Question

We consider a zero-coupon bond with a maturity of 10 years, face value of CHF 100 million and YTM of 10%.

Question:

1. Compute the price of the bond and its modified duration (sensitivity).

2. How does the price of the bond change when the YTM decreases to 9%? Compute the change by using the sensitivity and the direct price calculation and justify the differences.

3. Compute the convexity of the bond. Compute the change in the price of the bond when the YTM decreases to 9% while taking into account the convexity. Compare your results to the ones in question and justify the differences.

Answer 1

1)

Bond Price = Face Value / (1 +YTM)^{no of periods}

Bond Price = CHF 100,000,000 / (1 + 10%)¹⁰

Bond Price = CHF 38,554,328.94

Macaulay Duration of the Bond = Maturity since this is a zero coupon bond

Macaulay Duration of the Bond = 10 years

Modified Duration = Macaulay Duration / (1 + YTM)

Modified Duration = 10 / (1 + 10%)

Modified Duration = 9.09%

2)

Change in the bond price at YTM = 9%

Bond Price = Face Value / (1 +YTM)^{no of periods}

Bond Price = CHF 100,000,000 / (1 + 9%)¹⁰

Bond Price = CHF 42,241,080.69

Change in Bond price = (Bond price at YTM 9% - Bond price at YTM 10%) / Bond price at YTM 10%

Change in Bond price = (CHF 42,241,080.69 - CHF 38,554,328.94) /  CHF 38,554,328.94

Change in Bond price = 3,686,751.75 / CHF 38,554,328.94

Change in Bond price = 9.56%

Change in Bond price predicted by modified duration

Change in Bond price = - Modified Duration * Change in yield

Change in Bond price = - 9.09 * (9% -10%)

Change in Bond price = 9.09%

Error = Change in Bond price based on valuation - Change in Bond price predicted by duration

Error = 9.56% - 9.09%

Error = 0.47%

3)

Convexity = 1 / (Bond price * (1 + YTM)²) [(Cash flow / (1 + YTM)^period) * time period * (1 + time period)]

There is only one cash flow that is made at maturity in this zero coupon bond

Convexity =  1 / (CHF 38,554,328.94 * (1 + 10%)²) [(CHF 100,000,000 / (1 + 10%)¹⁰) * 10 * (1 + 10)] .

Convexity = 90.91

Change in Bond price predicted by modified duration & convexity

Change in Bond price = - Modified Duration * Change in yield + Convexity * (Change in yield)²

Change in Bond price = - 9.09 * (9% -10%) + 90.91 * (9% - 10%)²

Change in Bond price = 9.54%

Error = Change in Bond price based on valuation - Change in Bond price predicted by duration & convexity

Error = 9.56% - 9.54%

Error = 0.02%