Question

A 4-year 12% coupon bond has a yield of 10%.
(a) What are its Macaulay Duration, Modified duration, and convexity (I do not mean effective convexity)
(b) What is the actual price change, Modified Duration predicted price change and Modified Duration + convexity predicted change in price for an increase of 50 basis point in the yield.
Assume a flat term structure before and after the increase and annual coupons. (Note: For convexity do not use effective convexity measure)

Answer 1

Bond Valuation:

a) Given:

Face value of bond = $ 1000 (in the absence of any information face value is assumed $ 1000)

Coupon rate = 12%

Yield to maturity = 10%

Maturity years = 4 years

Maturity Value = $1000 (assumed maturity at par in the absence of any information)

Calculation of fair price of bond

Fair price = [Interest / (1 + Kd)n + Interest / (1 + Kd)n......... + Maturity Value / (1 + Kd)n)]

Fair Price = [120 / (1 + 0.1)1 + 120 / (1 + 0.1)2 + 120 / (1 + 0.1)3 + 120 / (1 + 0.1)4 + 1000 / (1 + 0.1)4]

Fair Price = (120 x 3.17) + (1000 x 0.683)

Fair Price = $1063.40

Hence, fair price of bond is $1063.40.

Macaulay's Duration

Duration of bond = 1/Fair price of bond x [(1 x Interest/(1 + Kd)1 + 2 x Interest/(1 + Kd)2 + 3 x Interest/(1 + Kd)3 + 4 x Interest/(1 + Kd)4 + 4 x maturity value / (1 + Kd)4]

Duration of bond = 1/1063.40 x [(1 x 120/ (1 + 0.1)1 + 2 x 120/ (1 + 0.1)2 + 3 x 120/ (1 + 0.1)3 + 4 x 120/ (1 + 0.1)4 + 4 1000 / (1 + 0.1)4]

Duration of bond = 1/1063.40 x (120 x 0.909 + 240 x 0.826 + 360 x 0.751 x 480 x 0.683 + 4000 x 0.683)

Duration of bond = 1/1063.40 x (109.08 + 198.24 + 270.36 + 327.84 + 2732)

Duration of bond = 3637.52 / 1063.40

Duration of bond = 3.42 years

Hence, duration of bond is 3.42 years.

Modified duration of bond

Modified duration = Macaulay's duration / (1 + yield to maturity)

Modified duration = 3.42 / (1 + 0.1)

Modified duration = 3.11 years

Hence, modified duration of bond is 3.11 years.

Convexity of bond

Convexity is a measure of the curvature of the changes in the price of a bond, in relation to changes in interest rates , addresses this error, by measuring the change in duration, as interest rates fluctuate. The formula is as follows:

C = square of d (B(r))/ B x d x square of r

where,

C = convexity

B = the bond price

r = the interest rate

d = duration

Putting the values in this formula:

C = square of 3.42 x 1063.40 x 0.1 / 1063.40 x 3.42 x square of 0.1

C = 1243.80 / 36.37

C = 34.20

Hence, convexity at yield to maturity 10% is 34.20.

b) Calculation of fair price of bond if yield to maturity increases by 50 basis points that is revised YTM is 10.5%

Fair price = [Interest / (1 + Kd)n + Interest / (1 + Kd)n......... + Maturity Value / (1 + Kd)n)]

Fair Price = [120 / (1 + 0.105)1 + 120 / (1 + 0.105)2 + 120 / (1 + 0.105)3 + 120 / (1 + 0.105)4 + 1000 / (1 + 0.105)4]

Fair Price = (120 x 3.14) + (1000 x 0.671)

Fair Price = $1047.80

Hence, fair price of bond is $1047.80.

Macaulay's Duration with YTM 10.5%

Duration of bond = 1/Fair price of bond x [(1 x Interest/(1 + Kd)1 + 2 x Interest/(1 + Kd)2 + 3 x Interest/(1 + Kd)3 + 4 x Interest/(1 + Kd)4 + 4 x maturity value / (1 + Kd)4]

Duration of bond = 1/1047.8 x [(1 x 120/ (1 + 0.105)1 + 2 x 120/ (1 + 0.105)2 + 3 x 120/ (1 + 0.105)3 + 4 x 120/ (1 + 0.105)4 + 4 1000 / (1 + 0.105)4]

Duration of bond = 1/1047.8 x (120 x 0.905 + 240 x 0.819 + 360 x 0.741 x 480 x 0.671 + 4000 x 0.671)

Duration of bond = 1/1047.8 x (108.6 + 196.56 + 266.76 + 322.08 + 2684)

Duration of bond = 3578 / 1047.8

Duration of bond = 3.41 years

Hence, duration of bond is 3.41 years.

Modified duration of bond

Modified duration = Macaulay's duration / (1 + yield to maturity)

Modified duration = 3.41 / (1 + 0.105)

Modified duration = 3.09 years

Hence, modified duration of bond is 3.09 years.

Convexity of bond

Convexity is a measure of the curvature of the changes in the price of a bond, in relation to changes in interest rates , addresses this error, by measuring the change in duration, as interest rates fluctuate. The formula is as follows:

C = square of d (B(r))/ B x d x square of r

where,

C = convexity

B = the bond price

r = the interest rate

d = duration

Putting the values in this formula:

C = square of 3.41 x 1047.80 x 0.105 / 1047.8 x 3.41 x square of 0.105

C = 1279.31 / 39.39

C = 32.48

Hence, convexity at yield to maturity 10.5% is 32.48.

Coupon	Convexity
10%	34.20
10.5%	32.48

As we can see, the higher the coupon, lower the convexity because 10% bond is more sensitive to interest rate changes than a 10.5% bond. A high convexity bond is more sensitive to changes in interest rates and should consecutively witness larger fluctuations in price when interest rate move.

The opposite is true for low convexity bonds whose prices don't fluctuate as much when interest rates changes.