is the proportion of the current coupon period remaining and t is the number of outstanding future payments or cash flows.

This equation assumes firstly that future values are compound discounted to a net present value and secondly that there are regular periods between coupon payments.

We started by breaking the equation down into four component parts. In previous issues, we have shown how one component takes into account the time until the next cash flow and how another recognises the amount of all future coupon payments.

This month, we look at the third component, which discounts the principal to a present value. This is the component:

\( \frac{1}{1\: + \: \frac{i}{n}t\: – \: 1}\)#### Recognising the future value of the principal

The value of a bond is dependent upon two types of cash flow – the principal, which will be repaid at maturity, and the coupon payments, which are assumed, in this model, to be fixed and paid at regular intervals. A bond investor will only pay the net present value of all future cash flows, both the repayment of principal and the coupon payments.

The future value of the principal is dependent on the face value of the bond (usually 100), on the yield (i) and the time to redemption (t). The number and timing of coupon payments a year are also factors. This is because the bond investor will calculate the value of the bond on the assumption that the investment could be reinvested at the current yield whenever a coupon payment is made.

These factors are taken into account by the expression \( \frac{1}{1\: + \: \frac{i}{n}t\: – \: 1}\)

This is the same discount factor that was explained in last month’s issue. Using such a discount factor, it is possible to discount the future cash flow (in this case the redemption of principal) to a net present value.

When discounting the value of the principal, t-1 is used (rather than t). This relates to the fact that an investor is prepared to pay the net present value of all future cash flows. The t-1 recognises the assumption that the periods between the first and the last coupon payments are even. If there are ten coupon payments to come, then there are nine (t-1) periods between the first and the last coupon payments to be discounted. The period to the first coupon payment is accounted for by the first expression in the equation, which was explained in Treasury Today March 2003.

We can explain this by considering the value of a bond with only one cash flow to come, i.e. t=1, using the equation on above:

\( Price = \frac{ 100 \:}{1\: + \: \frac{i}{n}w} \frac{c}{n} \: \times \: \frac{1 \: -\: \frac{1}{1\: + \: \frac{i}{n}1}}{1 \: -\: \frac{1}{1\: + \: \frac{i}{n}}} \: + \: \frac{1}{1\: + \: \frac{i}{n}0} \)This equates to:

\( Price = \frac{ 100 \:}{1\: + \: \frac{i}{n}w} \frac{c}{n} \: + \: 1 \)as the last coupon payment is also the first coupon payment and x^{0} = 1.

This shows that the bond holder will receive a final coupon payment – future value C/n x 100 – and the redemption of principal (assumed to be 100). These figures will then be discounted to reflect the time to the final coupon payment by the first component in the equation. This was explained in Treasury Today March 2003. This is the sum that an investor will be prepared to pay to receive that final payment.

#### An illustration

To show how this equation works, let us assume that an investor wants a price for a bond which pays semi-annual coupons at a coupon rate of 6%. The current yield is 5% and there are ten further cash flows due, with the next one due in three months’ time, i.e. w=1/2.

Using the equation above:

\( Price = \frac{100}{1\: + \: 0.025\:^{1/2}} \frac{0.06}{2} \: \times \: \frac{1 \: -\: \frac{1}{1\: + \: 0.025\:^{10}}}{1 \: -\: \frac{1}{1\: + \: 0.025}} \: + \: \frac{1}{1\: + \: 0.025\:^9} \)

\(Price = 98.773 \:0.2691 + 0.8007 = 105.67\)

It is important to recognise that the results given by this equation are not exactly the same as those which would be achieved by adding the net present values of the future coupon payments individually. Calculating each cash flow individually will provide a more accurate value of a bond, particularly when comparing different bonds with different characteristics or when calculating the value of bonds with a short maturity.

However, this equation remains important because it is the one usually used in the bond market to calculate price.