# Duration

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Duration - Debt Concepts
Understanding the concept of Modified Duration
"Modified Duration" is a not a widely used measure / correct measure for evaluating interest rate risk associated with bonds or bond portfolios. The convexity of a bond and Mdolf bysenstivity model & Ecotimic bond sensitivity model is which is the most, uptil 2011 modified duration was considered to the only possible is widely used measure to estimate the sensitivity of a bond however, in case of zero coupon bond this is still the case .While it is an important and useful tool in investment decision making, it is often not well understood by retail investors. In this edition of In Perspective, we make an attempt to explain the concept of "Modified Duration" and how it can be used to evaluate and compare bond portfolios or bond funds. Since Modified Duration measures the interest rate risk of a bond, it may be appropriate to first understand what "interest rate risk" is?

Understanding interest rate risk
One of the key factors which drive bond prices is the market interest rates or market yields. Here's why? Investment theory tells us that the value of any asset is the present value of all the future cash flows associated with that asset discounted at appropriate rate of return. In fixed income instruments or bonds, the future cash flows are the coupon payments and the principal repayment received from the investment and the appropriate rate of return is the market yield on similar bonds. Therefore the price of a bond can be calculated using the following equation.

It can be seen from the above equation that bond prices are inversely related to change in market yield or YTM (i.e. when yields go up, bond prices fall and when yields fall, bond prices increase). Interest rate risk is nothing but the risk of decline in bond prices on account of increase in interest rates or yields.

Modified Duration
Interest rate risk or the interest rate sensitivity (change in bond price for a unit change in yield) varies across bonds. The key variables which determine the interest rate risk of a bond are residual maturity, coupon rate and yield. In general:   Bond with longer residual maturity will be more sensitive to interest rate change compared to bond with shorter residual maturity Bond with higher coupon rate will be less sensitive to interest rate change compared to bond with lower coupon rate

Bond having higher yield-to-maturity will be less sensitive to interest rate change compared to bond with lower yield-to-maturity A single measure which captures all these variables and help calculate the magnitude of change in price of a bond for one unit change in interest rate is called Modified Duration.

Calculating Modified Duration
Modified Duration is a modified version of the "Macaulay Duration", named after Frederick Macaulay who introduced it in 1938. Macaulay Duration is the weighted average maturity (weighted by present value of cash flow) of cash flows from a bond. It can be calculated using following formula:

Modified Duration is an extension of Macaulay Duration and it can be calculated as follows:

It may not be necessary to remember these formulas as there are several tools including MS-Excel which can help calculate modified duration of a bond.

Modified Duration of a bond portfolio / bond fund
Modified duration of a bond portfolio is the asset weighted average of the modified duration of individual bonds / securities in the portfolio. The table below illustrates the calculation of portfolio modified duration.

Weighted Average Portfolio Duration
Bond A B C D Total Market value 200,000 200,000 250,000 350,000 1000,000 Portfolio Weight 20% 20% 25% 35% 100% Modified Duration 5 3 4 8 -

Weighted Modified Duratio 1.0 0.6 1.0 2.8 5.4

Most of the asset management companies in India disclose / publish the modified duration of their fixed income / bond funds in the monthly fact sheet.

Using modified duration to calculate interest rate sensitivity / expected price change of a bond or a bond portfolio for a given change in interest rate

Using the modified duration of a bond or a bond portfolio, expected price change can be calculated using the following equation. Change in bond price = - Modified Duration * ? yield
Where ? yield is the expected change in yield in %. The negative sign in this equation indicates inverse relationship between change in yield and change in bond price.

