Duration - math definitions

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This is an attempt to consolidate and hopefully simplify (with pictures?) the concept of duration. --LadyGeek 21:25, 10 February 2010 (UTC)

Duration
A measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. Duration is expressed as a number of years. Rising interest rates mean falling bond prices, while declining interest rates mean rising bond prices. The bigger the duration number, the greater the interest-rate risk or reward for bond prices. The duration number is a complicated calculation involving present value, yield, coupon, final maturity and call features. Fortunately for investors, this indicator is a standard data point provided in the presentation of comprehensive bond and bond mutual fund information. It is a common misconception among non-professional investors that bonds and bond funds are risk free. They are not. Investors need to be aware of two main risks that can affect a bond's investment value: credit risk (default) and interest rate risk (rate fluctuations). The duration indicator addresses the latter issue. Short-term, intermediate-term and long-term bond funds will have different durations. For example, Vanguard's short-, intermediate- and long-term bond index funds generally have durations of around three years, six years and 11 years, respectively

Bonds: Advanced Topics
The duration of a bond, or a bond fund, is a measure of its sensitivity to interest rates. The duration is approximately the amount by which the price of the bond will fall for every 1% rise in interest rates. This is only approximately correct because of convexity (see below). A bond fund with a ten-year duration is twice as sensitive to interest-rate risk as a bond fund with a five-year duration. (In practice, the relative risk is somewhat reduced because long-term rates are usually less volatile than short-term rates.)

The formal definition of duration is the average value of the time to each future payment, weighted by the proportion of the present value of the bond made up by the present value of each future payment. A zero-coupon bond is a promise to make a single payment at a future time, so its duration is equal to its maturity--the time until that payment is made. A conventional bond, or bond portfolio, or bond fund, makes multiple payments; each payment has a present value and a duration as if it were a zero-coupon bond, and the average of all durations is weighted by the present value of the payments divided by the present value of the whole bond. A bond which pays coupons will always have duration less than its maturity. High interest rates make the duration much shorter than the maturity, since a greater proportion of the bond's total value comes sooner. In low interest rate environments, coupon-paying bonds more closely resemble zero-coupon bonds (duration is closer to the maturity), since the present value of the principal returned at maturity is generally much larger than the coupon payments.

Incorporate Bernstein's definitions here (see below). Needs a picture!): Duration has another useful summary property, which is that if the yield curve shifts in parallel, then duration is the point of indifference to interest rate changes. For example, if a bond/portfolio/fund with a duration of 5 years experiences a market interest rate increase of 1%, its value will drop by approximately 5%; however, since the same coupon payment now represents a higher percentage of the bond's value, its yield is higher (it will match the market rate), and the higher yield plus higher market interest on coupon payments will exactly offset the lower NAV after 5 years.  To be absolutely assured of receiving a given sum on a future date (assuming parallel shifts of the yield curve), therefore, you should gradually reduce the duration as the date approaches.  A zero-coupon bond reduces duration by exactly the amount of time that passes, and is therefore the risk-less choice for meeting a future obligation. However, few investors have such exact demands on their capital. For most purposes, shifting from intermediate- or longer-term bonds to shorter-term bonds as the need for capital approaches will cause little risk; also, in practice non-parallel shifts of the yield curve are not likely to cause very large changes in returns.

Bonds and bond funds also compute an effective duration which takes into account the possibility that the bond will be called (paid back early).

Investopedia's article on bond duration has the mathematical formulas.

Individual Bonds vs a Bond Fund
Define Bernstein above, then refer to link from here: It's useful to focus on the duration of your bond fund, such as the Vanguard TIPS Fund, which currently has a duration of 5.0 years. William Bernstein provides an insightful definition of duration as the "point of indifference" for the owner of a bond fund in dealing with interest rate changes. If interest rates rise after purchasing a bond fund, the NAV of the fund falls, which hurts the investor. However, the dividends that the bond fund throws off can now be reinvested at a higher rate. The duration is the length of time that an investor needs to hold the fund for the increased yields to compensate for the decrease in NAV. In that sense, duration represents the length of time it would take for the total value of the fund, with dividends reinvested, to be worth exactly what it would have been worth had interest rates not risen. So, you should always hold bond funds with a duration equal to or shorter than the expected need for your money (note that holding the duration shorter than your need for the money leaves you exposed to the risk of lower returns if interest rates fall).

Of course, as discussed above, this definition of duration applies equally to bond funds and to an individual portfolio of bonds. The relationship does not exactly hold if the yield curve shifts in a non-parallel fashion, but the difference is expected to be small.

Interest rates change continuously, not just at a single point in time. Therefore, as the time when you will need the money approaches, you must reduce your duration accordingly, to protect you against any further increases in prevailing rates. The ways in which this can be done are discussed below.

Bond fund versus individual bonds, definitive answer
Various posts. From Linuxizer: The problem with popular explanations of duration is that: a) It is a simplifying formula. A darn good one, but nevertheless summarizing the behavior of a variety of different objects under a variety of circumstances with a single number will inevitably lead to assumptions. b) It has multiple meanings. I find it very useful to think of duration as (1) the value-weighted average of cash flows, but Fabozzi is saying that definition is less useful for most purposes. Instead, he says, think of it as the (2) approximate amount a bond's price changes when the market rate changes--approximate because convexity comes into play. But wait, Bernstein says! Duration is (3) the point of indifference to a single interest rate change. And since it's the point of indifference to a single rate change, it is also (4) the point of indifference to market changes assuming you keep your duration declining has your investment horizon changes.

That's four definitions, all true, and all useful for solving different problems.

