Question

Company Y issues $1 million face value bond that matures in 3 years. The bond has a coupon rate of 5%. The required rate of return is 8% (compounded semiannually). Calculate the following.
a) The price of the bond today.

b) The price elasticity of the bond, assuming that the required rate of return increases to 9%. Is the elasticity higher or lower when the bond has a lower coupon rate?

c) The modified duration of the bond. Use it to estimate the change in the bond price when the required rate of return increases by 1%.

Answer 1

Bond Pricing, Elasticity & Modified Duration

From the information given in the question, we have following things about the Bond issued by company.

Face Value of the Bond = $1mn

Coupon Rate = 5%

Time to Maturity = 3 years.

Required return (r) = 8%

(a). We have to calculate the price of the bond using PV method:

Annual Coupon = 5% * 1mn = $50,000 per year.

Price of the Bond = Coupon /(1+r)1 + Coupon/(1+r)2 + Coupon/ (1+r)3 + Face value /(1+r)3

Therefore, Price = $50000/(1+0.08)1 + $50000/(1+0.08)2 + $50,000/(1+0.08)3 + $1mn / (1+0.08)3

Or             Price = $46,296.29 + $42,866.94 + $39,691.61 + $793,832.24 = $922,687.10

Price of the Bond = $922,687.10

(b). Now we have to calculate price of the Bond when required rate of return increases to 9%

Annual Coupon = 5% * 1mn = $50,000 per year.

Price of the Bond = Coupon /(1+r)1 + Coupon/(1+r)2 + Coupon/ (1+r)3 + Face value /(1+r)3

Therefore, Price = $50000/(1+0.09)1 + $50000/(1+0.09)2 + $50,000/(1+0.09)3 + $1mn / (1+0.09)3

Or             Price = $45,871.56 + $42,084 + $38,609.20 + $772,183.50 = $898,748.24

Price of the Bond at 9% = $898,748.24

Bond Elasticity(sensitivity) using 8% required rate.

Sensitivity = Sum of (n * PV of coupons Every year from year 1 to 3) / (1+i)

Where, n = respective year i.e., 1,2, and 3

Pv of coupons we have calculated above for 8% and 9%

Therefore, sensitivity = (1*$46,296.29   + 2*$42,866.94 + 3*$39,691.61 + 3*$793,832.24) / (1+0.08) = $2,437,594.185

Using sensitivity, we have to calculate the price change due to change in 1% of required rate of return.

PV at 8% = $922,687.10

Sensitivity at 8% = $2,437,594.185

Approximate decrease in PV = Sensitivity * Increase in required return = $2,437,594.185 * 1% = $24375.94

PV at 9% using sensitivity = PV at 8% - Approximate decrease in PV = $922,687.10 - $24,375.94 = $898,311.16

Hence the price of the Bond at 9% using elasticity is coming as $898,311.16 which is close to the price that we have calculated above using PV method.

(c). Duration of Bond if the required rate increases to 1%.

To calculate Duration or Volatility of the Bond we first need to calculate Macaulay Duration of the Bond.

Macaulay Duration = Sum of (n * Pv of coupons every year) / (Price of the Bond)

Sum of (n* Pv of coupons) = 1*$45,871.56 + 2* $42,084 + 3*$38,609.20 + 3*$772,183.50 = $2,562,417.66

Price of the Bond at 9% = $898,748.24

Therefore, Macaulay Duration = 2,562,417.66 / 898,748.24 = 2.85 years

Now Duration of the bond = Macaulay Duration / (1+r)

Therefore, Duration = 2.85 / (1+0.09) = 2.616 years