Question

Price Volatility I: Given a $100 par value with a coupon rate of 8% paying semiannually, a term to maturity of 6 years; and an initial yield of 7%.

I.) What is the approximate duration using the shortcut formula by changing yields by 10 basis points?

ii.) What is the approximate convexity using the shortcut formula by changing yields by 10 basis points?

Answer 1

i) Step 1: Calculating current price of bond (P₀) when initial yield is 7%

As coupons are paid semi annually, Semi annual coupon payment = (Coupon rate x par value) / 2 = (8% x 100) / 2 = 8/2 = $4

Semi annual yield = Annual initial yield / 2 = 7% / 2 = 3.5%

No of half years to maturity = 2 x no of years to maturity = 2 x 6 = 12

We can find the current price for initial yield of bond using PV function in excel

Formula to be used in excel. =PV(rate,nper,-pmt,-fv)

Using PV function in excel, we get Price of bond at initial yield of 7% = P₀ = $104.831667

Step 2: Calculating price of bond when yield increases by 10 basis points

Annual Yield = Initial yield + 10 basis points = 7% + 0.10% = 7.10%

Semi annual yield = Annual yield / 2 = 7.10% / 2 = 3.55%

We can find the current price for yield of 7.10% using PV function in excel

Formula to be used in excel. =PV(rate,nper,-pmt,-fv)

Using PV function in excel, we get Price of bond when yield is 7.10% = P₊ = 104.335732

Step 3: Calculating price of bond when yield decreases by 10 basis points

Annual Yield = Initial yield - 10 basis points = 7% - 0.10% = 6.90%

Semi annual yield = Annual yield / 2 = 6.90% / 2 = 3.45%

We can find the current price for yield of 6.90% using PV function in excel

Formula to be used in excel. =PV(rate,nper,-pmt,-fv)

Using PV function in excel, we get Price of bond when yield is 6.90% = P_- = $105.330507

Step 4 . Calculating modified duration of bond

Approximate modified duration = [P_-   - P₊] / [ 2 x change in yield x P₀ ] = [105.330507 - 104.335732] / [ 2 x 0.10% x 104.831667] = 0.994775 / 0.209663 = 4.7446 years

Hence Approximate modified duration of bond = 4.7446 years

Approximate Macaulay duration = Approximate modified duration of bond x (1+ initial semi annual yield) = 4.7446 x (1+3.5%) = 4.7446 x 1.035 = 4.9106

Approximate Macaulay duration = 4.9106 years

2. Approximate convexity of bond = [P_-   + P₊ - 2 x P₀ ] / [(change in yield)² x P₀ ] = [105.330507 + 104.335732 - 2 x 104.831667] / [ (0.10%)² x 104.831677 ] = [105.330507 + 104.335732 - 209.663334] / [ 0.10% x 0.10% x 104.831677 ] = 0.002905 / 0.0001048 = 27.7194

Hence Approximate convexity of bond = 27.7194