Question

Consider a 10% corporate bond maturing in exactly 12 months. The bonds face value is $1,000 and is trading to yield 12%. Please solve the following with work.

a) Duration

b) Convexity

c) Modified Duration

d) Dollar Duration  

Answer 1

Coupon rate = 10% | Face value = $1,000 | Time to maturity = 12 months | YTM = 12%

Coupon payment = Coupon rate * Face value = 10% * 1000 = $100

As bond is maturing in 12 months, One coupon and the face value payment is remaining for the bond.

Cash flow after 12 months = Coupon Payment + Face value = 100 + 1000 = $1100

Using the cashflow and YTM, we can calculate the Price of Bond.

Price of the Bond = CF / (1 + YTM)

Price of the Bond = 1100 / (1+12%)

Price of the Bond = $982.14

a) Duration or Macaulay Duration formula =(Summation ( Number of period * CF / (1+YTM)^{Number of period} )) / Price

Number of period = 1 | Cashflow = 1100

Duration of the Corporate Bond = (1 * 1100 / (1+12%)) / 982.14

Duration of the Corporate Bond = 982.14 / 982.14

Duration of the Corporate Bond = 1 year

b) Convexity formula = (1 / (Price * (1+YTM)²))*(Summation of (Cashflow * (t² + t) / (1+YTM)^t))

Price of the bond = 982.14 | YTM = 12% | t = 1 | Cashflow = 1100

Convexity of the Corporate Bond = (1 / 982.14 * (1+12%)²) * (1100 * (1² + 1) / (1+12%)¹)

Convexity of the Corporate Bond = 0.000811688 * 1,964.29

Convexity of the Corporate Bond = 1.59

c) Modified Duration formula = Macaulay Duration / (1+YTM)

Macaulay Duration of Corporate bond = 1 (part a)

Modified Duration of the bond = 1 / (1+12%)

Modified Duration of the bond = 0.8929 or 0.89

c) Dollar Duration Formula = - Modified Duration * Price of the bond

Dollar Duration of the bond = - 0.89 * 982.14

Dollar Duration of the bond = - 874.11