# Duration - math definitions

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

## Math

- Macaulay Duration from Wolfram research. A demo of Macaulay 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

- 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
"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

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

Enable browser cookies.

- MATLAB, developed by Mathworks
- Financial Toolbox