Question

You are holding a 2-year 10% (annualized) coupon bond with face value $1,000 now. The interest rate now is 5% (semi-annual). However, the interest rate increases to 5.5% (semiannual) tomorrow. What is the Macaulay Duration now? What is the Modified Duration now? When the interest rate (semi-annual) increases to 5.5% tomorrow, what is the actual price change in this bond? And what is the bond price change using modified duration approximation? Which one is larger in absolute value?

******* Need answer not in excel form *********

Answer 1

1) Macaulay Duration: Macaulay Duration is calculated as the present value of the cash flows of the bond times weights by lenght of tiem of such cash flows divided by the current market price of the bond

The cash flows for the coupon bond is $100 for each of the 2 years (10% of $1,000) and $1,000 at end of 2nd year as maturity. Hence the cash inflows for the bond is as follows:

Period 1 = $100

Period 2 = $1,100 ($1,000 as maturity + $100 as interest)

The current interest rate has increased to 5.5% semi-annual. Hence the discount factors for these periods shall be calculated as follows (as Period 1 and Period 2 is at end of each year):

Period 1 = 1/{(1+5.5%)²}, = 0.8985

Period 2 = 1/{(1+5.5%)⁴}, =0.8072

The current market price of the bond shall be as follows:

$100*0.8985 + $1,100*.8072

=$89.85 + $887.94 = $977.79

Hence, Macaulay duration is (1*89.85+ 2*887.94)/977.79

= 1.91

2) Modified Duration: Macaulay duration /(1+Yeild to Maturity/number of coupon payments in a year)

In our example, yeild to maturity now with interest rate moving to 5.5% semi - annual shall be =(1+5.5%)² -1 =11.30%

Hence, modified duration is 1.91/(1+11.3%) =1.72

3) Price change in the bond

Price of bond when the interest rate was 5% semi-annual was as follows:

Discount factor for Period 1 = 1/(1+5%)²= 0.9070

Discount factor for Period 2 = 1/(1+5%)⁴ = 0.8227

Hence value of bond was $100*0.9070 + $1,100*0.8227

=$90.7+$904.97

=$995.67

Hence price change is $995.67-$977.79 (as calculated above) =$17.88

4) price change using modified duration

Modified duration 5% semi-annual rate is to be calculated

Macaulay duration during 5% semi-annual rate is as follows:

(1*$90.7+2*$904.97)/$995.67

=1.91

Yield to maturity at 5% semi annual interest rate is (1+5%)² - 1 =10.25%

Hence modifed duration is 1.91/(1+10.25%) =1.73

So for a 0.5% change in interest rate semi -annual, it is 11.3% -10.25% effective rate change annually = 1.05%,

hence change in bond price chall be 1.05% *1.73 = 1.8165%

Price of bond was $ 995.67

Change in price will be $995.67 *1.8165% =$18.08   

Change using modified duration is larger in absolute value