Duration - math definitions

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Duration - math definitions provides a mathematical background on the concept of duration.


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.

William F. Sharpe

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


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

      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


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

But the term inside the parentheses preceded with "sum" 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


   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


    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.


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