Math
"Duration is associated with the slope of the price-yield curve. The absolute value of slope at any point on the price-yield curve is the Macaulay duration times the price of the security, divided by one plus the periodic yield."
 * Macaulay Duration from Wolfram research. A demo of Macaulay Duration

William F. Sharpe

 * Duration, Interest Rates and Bond Yields page from William F. Sharpe's Macro-Investment Analysis text. (Permission to use material is elucidated here.)

The maturity of a bond provides important information for its valuation. The values of longer-term bonds are generally affected more by changes in interest rates, especially longer-term rates. However, for coupon bonds, maturity is a somewhat crude indicator of interest rate sensitivity. A high-coupon bond will be exposed more to short and intermediate-term rates than will a low coupon bond with the same maturity, while a zero-coupon bond will be exposed only to the interest rate associated with its maturity. To provide a somewhat better measure than maturity, Analysts often compute the duration of a set of cash flows.

Let df be a {1*periods}vector of discount factors and cf a {periods*1} vector of cash flows. The duration of cf is a weighted average of the times at which payments are made, with each payment weighted by its present value relative to that of the vector as a whole. In the previous example, the bond has cash flows cf :

Yr1     6 Yr2     6 Yr3   106

The market discount function df is:

Yr1      Yr2       Yr3 0.9400   0.8800    0.8200

The present values of the cash flows are v = df.*cf'' :

Yr1      Yr2       Yr3 5.6400   5.2800   86.9200

To compute weights we divide by total value, w = v/(df*cf), giving:

Yr1      Yr2       Yr3 0.0576   0.0540    0.8884

In MATLAB, the expression [1:3]' produces the {periods*1} vector of time periods:

Yr1     1 Yr2     2 Yr3     3

The duration, given by d = w*([1:3]'), is 2.8307 years -- somewhat less than the maturity of 3 years.

Well and good, but what use can be made of duration? In some circumstances, quite a bit. In others, somewhat less. We make the calculation to better understand the reaction of the value of a vector of cash flow to a change in one or more interest rates. In practice, of course, many such rates along the term structure may change at the same time. In general, if the discount function changes from df1 to df2, the present value of cash flow vector cf will experience a change in value equal to:

dV = (df2 - df1)*cf

How can one number summarize the effect on value of a change in potentially many different interest rates along the discount function?

Of necessity, a change in the yield-to-maturity of a bond will cause a predictable change in the value of that bond or set of cash flows, since there is a one-to-one relationship between the two. The relationship holds as well for most cash flow vectors. In such case the term internal rate of return is utilized, instead of yield-to-maturity. If there are sufficiently many positive and negative cash flows in a vector, the internal rate of return may not be unique, causing potential mischief if one relies upon it. However, this cannot happen if the vector consists of a series of negative (positive) flows, followed by a series of positive (negative) flows -- that is, if there is only one reversal of sign.

In practice, a bond's duration is usually calculated with a discount function based on its own yield-to-maturity, that is:

[ 1/(1+y)  1/((1+y)^2)   1/((1+y)^3) ]

Now, consider c(t), the cash for the t'th period. Using the bond's yield-to-maturity, Its present value is:

v(t) = c(t)/((1+y)^t)

If there is a very small change dy in y, the change in v(t) will be:

dv(t) = (c(t)*(-t*(1+y)^(-t-1))) * dy

or       dv(t)  =  (v(t)*-t) * (dy/(1+y))

Summing all such terms we have the total change in value dv :

dv = sum(dv(t)) = - sum(v(t)*t) * (dy/(1+y))

Finally, the proportional change in value, dv/v

is:

dv/v = sum(dv(t)/v) = - sum((v(t)/v)*t) * (dy/(1+y)

But the term inside the parentheses preceded with &quot;sum&quot; is the duration, calculated using the bond's own yield-to-maturity. Thus we have:

dv/v = - d * (dy/(1+y))

Sometimes the duration is divided by (1+y) to give the modified duration. Letting md represent this, we have:

dv/v = - md * dy

Thus the modified duration indicates the negative percentage change in the value of the bond per percentage change in its own yield-to-maturity. The minus sign indicates that an increase (decrease) a bond's yield-to-maturity is accompanied by a decrease (increase) in its value.

Duration (modified or not) is of no interest unless one can establish a relationship between a bond's own yield-to-maturity and some market rate of interest. For example, assume y = y20+.01, where y20 is the interest rate on 20-year zero coupon government bonds. In this case:

dy = dy20

and:

dv/v = -  md * dy20

which relates the percentage change in the bond's value to the change in a market rate of interest.

The concept of duration that is especially relevant for Analysts who counsel the managers of defined-benefit pension funds. Many such funds have obligations to pay future pensions that are fixed in nominal (e.g. dollar) terms, at least formally. Moreover, the bulk of the cash flows must be paid at dates far into the future. The present value of the liabilities of such a plan can be computed in the usual way and its yield-to-maturity (internal rate of return) or discount rate , determined, using market rates of interest. In many cases, the discount rate will be very close to a long-term rate of interest (e.g. that for 20-year bonds). Since term structures of interest rates tend to be quite flat at the long end, any change in the long-term rate of interest will be accompanied by a roughly equal change in the discount rate for a typical pension plan of this type. Thus the duration of the plan's cash flows provides a good estimate of the sensitivity of the present value of its liabilities to a change in long-term interest rates. Any imbalance between the duration of the assets in a pension fund held to meet those liabilities and the duration of the liabilities may well provide an indication of the extent to which the fund is taking on interest rate risk.

MATLAB

 * MATLAB, developed by Mathworks


 * Financial Toolbox

Functions

 * bnddurp Bond duration given price (SIA compliant)
 * bnddury Bond duration given yield (SIA compliant)

Examples

 * Sensitivity of Bond Prices to Parallel Shifts in the Yield Curve (3-D graph)
 * Sensitivity of Bond Prices to Changes in Interest Rates