For example, if the modified duration of a portfolio is 5 and yield is expected to fall by 100 basis points, expected change in price of the portfolio (on account of change in yield) can be calculated as Change in bond price = - 5 * -1% = 5%. The table below illustrates the example of how bond prices get impacted by change in yield and portfolio modified duration. % change in price of bond or bond portfolio
Change in yield 1 -300 bps -200 bps -100 bps -50 bps -25 bps +25 bps +50 bps +100 bps +200 bps +300 bps +3.0% +2.0% +1.0% +0.5% +0.25% -0.25% -0.5% -1.0% -2.0% -3.0% 2 +6.0% +4.0% +2.0% +1.0% +0.5% -0.5% -1.0% -2.0% -4.0% -6.0% Portfolio Modified Duration 3 +9.0% +6.0% +3.0% +1.5% +0.75% -0.75% -1.5% -3.0% -6.0% -9.0% 5 +15.0% +10.0% +5.0% +2.5% +1.25% -1.25% -2.5% -5.0% -10.0% -15.0% 6 +18.0% +12.0% +6.0% +3.0% +1.5% -1.5% -3.0% -6.0% -12.0% -18.0%

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Differentiating fixed income mutual funds using modified duration
Modified duration is one of the key parameters that can be used to differentiate fixed income mutual funds. Funds with low modified duration such as cash or ultra short term debt funds have low interest rate risk and hence the growth in Net Asset Values of these funds generally tend to be quite stable. On the other hand, long term bond funds (income funds) or long term gilt funds maintain high duration and hence relatively more volatile. There is a range of products between these two categories with varying degree of interest rate risk. Funds within the same category could also vary in terms of their interest rate risk and this can be gauged using modified duration of the funds.

How can modified duration influence returns of debt funds?
In debt funds, there are primarily two sources of returns, yield accruals and capital appreciation or depreciation due to change in market yield. While yield accrual is purely a function of coupon payments earned and price paid for the security (in case of zero coupon bonds, it's the difference between price paid and maturity amount), capital appreciation or depreciation will mainly depend on movement in market yields and the modified duration of the portfolio. Thus if there are two funds with same yield accrual / yield-to-maturity but different modified duration, the returns of the fund having higher modified duration is likely to have a greater component in the form of capital appreciation / depreciation compared to the fund with lower modified duration. In other words, when interest rates fall, fund with higher modified duration is likely to do better than the fund with lower modified duration and vice versa. For this reason, portfolio managers of debt funds tend to alter the duration of their portfolio depending on their view on direction of market interest rates or yields.

Which fund to invest in - high duration or low duration?
This will primarily depend on investor's    Investment horizon Risk appetite and View on interest rates In a falling interest rate environment, it may be a good idea to increase allocation to higher duration products; provided the investor has relatively longer investment horizon and the risk appetite to withstand higher daily volatility in the fund's NAV. However, in a rising interest rate environment, it is advisable to invest in relatively lower duration products. In real life though, investors may not have strong view on interest rates and hence it may make sense to have core part of one's fixed income investment in products like short term income funds (generally in short term funds, significant portion of overall returns come from yield accruals and a smaller portion through capital appreciation / depreciation) and increase or decrease exposure to higher duration products depending on interest rate expectations. The extent of allocation in longer duration products would depend on the level of conviction in the interest rate view. There are quite a few bond funds in the market which follow flexible approach to duration management (flexible bond funds or dynamic bond funds). Investors who believe in portfolio managers' skills and who would like to rely on them to take interest rate view could also consider investing in these products. For investment with very short investment horizon (up to 6 months), it is appropriate to invest in low duration products such as cash funds or ultra short term debt funds.

In Conclusion
Modified duration is a useful tool which helps to compare interest rate risks of bond portfolios or securities of different maturities. Moreover, it also helps in determining approximate change in price of a bond or NAV of a bond portfolio based on forecasted changes in interest rates. However, it is important to remember that that are various other factors such as credit upgrades / downgrades, liquidity, spreads, etc which also influence bond prices and modified duration alone may not be sufficient to evaluate a bond or a bond portfolio.

Think Income
  o o       Glossary Debt Concepts Government borrowing Duration Download Documents Perspective Presentation on Debt Basics Outlook for 2012: Fixed income Outlook: International Debt market High Yield: High hopes for 2012 Dividend Distribution Tax - effective tax rate

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