Duration

Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds.
One method of measuring interest rate risk is by the full valuation approach, which simply
calculates what bond prices will be if the interest rate rose by specific amounts. The full
valuation approach is based on the fact that the price of a bond is equal to sum of the present
value of each coupon payment plus the present value of the principal payment.

Bond Value = Present Value of Coupon Payments + Present Value of Par Value

Another method, which is less computationally intensive, is by calculating the duration of a

bond, which is the weighted average of the present value of the bond’s payments, and can be
viewed as the average, or effective, maturity of a bond. Graphically, the duration of a bond can
be envisioned as a seesaw where the fulcrum is placed so as to balance the weights of the present
values of the payments and the principal payment. The longer the duration, the longer is the
average maturity, and, therefore, the greater the sensitivity to interest rate changes.

Although the effective duration is measured in years, it is more useful to interpret duration as a
means of comparing the interest rate risks of different securities. Securities with the same
duration have the same interest rate risk exposure. For instance, since zero-coupon bonds only
pay the face value at maturity, the duration of a zero is equal to its maturity. It also follows that
any bond of a certain duration will have an interest rate sensitivity equal to a zero-coupon bond
with a maturity equal to the bond’s duration.

Duration is also often interpreted as the percentage change in a bond’s price for a 1% change in
its yield to maturity (YTM). So, for instance, the price of a bond with a 10-year duration would
change by 10%.

Duration can be estimated by the following equation:

Duration Approximation Formula

P0 = Bond price.
P- = Bond price when interest rate is incremented.
P+ = Bond price when interest rate is decremented.
∆y = change in interest rate in decimal form.

The interest rate is shocked up and down by a specific amount to obtain the new bond prices.
Note that even if the interest rates are shocked by an amount different from 1%, duration is still
interpreted as the percentage change in bond price for a 1% change in the YTM.

Macaulay Duration
It was Frederick Macaulay who developed the concept of duration, equating it to the average
time to maturity or the time required to receive half of the present value, discounted by the
bond’s yield to maturity, of the bond’s cash flow. The Macaulay duration is calculated by 1st
calculating the weighted average of each cash flow at time t by the following formula:

wt = weighted average of cash flow at time t.

CFt = Cash flow at time t.
y = yield to maturity

Then these weighted averages are summed:

Macaulay Duration Formula

T = number of cash flow periods.

Hence, the Macaulay duration measures the effective maturity of a bond, and can also be used to
calculate the average maturity of a portfolio of fixed-income securities.

Modified Duration

Modified duration is a modification of the Macaulay duration to estimate interest rate risk,
calculating the change in a bond’s price to a change in its yield to maturity by the following
formula:

Modified Duration Formula

Dm = Modified Duration
DMac = Macaulay Duration
y = yield to maturity
k = number of payments per year

The modified duration formula is valid only when the change in yield will not alter the cash flow
of the bond, such as may occur, for instance, if the price change for a callable bond increases the
likelihood that it will be called.

It is also only valid for small changes in yield, because duration itself changes as the yield
changes. It is a 1st derivative of the price-yield curve, which is a line tangent to the curve at the
current price-yield point.
 Duration and Modified Duration Formulas for Bonds using Microsoft Excel
Duration = DURATION(settlement,maturity,coupon,yield,frequency,basis)

Modified Duration = MDURATION(settlement,maturity,coupon,yield,frequency,basis)

 Settlement = Date in quotes of settlement.

 Maturity = Date in quotes when bond matures.
 Coupon = Nominal annual coupon interest rate.
 Yield = Annual yield to maturity.
 Frequency = Number of coupon payments per year.
o 1 = Annual
o 2 = Semiannual
o 4 = Quarterly
 Basis = Day count basis.
o 0 = 30/360 (U.S. NASD basis). This is the default if the basis is omitted.
o 1 = actual/actual (actual number of days in month/year).
o 2 = actual/360
o 3 = actual/365
o 4 = European 30/360

1. Example—Calculating Modified Duration using Microsoft Excel

Calculate the duration and modified duration of a 10-year bond paying a coupon rate of 6%, a
yield to maturity of 8%, and with a settlement date of 1/1/2008 and maturity date of 12/31/2017.

Duration = DURATION("1/1/2008","12/31/2017",0.06,0.08,2) = 7.45

Modified duration = MDURATION("1/1/2008","12/31/2017",0.06,0.08,2) = 7.16

Note that modified duration is always slightly less than duration, since the modified duration is
the duration divided by 1 plus the yield per payment period.

