Why you need to worry about “duration” if your mutual funds invest in bonds

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SEBI’s Mutual Fund Scheme Categorization requires fund houses to use a term known as Macaulay Duration to differentiate among debt mutual funds. Here is a simple explanation of what is Macaulay Duration and the more important modified duration and why you need to understand their significance if your mutual fund invests in bonds (debt mutual funds, hybrid funds, asset-allocation funds, balanced funds).

Here is a discussion on the pros and cons of SEBI’s Mutual Fund Scheme Categorization. If debt mutual funds are new to you, here is a simple explanation: What is a Debt Mutual Fund?

Download free e-book: A Beginner’s Guide To Investing in Debt Mutual Funds

Please note that the following is an extremely simple account of duration concepts meant for new mutual fund investors.

What is “duration” in bond and debt mutual fund investing?

Duration is a measure of risk. The Macaulay Duration is a measure of how long it will take for you to recoup your investment (at the price that you purchased).

The modified duration is a measure of much the price of a bond and therefore the NAV of a debt mutual fund will change if interest rates change by 1%.

Why do bond prices vary in the market?

Simple demand and supply. If new bonds are available at higher interest rates, the demand for older bonds will decrease and they will have to be priced lower than their face value.

A bond initially priced ar Rs. 100 offering 10% interest rate will have an initial yield of

10/100 = 10%, where 10 is the interest paid out on the Rs. 100.

If newer bonds are available for 11%, the older bonds will have to be priced lower, say Rs. 90.

Then for a new buyer, the yield is 10/90 = 11.11%. So now the old bond is bit more attractive than the new 11% bonds.

That is, as the price drops, the bond yield increases and vice versa.

Price vs Yield for a short-term bond

This is the bond price vs yield for a two year bond with a 10% coupon rate. I shall not discuss the math behind this relationship here.

bond price vs bond yield for a two year bond

Price vs Yield for a long-term bond

Notice how the above graph changes when the duration is increased.

bond price vs bond yield for a 20 year bond

The two graphs illustrate why the idea of duration is important to debt mutual fund investors.

For a short-term bond, when the yields change (due to market supply vs demand), the corresponding price change is small and therefore the risk is also small.

For a long-term bond, the price change is a lot stronger or steeper (esp. towards lower yields). This is the reason why I keep saying stay away from dynamic bond funds and long-term gilts. If rates do not decrease or worse if rates start increasing after you buy such funds, you will see significant losses.

Duration is a measure of the risk in bond price

What is Macaulay Duration?

Let us find out with a series of examples.

Suppose a Rs. 1000 bond was issued at a coupon rate (interest rate) of 8%. So Rs. 1000 is the original price of the bond and the interest payment each year is Rs. 80.

We wish to buy this bond and there is one more year for maturity. The current yield is 8% (same as original rate) and therefore the current price is also Rs. 1000.

After 1Y, the bond issuer will have to pay us Rs. 80 interest + the face value of Rs. 1000.

So now we ask, what is the amount needed to generate 1080 after 1Y?

The answer is 1080/(1+8%) = 1000, where 8% is the current yield of the bond.

The Macaulay duration is defined as the amount of time it takes to recoup our investment.

So we paid Rs. 1000 at a yield of Rs. 8%. This will be locked in for a year. So it will take one year to recoup the original investment. The Macaulay duration is 1 year. This is of course trivial. Please hang on. This situation is tabulated below.

Example 1:

Example 2:

Let consider the same bond but with two years to maturity and the current yield is 8% and therefore price is Rs. 1000.

After 1Y, we will receive the interest of Rs. 80. After 2Y: Rs 80 + Rs. 1000 = Rs. 1080.

Now out of the Rs. 1000 we paid, say

Rs. X will become Rs. 80 after 1Y at the current yield of 8%. X = 80/(1+8%) = Rs. 74.07

Rs. y will become Rs. 1080 after 2Y at the current yield of 8%.

Y = 1080/(1+8%)^2 = Rs. 925.93

Rs. X + Rs. Y = Rs. 1000 = amount we paid to buy the bond.

Let us look at this in another way.

