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 at the end of the 10th year, it will also return the principal amount of$1000.

Now, let's say the bond is sold at a discount for $900 instead of its face value of$1000. This means the buyer pays $900 for the bond, receives$50 each year for 10 years, and gets $1000 at the end of the 10th year.

The yield to maturity (YTM) in this case is the interest rate that makes the present value of the bond's payments equal to its price. It can be calculated using the formula:

YTM = [C + (F - P)/n] / [(F + P)/2]

where: C = annual coupon payment = $50 F = face value of the bond =$1000 P = purchase price of the bond = $900 n = number of years to maturity = 10

Substituting these values into the formula, we get:

YTM = [$50 + ($1000 - $900)/10] / [($1000 + $900)/2] = 6.11%

So, in this case, the YTM (6.11%) is higher than the coupon rate (5%). This is a general rule: when a bond is sold at a discount, its YTM is higher than its coupon rate. This is because the investor is getting the bond for less than its face value, so the return on investment is higher.