Barbican Consulting Limited

# Duration

**Duration Explained**

Duration is used to measure interest rate risk. It is** **the weighted average life of the bond's present values and is expressed in years.

As an approximation, a financial instrument that has a duration of 1 year will lose 1% of its value for a 1% increase in interest rates. A financial instrument with a duration of 3 years will lose 3% of its value for a 1% increase in interest rates. The greater the portfolio duration the greater the interest rate risk you run.

Duration is not widely used by traders but is often used by portfolio managers to limit the interest rate exposure that is being taken.

A useful characteristic of duration is that it is additive. Therefore the duration of a portfolio equals the weighted average duration of the instruments it contains.

Duration should not be confused with maturity. A bond with a maturity of 5 years will have a duration slightly shorter than 5 years (unless it is a zero coupon bond).

**Calculating duration**

Here is a simple example.

This three year bond has four cash flows over its life. That's three coupons of 500,000 and one principal repayment of 10,000,000.

These coupons are discounted to give their present value. The interest rate used to discount would normally be the bond's yield. In this example, 5% is used.

The individual present values are then multiplied by the year in which they arise (PV x years). To arrive at the duration, the "PV years" are divided by the net present value of the bond (10,000,000). The summation of the individual durations provides the bond's duration, in this case 2.86 years.

**Using duration**

If we invested $10,000,000 in this bond and interest rates increased by a flat 1% our investment would lose approximately $286,000 ($10,000,000 x 2.86%). Alternatively a 0.01% increase in rates would cost us $2,860.

If you used the example to calculate the duration of a longer dated bond its duration and therefore its risk would be higher.

Institutions that use duration often assign a target duration to their investment portfolios. They specify that an asset portfolio should be subject to a minimum and a maximum duration, (for example 2 years as the minimum and 4 years as the maximum duration). Active management can then occur within these two constraining limits.

When interest rates are expected to increase, the duration of a portfolio is reduced. This is achieved by buying shorter maturity assets or by hedging with swaps or futures. When interest rates are expected to fall the duration of a portfolio is increased. This can be achieved by buying longer dated assets, receiving fixed interest on long dated swaps or buying bond futures.

**Modified duration**

Bonds that trade at higher yields exhibit a lower sensitivity to changes in interest rates; (sometimes this is referred to as “convexity”).

This means when a bond's yield is high a further 1% change in interest rates has a smaller effect on the value of the bond than when its yield is low.

Highly rated bonds with low yields are more sensitive to interest rate changes than lower rated bonds with higher yields. Modified duration takes this into account.

To calculate modified duration the bond's duration is divided by 1 + the bond's yield.

In our earlier example the duration is 2.86 years.

The modified duration is 2.86 / (1 + 0.05%) = 2.72 years

This modified duration calculation shows us that a 0.01% change in interest rates would alter our portfolio value by $2,720.

If this bond traded with a 10% yield its modified duration would be 2.86 / 1.10 = 2.60 years.

The higher yielding bond would therefore *only* change in value by $2,600 for a 0.01% change in interest rates. This is important if you use duration as a measure of interest sensitivity or for the calculation of hedge ratios.

**Strengths and weaknesses**

Duration is a useful tool for portfolio management. It will help you manage and control the interest rate risk associated with parallel shifts in the yield curve. It has the useful property that the durations of two portfolios can be added together on a weighted basis in order to determine the overall portfolio duration.

But duration will not tell you the risk you face from a non-parallel change in interest rates. It is a weakness that those using duration must consider.

It also does not provide any statistical probability of loss although it could be used to give you an approximation of losses under extreme interest rate movements.

*First Published by Barbican Consulting Limited 2010*