Duration is a measure of interest rate risk exposure of a financial asset. (Usually when people refer to duration they are talking about bonds, but if you read the Time Value of Money articles you'll know that the concept applies to all financial assets).

Duration measures the sensitivity of a security's price to interest rates.

Loosely defined, duration is how long it would take for you to get your money back (the "point of indifference") if a rise in interest rates causes your bond portfolio to drop in value. Duration is a measure of the mean length of time that would pass before a stream of known and fixed payments would return their present value. The longer the pay-back period, the more sensitive the value of the asset or liability is to interest rate changes.

For an asset with a single payment (a zero coupon bond), duration equals maturity. When there are interim payments, duration will be less than maturity.

Essentially, when interest rates go up, assets fall in value. On the other hand, when interest rates go up you can reinvest your maturing assets at a higher rate, so in the long run you'll be better off if interest rates go up.

When an asset pays an income stream, you can reinvest the coupons, dividends, rent or whatever to buy more securities. If prices fall in response to a rise in interest rates, you'll be able to buy assets cheaply, obviously at some time in the future you'll actually be better off due to this fall in prices. The time in the future when you will be better off is the "duration" of the asset.

If you apply the concept of duration to the stock market, you get an interesting perspective on crashes. If the dividend yield is 3.5% (as was the average dividend yield of the ASX200 in June 2002), then a 50% fall in prices would raise dividend yields to 7%. (Assuming that a market crash would not be accompanied by a fall in dividends, which is a fairly reasonable assumption because even in the Great Depression absolute dividends only fell by a quarter).

With your portfolio now returning 7%, you'll be able to buy more assets which return 7%. After reinvesting income pretty soon you'll actually end up better off. The lesson in this is that if you are just starting saving now, and you have a truly long term outlook on markets, the more prices fall right now and in the near future the better for you.

Another interesting lesson is that when you regularly invest (dollar cost average: see the portfolios FAQ for more), you'll in effect reduce the duration risk of your portfolio. If you invest a single lump sum right now, and the market crashes next week, it may take you a long time to get your money back. On the other hand, if you instead spread out your investments over the next year obviously you'll be better off having deferred some of your investments.

There are many formulae and many definitions for duration, some of them very sophisticated. This formula is one of the more commonly applied and gives the duration of a typical cash flow producing asset like a bond:

where:

D= duration of the bond;
CF_t= interest or principal payment at time t;
t = time period in which principal or coupon interest is paid;
n = number of periods to maturity;
i = the yield to maturity (market interest rate).

The denominator is the price of the bond and is just another form of the present value formula.

The formula may look a little tricky, so I'll just give a worked example:

A 5 year bond paying a coupon of 5% suddenly takes a dive when prices fall in response to a rise in interest rates from 6% to 10%. Using the present value formula we see that this results in a fall in price from 95.79c in the dollar (present value of a 5 year bond paying a 5% yield to maturity when interest rates are 6%) to only 81.05c in the dollar (present value of a 5 year bond paying 5% coupons when interest rates are 10%.

We calculate how long it is to be before we get our money back, reinvesting the coupons in either new 10% bonds trading at par or heavily discounted 5% bonds selling at 81.05% of par.

We work out the numerator first,
5%*1/(1+10%)^1 + 5%*2/(1+10%)^2 + 5%*3/(1+10%)^3 + 5%*4/(1+10%)^4 + 105%*5/(1+10%)^5 = 3.637

The denominator is,
5%/(1+10%)^1 + 5%/(1+10%)^2 + 5%/(1+10%)^3 + 5%/(1+10%)^4 + 105%/(1+10%)^5 = 0.8105

So the duration is 3.637/0.8105 = 4.49 years.

If we redo the exercise with a 30 year bond, 5% coupon, market interest rates are 10%, we find the duration is 11.43 years. You can save yourself a lot of tedious adding up by using a spreadsheet, which is handy when doing durations for long maturity securities.

Uses of duration

Duration is used as a way to ensure that a goal to be met in the future will not be affected by interest rate changes. For example, if you were saving up for a new house in 5 years time, and assuming that you wanted to hedge away interest rate risk, you'd construct your portfolio so the duration was 5 years.

The simplest way to do that would be to invest in zero coupon bonds, or a similar investment with a 5 year maturity. As I mentioned before, the duration of a single payment asset such as a zero is equal to the maturity.

A more sophisticated way is used by fund managers who need to plan to meet fixed commitments, for example endowment funds and pension funds. The manager tries to ensure that the duration of the portfolio is equal to the holding period, this is called the "duration-matching" approach. By carefully constructing the portfolio, capital gains or losses caused by movements in interest rates will be offset by changes in the rate at which funds can be reinvested.

Another concept that I should introduce at this time if we are going to get into duration matching is the so-called "duration gap".

The formula for duration gap is:

.

Where:

D_G is the duration gap (which you want to be zero if you are duration matching);

D_A is the duration of the assets;

MV_L is the market value of the liabilities;

MV_A is the market value of the assets;

D_L is the duration of the liabilities.

To calculate the duration of two assets, you take their weighted averages. So if your portfolio was 30% invested in an asset with a 3 year duration and 70% invested in an asset with a 5 year duration, the duration of the two assets would be .3 * 3 + .7 * 5 = 4.4 years.

Say you had future liabilities of $10,000 in 5 years, $15,000 in 7 years and $20,000 in 10 years. Interest rates are 7%. The present value of these liabilities is $28,926 (market value) and the duration is 7 years.

You've got assets presently worth $30,000 and you can get a 7% return on these assets. A simple way to find these goals would be to buy zero coupon bonds that pay $10,000 in 5 years, $15,000 in 7 years and $20,000 in 10 years. If, however, we invested in a portfolio of assets other than zero coupon bonds we might be able to get a better return. What we need to do is invest in a portfolio of assets such that the duration gap is zero.

Obviously we have enough money because our present assets exceed the value of the liabilities, but we would probably still wish to hedge away interest rate risk by ensuring that the duration gap is minimised, preferably zero.

So far we've got MV_L = $28,926, MV_A = $30,000, D_L = 7. D_G is going to be zero, so we solve for our required asset duration:
0 = D_A - $28,926/$30,000 x 7.
D_A = $28,926/$30,000 x 7 = 6.75 years.

Say we invested $15,000 in a property securities fund that was expected to pay a 7% rental yield for 9 years, growing at 3%pa, and then pay us back $19,000 at the end. Cash flows would be -$13,950, $1081.50, $1113.95, $1,147.36, $1,181.78, $1,217.24, $1,253.75, $1,291.37, $20,330.

Applying the duration formula in a spreadsheet, I calculate that the duration of this asset is 7.19 years.

We need to work out the duration of whatever other assets we are going to invest in, so the weighted average durations will be .5 * 7.19 + .5 x D = 6.75. We find that D, the duration of the rest of the portfolio needs to be 6.31.

The rest of our portfolio would be made up of shares and bonds, we'd have to calculate the duration of each of the investments we are getting into so that the overall portfolio duration gap is equal to zero.

That isn't a particularly simple task when you've got dozens of assets to watch and except for zero coupon securities durations change every day with fluctuating interest rates, just as present values change. It is difficult to assess the duration of loans on which customers have the option to prepay or make irregular drawings, for example line of credit loans. Because banks have difficulty hedging risks, they demand a higher return from that loan, this is why line of credit mortgages have a higher interest rate than traditional mortgages and why normal bank accounts don't pay as well as term deposits. This is perhaps beyond the scope of what a normal investor does, but this is the sort of thing financial institutions need to do to minimise their risks.