Question

The 12% rate bond of Rs 1,000 face value has a current market value of Rs 1,044.57 and a yield to maturity of 10.8%. Maturity in 5 years. Calculate duration and modified duration

Answer 1

To find the bond duration

The formula is given by

Duration $=\frac{1+Y}{Y}-\frac{(1+Y)+T(C-Y)}{C[(1+Y)^{T}-1]+Y}$

where:

C= coupon rate

Y= Yield to maturity

T = Maturity

From the question, we have our parameters

C=12%, Y=10.8% and T=5 yrs

Converting from percentage to decimal

C=0.12, Y=0.108 and T=5yrs

Now, substituting the parameters into the equation

Duration $=\frac{1+Y}{Y}-\frac{(1+Y)+T(C-Y)}{C[(1+Y)^{T}-1]+Y}$

$=\frac{1+0.108}{0.108}-\frac{(1+0.108)+5(0.12-0.108)}{0.12[(1+0.108)^{5}-1]+0.108}$

$=\frac{1.108}{0.108}-\frac{1.1 08+5(0.012)}{0.12[(1.108)^{5}-1]+0.108}$

$=10.259 -\frac{1.1 08+0.06}{0.12[0.6699]+0.108}$

$=10.259 -\frac{1.168}{0.080+0.108}$

$=10.259 -\frac{1.168}{0.188}$

$=10.259-6.213\\=4.046$

Hence, the Duration is $4.046$

Now, to find the MODIFIED DURATION by the formula

modD $=\frac{Macaulay Duration}{1+(\frac{YTM}{n})}$

Thus, we must firstly find the value for the Macaulay Duration.

The tabulated data in the image below shows the Time period(t), Cash flow, PC of cash flow and the PC of Time-weighted Cash flow.

From the image, the summation of the PV of Time-weighted Cash flow is $4240.39$ and that of the PV of cash flow is $1044.58$

The Macaulay Duration for the 5 yrs bond is calculated as

MacDur $=\frac{4240.39}{1044.58}\\$

$=4.06$

Now, the Modified Duration given by the expression

modDur$=\frac{Macaulay Duration}{1+(\frac{YTM}{n})}$

Where $n=1$ and $YTM=10.8$ %

modDur$=\frac{4.06}{1+(\frac{0.108}{1})}$

$=\frac{4.06}{1+0.108}$

$=\frac{4.06}{1.108}\\=3.66$

HENCE, the Modified Duration is $3.66$