X/1000 of the original investment will be locked in for 1Y, or

Rs. (X/1000) x 1 Year = Fractional lock-in period of Rs. X = 0.07 years (see table below)

In other words, if Rs. 1000 is locked in for one year, how long should Rs. X be locked in for the same yield in investment:

This is (X/1000) x 1 Year = 0.07 years

Similarly, Y/1000 for the original investment will be locked in for 2Y, or

Rs. (Y/1000) x 2 years = Fraction lock-in period of Rs. Y. =1.85 years

Or, If Rs. 1000 is locked in for two years, how long should Rs. Y be locked for the same yield in investment:

This is (Y/1000) x 2 Years 1.85 years

Macaulay Duration = 0.07 + 1.85 = sum of fractional lock-in periods = 1.93 years

Macaulay duration for a two year bond

That is, after 1.93 years, you would have effectively recouped your investment even though you will get your money back only after 2Y.

Example 3

One more example of a 5 year bond and the current yield is the same as coupon rate.

Macaulay duration for a five year bond

The Macaulay duration of 4.31 years = sum of entries in column I. That is the fractional lock-in periods for each payment.

Example 4: 

Now let us assume that the 5Y bond is available at a reduced yield of 5%. This means we have to pay more for the bond. It is more valuable.

Macaulay Duration for a 5 year bond with yield lower than coupon rate

The Macaulay duration has increased but only by a small amount. Note that 4.36 years is the time required to recoup the original investment of Rs. 1129.88.

Example 5:

Now consider a 15-year bond with current yield the same as coupon rate.

Macaulay duration of a 15 year bond with same yield as coupon rate

Example 6:

Now if the yield was lower by 3% ==> 5%, the Macaulay duration is 10 years

If the tenure of the bond is increased by 3 times, its sensitivity to changes in yield increases by 7 times!

Higher the tenure of the bond, the more sensitive the Macaulay duration to changes in yield.

In fact, the sensitivity is rather less than what is actually seen. So to depict the price sensitivity to yield changes better, we need another measure the modified duration.

The modified duration is the rate at which the price changes with yield. With a little math (that we will spare ourselves from) we can show

Modified duration= Macaulay Duration/(1+ current yield) (for annual payments).

If you remember your school calculus

(Change in bond price)/(change in interest rate or yield) =  rate of change

The Modified duration =-1 x (Change in bond price)/(change in interest rate or yield)

The negative sign is because the slope of the bond price vs bond yield curve (see above) is negative.

So we have,

(Change in bond price) = -1 x Modified Duration x (change in interest rate)

 For debt funds, the formula (approximately) is

Change in NAV = -1 x Modified Duration x (change in interest rate)

If the modified duration is 5 years and if the interest rates have increased by 1%,

change in interest is +1% (it would be -1% if the rates had dropped by 1%)

So the change in bond price or NAV =(-1) x 5 x (+1%) = -5%

Higher the modified duration, higher the Macaulay duration, higher the sensitivity to interest rate movement. This sensitivity is for both corporate bonds and gilts: Debt Mutual Funds: Credit Risk and Interest Rate Risk Can Co-exist!

 Do not invest in debt mutual funds without understanding risks! Download our free e-book: A Beginner’s Guide To Investing in Debt Mutual Funds

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About the Author M Pattabiraman author of freefincal.comM. Pattabiraman(PhD) is the author and owner of freefincal.com.  He is an associate professor at the Indian Institute of Technology, Madras since Aug 2006. Pattu” as he is popularly known, has co-authored two print-books, You can be rich too with goal based investing (CNBC TV18) and Gamechanger and seven other free e-books on various topics of money management.  He is a patron and co-founder of “Fee-only India” an organisation to promote unbiased, commission-free investment advice. Pattu publishes unbiased, promotion-free research, analysis and holistic money management advice. Freefincal serves more than one million readers a year (2.5 million page views) with numbers based analysis on topical issues and has more than a 100 free calculators on different aspects of insurance and investment analysis. He conducts free money management sessions for corporates  and associations(see details below). Previous engagements include World Bank, RBI, BHEL, Asian Paints, TamilNadu Investors Association etc. Contact information: freefincal {at} Gmail {dot} com (sponsored posts or paid collaborations will not be entertained)
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