Bond Duration Calculation

As the name suggests it is the time frame an investor has to wait till he receives the cash payments. In case of a zero-coupon bond the duration automatically is same as the maturity years but in case of a bond with a coupon the duration is less than the number of years to maturity as the investor receives some of the cash before the maturity.

Let us take an example of a 7 year old bond with a coupon of 5% with a price of $95 and a yield of 12 %. We can demonstrate the cash flows and timings of the bond as follows

                                 5%           5%          5%          5%           5%          5%           105%

                            Year 1       Year 2   Year 3     Year 4      Year 5        Year 6      Year 7

                                                                               

               

We can calculate the average time to the cash flows weighted by the cash flows themselves:

       (5 *1) + (5*2)+ (5*3)+ (5*4)+  (5*5)+ (5*6)+ (105*7)  

                                (5+5+5+5+5+5+5+105)

=     6 years

                            

Macaulay’s Duration is very similar in calculation except that we are going to use the present value of each cash flow by discounting each cash flow by the yield of 11.063%

Principal	100
coupon	5%
yield	11.063%
Years	coupon	Pv of CF	PV of CF*time to Cash flow
1	5	4.50	4.50
2	5	4.05	8.11
3	5	3.65	10.95
4	5	3.29	13.14
5	5	2.96	14.79
6	5	2.66	15.98
7	105	50.37	352.62
	sum	71.49	420.10
	duration	PV of CF*time to Cash flow/PV of CF	5.88

*PV Present value CF cash flow

Duration demonstrates the price-sensitivity of the bond as well as can be used for Immunization.

When you invest in bonds and there is a fall in the yields you lose because you won’t be able to earn as much you had expected on the reinvestment on the coupons received. If the bonds are held to maturity your total returns are less than you had expected this is called Reinvestment risk.

Modified Duration

The calculation is based on the fact that the sensitivity of the bond price to yield changes is dependent on how steep the price/yield curve slopes.

                                        Modified Duration = Macaulay’s Duration/ (1 + YTM/ N)

Where YTM = yield to maturity

             n = no. of coupons per year

So for the previous example of the bond we had considered the Modified duration will be

n = 1 ( as the bond has an annualized coupon)

YTM =11.063%

5.88 / (1+.11063) = 5.29

Modified Duration is also known as Volatility in some markets.

The change in price of a Bond can be approximated to

                              Δ Price = - (dirty price * Δ yield *modified duration)

For example,

If we assume that the yield rose from 11.063 % to 12.063% 1% yield the

Δ Price = - (95*.001*5.29) = -5.07

So, we can expect the price to drop to (95-5.07) = 89.97

Therefore, we can see that higher the duration the higher the bond is sensitivity to the yield change.

Portfolio Duration

We can use the Modified Duration for a single bond or a portfolio of bonds. A portfolio’s modified duration gives the sensitivity of the portfolio to the yield changes. A portfolio’s modified duration will give the sensitivity of the portfolio’s Value to the change in yield. With this knowledge an owner of a portfolio can match the modified duration between assets and liabilities to hedge the risk.

Let us consider an example for the following portfolio

	Face Value	Dirty Price	Modified Duration
Bond A	2 million	95.45	5.35
Bond B	1 million	105	7.20
Bond C	3 million	98.2	3.45

= - Δ yield * ((2 million *95.45/100*5.35) + (1 million *105/100*7.20) + (3million*98.2/100*3.45))

Now we have a Bond D to short and immune the above portfolio

	Dirty Price	Modified Duration
Bond D	110.2	9.75

To offset the change in value for a small parallel yield change we need to short D equal to

= - Δ yield *(Face value of bond D *110.2/100*9.75)

ð  Face value of bond D = 2.6 million

All the above calculation is based on the assumption that the movement of the yield curve is parallel along the yield curve. In real world that is not the case the yield curve does not fall or rise as a straight line this is known as Yield curve risk. The difference between the actual and the estimation depends on the degree of the curvature and this is known as Convexity (which I will hopefully discuss in my next article).