Show with an example the relationship between yield to maturity and the coupon rate of a bond when it is sold at a discount.'

Sure, let's take an example to illustrate the relationship between yield to maturity (YTM) and the coupon rate of a bond when it is sold at a discount.

Let's say we have a bond with a face value of $1000, a coupon rate of 5$50 (5% of $1000) each year for 10 years, and then return the face value of$1000 at the end of the 10th year.

Now, let's say the bond is sold at a discount for $900 instead of its face value of$1000. This means the investor is buying the bond for less than its face value.

The yield to maturity (YTM) is the total return the investor can expect if they hold the bond until maturity. It takes into account both the annual coupon payments and the difference between the purchase price and the face value.

In this case, the investor is getting $50 each year in coupon payments, and they will also get an additional$100 at the end of the 10 years when the bond matures and they receive the face value of $1000. This additional$100 is spread out over the 10 years, effectively increasing the annual return of the bond.

Therefore, when a bond is sold at a discount, its yield to maturity is higher than its coupon rate. This is because the investor is getting the regular coupon payments plus an additional return from the increase in the bond's value when it matures at its face value.

In our example, the coupon rate is 5%, but the yield to maturity would be higher than 5% because the bond was purchased at a discount. The exact YTM can be calculated using a financial calculator or a bond yield formula.