Convexity adds a term to the modified duration, making it more precise, by accounting for the
change in duration as the yield changes—hence, convexity is the 2nd derivative of the price-yield
curve at the current price-yield point.

Although duration itself can never be negative, convexity can make it negative, since there are
some securities, such as some mortgage-backed securities that exhibit negative convexity,
meaning that the bond changes in price in the same direction as the yield changes.

Effective Duration for Option-Embedded Bonds

Because duration depends on the weighted averages of the present value of the bond’s cash
flows, a simple calculation for duration is not valid if the change in yield could result in a change
of cash flow. Valuation models must be used in calculating new prices for changes in yield when
the cash flow is modified by options. The effective duration (aka option-adjusted duration) is
the change in bond prices per change in yield when the change in yield can cause different cash
flows. For instance, for a callable bond, the bond will not rise above the call price when interest
rates decline because the issuer can call the bond back for the call price, and will probably do so
if rates drop.

Because cash flows can change, the effective duration of an option-embedded bond is defined as
the change in bond price per change in the market interest rate:

Effective Duration Formula

∆i = interest rate differential

∆P = Bond price at i + ∆i – bond price at i - ∆i.

Note that i is the change in the term structure of interest rates and not the yield to maturity for the
bond, because YTM is not valid for an option-embedded bond when the future cash flows are
uncertain.

Duration Formulas for Specific Bonds and Annuities

There are several formulas for calculating the duration of specific bonds that are simpler than the
above general formula.

The formula for the duration of a coupon bond is the following:

Duration Formula for Coupon Bond

y = yield to maturity

c = coupon interest rate in

decimal form

T = years till maturity

If the coupon bond is selling for par value, then the above formula can be simplified:

Duration Formula for Coupon Bond Selling for Face Value

 y = yield
to
maturity

T = years
till
maturity

The duration of a fixed annuity for a specified number of payments T and yield per payment y
can be calculated with the following formula:

Fixed Annuity Duration Formula

y = yield to maturity

T = years till maturity

A perpetuity is a bond that does not have a maturity date, but pays interest indefinitely.
Although the series of payments is infinite, the duration is finite, usually less than 15 years. The
formula for the duration of a perpetuity is especially simple, since there is no principal
repayment:

Perpetuity Duration Formula

y = yield to maturity

Portfolio Duration
Duration is an effective analytic tool for the portfolio management of fixed-income securities
because it provides an average maturity for the portfolio, which, in turn, provides a measure of
interest rate risk to the portfolio.

The duration for a bond portfolio is equal to the weighted average of the duration for each type
of bond in the portfolio:

Portfolio Duration = w1D1 + w2D2 + … + wKDK

 wi = market value of bond i / market value of portfolio

 Di = duration of bond i
 K = number of bonds in portfolio
To better measure the interest rate exposure of a portfolio, it is better to measure the contribution
of the issue or sector duration to the portfolio duration rather than just measuring the market
value of that issue or sector to the value of the portfolio:

Portfolio Duration Contribution = Weight of Issue in Portfolio x Duration of Issue

Convexity
Duration is only an approximation of the change in bond price. For small changes in yield, it is
very accurate, but for larger changes in yield, it always underestimates the resulting bond prices
for non-callable, option-free bonds. This is because duration is a tangent line to the price-yield
curve at the calculated point, and the difference between the duration tangent line and the price-
yield curve increases as the yield moves farther away in either direction from the point of
tangency.

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st
derivative to the equation for the duration and the 2nd derivative to the equation for the price-
yield function, and is calculated by the following equation:

Convexity Formula

P = Bond price.

y = Yield to maturity in decimal form.

T = Maturity in years.

CFt=Cash flow at time t.

The equation for duration can be improved by adding the convexity term:

Calculating the Change in Bond Prices with Interest Rates Using Duration + Convexity
Adjustment

∆y = yield change

∆P = Bond price change

Convexity can also be estimated with a simpler formula, similar to the approximation formula
for duration:

1. Convexity Approximation Formula

Convexity P+ + P- - 2P0 P0 = Bond price.
 P = Bond price when interest rate is incremented.
───────────── -
= P+ = Bond price when interest rate is decremented.
2 x P0(Δy)2
∆y = change in interest rate in decimal form.

Note, however, that this convexity approximation formula must be used with this convexity
adjustment formula, then added to the duration adjustment:

1. Convexity Adjustment Formula

∆y = change in interest rate in decimal
Convexity Adjustment = Convexity x 100 x (Δy)2
form.

