Question

A newly issued bond has a maturity of 10 years and pays a 7.5% coupon rate (with coupon payments coming once annually). The bond sells at par value. a. What are the convexity and the duration of the bond? Use the formula for convexity in footnote 7.

Answer 1

Bond

Explanation:

Bond price = B = Face value = F = $100 (Because the bond sells at par value.)

Time to maturity = n = 10 years

Number of periods in a year = p = 1 (annual coupon payments)

Coupon rate = C = 7.5% = 7.5%*100 = $7.5

Yield = r = 7.5% (since bond is selling at par hence the coupon rate = yield)

Duration and Convexity:

Duration is a used as a measure of risk because it provides the average length of time by when the bond holder will receive their payments made. Longer the duration, longer is the time to receive the payments and hence higher the risk.

Expanding the summation and solving for Convexity, we get,

Period (t) Cash Flow PV = Cash Flow / (1+r/p)^(t*p) (t^2+t)*PV

1 7.5 6.97674418604651 13.953488372093

2 7.5 6.48999459167117 38.939967550027

3 7.5 6.03720427132202 72.4464512558643

4 7.5 5.61600397332281 112.320079466456

5 7.5 5.22418974262587 156.725692278776

6 7.5 4.85971138848918 204.107878316546

7 7.5 4.52066175673412 253.157058377111

8 7.5 4.20526675045035 302.779206032425

9 7.5 3.91187604693056 352.06884422375

10 7.5 3.63895446226098 400.284990848708

10 100 48.5193928301464 5337.13321131611

Sum of (t^2+t)*PV 7243.91686803787

Convexity = sum/(B*(1+r)^2) 62.6839750614418

Convexity = 62.6839750614418 = 62.684 (rounded to 3 decimal places as requested in the question)