# Interest Rate Risk

Interest rate risk is the risk that changes in market interest rates will affect the value of bonds and other debt instruments. The changes in market interest rates may arise due to multiple factors: changes in Federal Reserve policy, movement in yield curve due to overall economic outlook, etc.

Debt instruments typically pay fixed periodic payments of interest and/or a final balloon payment at maturity. They are valued by discounting those future cash flows based on the applicable market interest rate relevant to the security. Following is the formula for bond price where C is the coupon payment, Y is the yield, F is the face value i.e. redemption value:

$$ Bond\ Price\\=C\times\frac{1−\left(1+\frac{Y}{m}\right)^{-n\times m}}{\frac{Y}{m}}+\frac{F}{\left(1+\frac{Y}{m}\right)^{n\times m}} $$

Let’s say a bond with 5 years to maturity has face value of $100 and semi-annual coupon payment of $2.5. If the market interest rate is 4.5%, the bond price is $102.25

$$ Bond\ Price \\ =$2.5 \times \frac{1−\left(1+\frac{4.5\%}{2}\right)^{-5\times2}}{\frac{4.5\%}{2}}+\frac{$100}{\left(1+\frac{4.5\%}{2}\right)^{5\times2}} \\ =$102.25 $$

If the rate rises to say 5.5%, the bond price drops to $97.84.

$$ Bond\ Price\\ =$2.5 \times \frac{1−\left(1+\frac{5.5\%}{2}\right)^{-5\times2}}{\frac{5.5\%}{2}}+\frac{$100}{\left(1+\frac{5.5\%}{2}\right)^{5\times2}}\\=$97.84 $$

It is established that whenever market interest rates drop, debt instruments fall in value. The reason behind this is that the holder of the bond is missing out on the high interest rates available in the market. If the bond-holder in the above example purchased the bond for $101 and wanted to sell it before maturity, he will have to settle for a lower market value of his bond. Even if he intends to hold the bond to maturity, the opportunity cost is still there, even tough he will receive the full face value of the bond.

Interest rate risk is measured by duration. A higher a bond duration means higher interest rate risk. As you can see from the bond price function, bond price fluctuates more when n is higher, i.e. when the bond has a longer maturity and higher coupon payment.