Duration and Convexity
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,bas
is)

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
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 underestimates the resulting
bond prices. 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 priceyield 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 Plus Convexity
∆y = yield change
∆P = Bond price change
Convexity can also be estimated with a simpler formula,
similar to the approximation formula for duration:
Convexity 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.
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.