Question

GE has an outstanding bond issue that is currently selling for $110.21 to yield 6.59%. The bond pays a semi-annual coupon of 8% and will mature in 10 years. Find the convexity of this bond using the approximate convexity formula. (You may use 10 basis point change in yield to approximate the convexity).

Answer 1

1. What is Yield?

Yield represent what the market required rate of return.

2. What is Coupon Rate?

Coupon rate is the interest rate on face value of bond. What bond gives to market.

As we see in the question, yield is less than coupon rate means market required rate of return is less than interest rate. It means bond gives super-normal return to economy so entire world will be invested in this bond, hence demand increase and supply is constant. Due to demand and supply concept, bond price will be trigger at such rate where super-normal return doesn't exist after such price.

So let us do first start with finding face value of bond;

Suppose Bond face value is 'X'.

Price= (PV of interest amount + PV of bond face value) at Yield rate

PVAF (Present value annuity factor)= Amount (Rate,n) (Calculation on calc= 1/(1+(rate/100)) then equal to counts 20 at last GT)

PVIF (Present value annuity factor)= Amount (Rate, n) (Calculation on calc= 1/(1+(rate)) then equal to counts 20)

n= 10 Years, 20 Semi Annual

Yield Rate (Semi-Annually)= 6.59/2= 3.295%

Price= (X*PVAF(3.295 %, 20) + X PVIF (3.295 %, 20))

110.21=(4% of X * 14.4797 + X *0.5229)

110.21=1.1021 X

X= $100/-

Bond Face Value is $ 100/-

Lets find the Bond price if yield increase by 10 basis point to 6.69%

Price= PV of interest amount and bond face value at YTM

Price= [(100*4%)*PVAF(3.345,20)] + 100*PVIF(3.345,20)

Price= (4*14.4138) + (100*0.5179)

Price= $ 109.4452/-

Lets find the Bond price if yield decrease by 10 basis point to 6.49%

Price= PV of interest amount and bond face value at YTM

Price= [(100*4%)*PVAF(3.245,20)] + 100*PVIF(3.245,20)

Price= (4*14.546) + (100*0.5280)

Price= $ 110.984/-

Convexity Approximation:-

= (Price when YTM increase + Price when YTM decrease - 2Current bond price) / 2 * Current Bond Price (% change in YTM)²

= (109.4452 + 110.984 - 2*110.21) / 2(110.21) (.001)²

^{= -0.0092/0.00022042}

=-41.74% Semi Annually, 41.74/2=-20.87% Annually

Convexity is -20.87% annually, it shows if YTM (Market expection) changes by 1% bond price will be changes by 20.87% on annual basis in opposite direction as minus sign indicates the opposite direction.