 # 2.1 Duration

## Definition

Duration is a characteristic of a bond. For fixed-coupon bonds, duration can be intuitively defined as the average maturity of all bond payments, where each payment is weighted by its value. In the fixed-income market, duration is an essential tool for risk management, as it measures the sensitivity of an asset price to movements in yields.

To understand the duration concept, consider a bond that pays \$50 in one year and \$50 in two years. The maturity of this bond is two years. This, however, only corresponds to the last payment. Using a lever analogy, duration is measured by the distance from the fulcrum, which is about one-and-a-half year. (This is, however, a somewhat extreme example. In practice, bonds make coupon payments that are much smaller than the principal at the end.)

The reason why duration is so important is that it also measures the sensitivity of the bond price to changes in yields:

Bond Return = - Duration x 1/(1+y) x Yield Change
Note that this is only an approximation, as duration replaces the curved relationship between prices and yields by a linear relationship. Also, this duration measure assumes that all yields (across all maturities) move in parallel fashion. In many cases, however, this approximation will work well. Plot bond price against yield

## Duration and Risk

Duration, or interest rate bets, can be increased either by investing in securities with longer duration (e.g. 30-year bonds), or by leveraging the portfolio.

Table 1 below compares measures of duration for bonds with maturities varying from 1 year to 30 years. Duration is based on 8% par fixed-coupon bonds. We observe that duration is increasing with maturity: the duration of a 5-year note is 4.0 years, and that of a 30-year bond is 11.3 years.

## Table 1. Maturity and Duration

### 8% Yield, 8% Coupon Bonds

MaturityDuration
(Years) (Years)
1 0.93
2 1.78
3 2.59
5 3.99
7 5.21
10 6.71
30 11.26

Therefore, to increase the duration of a portfolio, a manager can either invest in longer maturity issues, or leverage shorter maturities. Assume for instance that investors put \$100 million in a portfolio. This can be used to buy a 5-year note. Next, the note can be pledged as collateral (in a reverse repo in exchange) for cash. The portfolio manager has the cash but is obligated to purchase back the note at a fixed price in the future, and therefore is still exposed to price movements.

In the meantime, the cash can be used to invest in another \$100 million 5-year note. This process can be repeated a second time, for a total holding of \$300 million. As the initial investment was only \$100 million, the leverage ratio is 3:1. Therefore, any price movement will be accentuated by a factor of 3. In other words, the portfolio duration is now

Price Change = - 4 x \$300 million x 1/(1+y) x Yield Change

Price Change = - 12 x \$100 million x 1/(1+y) x Yield Change.
Relative to the initial \$100 million investment, the duration is now of 12 years. The risk of the 3:1 leverage portfolio is therefore similar to that of a 30-year bond. Back to Case
Contents (c) 1996 - Philippe Jorion
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