Question

A coupon has 20 years to maturity. Coupon rate is 10% annually with par value of $1000. Current yield to maturity is 10%. What is the convexity of the bond? Please estimate the convexity using changes in bond prices and use 200 basis points to calculate the convexity. (please show full working)

Answer 1

Convexity:

Convexity is the sensitivity of duration to changes in the interest rates when the price changes are not linear, especially with embedded options.

Formula:

C = [ (V_) + (V+) - (2* Vo) ] / [ (change in yield)^2 * Vo ]

Here Value of the bond for 200bps change in the yield has to be calculated.

• V_ is the value of the bond when the yield shifted downwards by 200bps. YTM = 10% - 2% = 8%
• V_ = $100/(1+0.08)^1 + $100/(1+0.08)^2 + ....+ $1100/(1+0.08)^20 = $1,196.36

V+ is the value of the bond with upward shift of 200bps. YTM = 10% + 2% = 12%

• V+ = $100/(1+0.12)^1 + $100/(1+0.12)^2 + ....+ $1100/(1+0.12)^20 = $850.61

Vo = Value of the bond at 10% YTM and 10% coupon rate. So its price is equal to par. Vo = $1,000

• change in the yield curve = 200bps = 2% = 0.02
• square of the change in the bps = (0.02)^2 = 0.0004

using values in the formula, we get, C:

• C = [ $1196.36 + $850.61 - {2*$1000)] / [ 0.0004 * $1000]
• C = $46.97 / $0.4
• C = 117.425 (answer)