Suppose a bond with a 10-year maturity and coupon rate of 7% is sold at par value (100% price). When an investor buys the bond he will obtain a 7% yield.

Assuming that, due to changes in market conditions, the yield for a 10-year bond of similar credit worthiness (quality) has now increased to 7.5%. If there is another issuer who wants to issue a 10-year bond, it must be priced to yield 7.5% to match the market conditions. So the coupon rate of the new bond will have to be 7.5% in order to be sold at par.

Now if the investor who bought the 7% coupon bond wants to sell it, he must sell it to yield 7.5% otherwise why should anybody buy a 10-year bond yielding 7% if there are bonds of the same maturity and quality yielding 7.5% available in the market? Since this is a fixed-rate bond, investors cannot force the issuer to raise the coupon rate from 7% to 7.5%, so the bond must be sold at a certain price other than the original par value so that the yield will be 7.5%. By calculation it turns out that the bond will have to be sold at 96.5259%, i.e. less than par value. In the bond market this is referred to as the bond being at a discount.

The opposite is true. Suppose that the yield changes to 6%. The price of the 7% coupon bond will be 107.4387%, which is at a premium.

The above Example illustrates that between interest rates (yields), coupon rates and bond prices, the following relationships hold:

- If the coupon rate = yield, then
**price = par value**. - If the coupon rate < yield, then price < par value (
**discount**) or If the coupon rate > yield, then price > par value (**premium**). - If interest rates increase, then the price of a bond decreases.

If interest rates decrease, then the price of a bond increases.

**Interest rate risk**

The degree of sensitivity of a bond price to interest rate changes is called the **interest rate risk**. A bond is said to be more sensitive to interest rate changes when a small change in the interest rates creates a relatively large swing in the bond price, and vice versa. Interest rate sensitivity is a function of the bond’s maturity, the bond’s coupon and the presence of an embedded option in the bond.

The link between interest rate risk and the characteristics of a bond can be described as follows:

**1. The interest rate risk increases with a longer maturity.**

Intuitively, we would say that we could predict the future better in the short term than the long term. So the longer the maturity, the higher the uncertainty between now and the maturity date therefore the higher the risk becomes.

**2. The interest rate risk increases with a lower coupon rate.**

We could intuitively also say that the lower the coupon an investor receives the less the cash cushion he has to offset future uncertainty.

**3. The interest rate risk depends on the embedded options:**

- When interest rates decline, the interest rate risk decreases with an embedded call option

The probability of the issuer calling the bond when interest rates decline becomes higher, so the price of the bond will converge to the call price and become less sensitive to small changes in interest rates.

- When interest rates increase, the interest rate risk decreases with an embedded put option

Likewise the probability of the investor demanding repayment of the bond when interest rates increase becomes higher, so the price of the bond will converge to the put price and then become less sensitive to small changes in interest rates.

The relationship between the price of a callable bond, an option-free bond and the price of the embedded call option is as follows:

Price of callable bond = price of option-free bond – price of embedded call option.

The implication of this relationship is that the price of the callable bond would have to be lower than the equivalent non callable bond since the issuer holds the right to exercise the option. The call option is valuable to the issuer at the expense of the investor.

All other factors held constant, the higher the market yield level of a bond, the lower the price sensitivity. The mathematical model will be explained in more detail in the next Study Session, but an illustration is provided below. This feature is called ‘tapering off’ and refers to a bond having lower price sensitivity at a higher level of interest rates.

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