Hence:

Bond Price Change Formula

Bond Price Change = Duration x Yield Change + Convexity Adjustment

Important Note! The convexity can actually have several values depending on the convexity
adjustment formula used. Many calculators on the Internet calculate convexity according to the
following formula:

2. Convexity Approximation Formula

P0 = Bond price.
P+ + P- - 2P0
Convexity P = Bond price when interest rate is incremented.
───────────── -
= P+ = Bond price when interest rate is decremented.
P0(Δy)2
∆y = change in interest rate in decimal form.

Note that this formula yields double the convexity as the Convexity Approximation Formula #1.
However, if this equation is used, then the convexity adjustment formula becomes:

2. Convexity Adjustment Formula

Convexity Adjustment = Convexity/2 x 100 x (Δy)2 ∆y = change in interest rate in decimal form.

As you can see in the Convexity Adjustment Formula #2 that the convexity is divided by 2, so
using the Formula #2's together yields the same result as using the Formula #1's together.

To add further to the confusion, sometimes both convexity measure formulas are calculated by
multiplying the denominator by 100, in which case, the corresponding convexity adjustment
formulas are multiplied by 10,000 instead of just 100! Just keep in mind that convexity values as
calculated by various calculators on the Internet can yield results that differ by a factor of 100.
They can all be correct if the correct convexity adjustment formula is used!

Convexity is usually a positive term regardless of whether the yield is rising or falling, hence, it
is positive convexity. However, sometimes the convexity term is negative, such as occurs when
a callable bond is nearing its call price. Below the call price, the price-yield curve follows the
same positive convexity as an option-free bond, but as the yield falls and the bond price rises to
near the call price, the positive convexity becomes negative convexity, where the bond price is
limited at the top by the call price. Hence, similar to the terms for modified and effective
duration, there is also modified convexity, which is the measured convexity when there is no
expected change in future cash flows, and effective convexity, which is the convexity measure
for a bond for which future cash flows are expected to change.

Price Value of a Basis Point

Sometimes the volatility of bond prices to interest rates is calculated as the absolute value of the
change in price when the interest rate changes by 1 basis point (0.01%), which is called, aptly
enough, the price value of a basis point (PVBP), or the dollar value of a 01 (DV01).

PVBP = |initial price – price if yield changes by 1 basis point|

(Math note: the expression |x| denotes the absolute value of x.)

Although bond prices increase more when yields decline than decrease when yields increase, a
change in yield of 1 basis point is considered so small that the difference is negligible. Since
duration is the approximate change in bond price for a 100 basis point change in yield, the price
value of a basis point is 1% of the duration percentage.

For instance, in the example of above, the duration for the bond was found to be 7.45%. Hence,
the PVBP is equal to:

PVBP = Duration x 1% = 7.45% x 0.01 = 0.0745 x 0.01 = .0745%

So a bond selling for par would change by:

Price Change = 100 x 0.0745% = 100 x 0.000745 = $0.0745

Hence, a bond with a par value of $1,000 would change in price by $0.75 (rounded) when the
yield changes by 1 basis point.

Yield Volatility (Interest Rate Volatility)

Duration gives an estimate of the interest rate risk of a particular bond by relating the change in
price to the change in yield, but neither duration nor convexity gives a complete picture of
interest rate risk because some bonds change in yield, and therefore price, more than other bonds,
for a given coupon rate and current yield, when the prevailing rates change, primarily due to
changes in the perceived probability of default risk.

For instance, U.S. Treasuries generally have lower coupon rates and current yields than corporate
bonds of similar maturities because of the difference in default risk. Therefore, U.S. Treasuries
should have higher durations than corporate bonds, and, therefore, change in price more when
market interest rates change. However, changes in perception of the risk of default may also
change, blunting or augmenting what duration would predict.

For instance, during the recent subprime mortgage crisis, many bonds were perceived to be more
risky than investors realized, even those that had received top ratings from the credit rating
agencies, and so, many securities, especially those based on subprime mortgages, lost value,
greatly increasing their yields, while yields on Treasuries declined as the demand for these
securities, which are considered to be free of default risk, increased in price caused, not by the
decline in market interest rates, but by the flight to quality—selling risky securities to buy
securities with little or no default risk. The flight to quality is augmented by the fact that laws
and regulations require that pension funds and other funds that are held for the benefit of others
in a fiduciary capacity be invested in investment grade securities. So when investment ratings
decline for a large number of securities to below investment grade, managers of funds held in
trust must sell the riskier securities and buy securities that are likely to retain an investment grade
rating or be free of default risk—in most cases, U.S. Treasuries.

Therefore, yield volatility, and therefore, interest rate risk, is greater for securities with more
default risk, even if their durations are the same.