Duration (finance)

Duration (finance) measures how sensitive the price of a fixed-income security is to changes in interest rates. It is used to compare the interest rate risk of different bonds and to build hedges, often alongside convexity and the price value of a basis point (DV01). Duration is expressed in years for timing-based definitions and as a rate sensitivity for risk measures, and is most accurate for small, parallel shifts in yield.

Macaulay duration, introduced in 1938, is the present-value weighted average time to a bond’s cash flows and provides the link between payment timing and rate risk. Modified duration converts that timing statistic into an expected percentage price change for a stated compounding convention and a small change in yield to maturity.

When yields differ by maturity, Fisher–Weil duration discounts each cash flow at its own zero-coupon rate so that the parallel-shift result holds on a given term structure. To analyse non-parallel curve moves, practitioners use key rate durations that isolate sensitivity at selected maturities. For bonds whose cash flows vary with rates, such as securities with embedded options, effective duration is estimated by repricing the instrument for small up and down shifts in the curve while allowing cash flows to change.

Definition and intuition

This section uses the following conventions. Times $t_{i}$ are in years. The nominal yield to maturity is $y$ with $m$ compounding periods per year. Cash flows are $C_{i}$. The price as a function of yield is $$P(y)\;=\;\sum _{i=1}^{n}{\frac {C_{i}}{{\bigl (}1+y/m{\bigr )}^{mt_{i}}}}\,.$$

Define the present value of each payment $PV_{i}=C_{i}/(1+y/m)^{mt_{i}}$ and weights $w_{i}=PV_{i}/P$ which sum to one. Macaulay duration is the present-value-weighted average time to the cash flows: $$D_{\text{Mac}}\;=\;\sum _{i=1}^{n}t_{i}\,w_{i}\;=\;{\frac {\sum _{i}t_{i}\,PV_{i}}{\sum _{i}PV_{i}}}\,.$$ It summarises payment timing. For a zero-coupon bond that pays only at time $T$, $D_{\text{Mac}}=T$. For a level-coupon bond it lies between zero and final maturity.

To link timing to price sensitivity, differentiate the price with respect to yield. Modified duration is the first-order sensitivity of price to a small parallel change in $y$: $$D_{\text{mod}}\;=\;-\,{\frac {1}{P(y)}}\,{\frac {\mathrm {d} P}{\mathrm {d} y}}\;=\;{\frac {D_{\text{Mac}}}{1+y/m}}\,.$$ For a small change $\Delta y$ the approximation is $${\frac {\Delta P}{P}}\;\approx \;-\,D_{\text{mod}}\,\Delta y\,.$$

With continuous compounding at rate $r$, pricing is $P(r)=\sum _{i}C_{i}\,e^{-rt_{i}}$ and $$D_{\text{cont}}\;=\;-\,{\frac {1}{P(r)}}\,{\frac {\mathrm {d} P}{\mathrm {d} r}}\;=\;\sum _{i}t_{i}\,w_{i}\;=\;D_{\text{Mac}}\,.$$ These relations keep notation consistent across compounding conventions.

Worked examples

• Zero-coupon bond

Assume maturity $T=3$ years and yield $y=5\%$ with annual compounding ($m=1$). Then $$D_{\text{Mac}}=T=3,\qquad D_{\text{mod}}={\frac {T}{1+y/m}}={\frac {3}{1+0.05}}\approx 2.857.$$ A 25-basis-point change in yield ($\Delta y=0.0025$) gives $${\frac {\Delta P}{P}}\approx -\,D_{\text{mod}}\,\Delta y\approx -\,2.857\times 0.0025\approx -\,0.71\%.$$

• Level-coupon bond

Consider a two-year bond with a 5% annual coupon and yield $y=6\%$ (annual compounding). Present values of the cash flows: $$PV_{1}={\frac {5}{1.06}}\approx 4.717,\qquad PV_{2}={\frac {105}{1.06^{2}}}\approx 93.450.$$ Price and cash-flow weights: $$P=PV_{1}+PV_{2}\approx 98.167,\qquad w_{1}={\frac {PV_{1}}{P}}\approx 0.048,\quad w_{2}={\frac {PV_{2}}{P}}\approx 0.952.$$ Macaulay duration: $$D_{\text{Mac}}=1\cdot w_{1}+2\cdot w_{2}\approx 1\cdot 0.048+2\cdot 0.952\approx 1.952.$$ Modified duration: $$D_{\text{mod}}={\frac {D_{\text{Mac}}}{1+y}}={\frac {1.952}{1.06}}\approx 1.842.$$ A 50-basis-point rise in yield ($\Delta y=0.005$) implies $${\frac {\Delta P}{P}}\approx -\,D_{\text{mod}}\,\Delta y\approx -\,1.842\times 0.005\approx -\,0.92\%.$$

Term-structure intuition

When the term structure is not flat, discounting each payment at its own zero-coupon rate preserves the weighting idea in Macaulay’s statistic and leads to the Fisher–Weil refinement for parallel shifts of the zero-rate curve. Non-parallel movements are analysed with key rate durations in later sections.

Formal derivation

Let a fixed-income instrument pay cash flows $C_{i}$ at times $t_{i}$ (years), $i=1,\dots ,n$. With a yield to maturity $y$ compounded $m$ times per year, the price as a function of yield is $$P(y)=\sum _{i=1}^{n}{\frac {C_{i}}{{\bigl (}1+y/m{\bigr )}^{mt_{i}}}}\,.$$

Write the present value of each payment as $PV_{i}=C_{i}/(1+y/m)^{mt_{i}}$ and define weights $w_{i}=PV_{i}/P(y)$ so that $\sum _{i}w_{i}=1$.

Differentiating $P(y)$ with respect to $y$ gives $${\frac {dP}{dy}}=\sum _{i=1}^{n}C_{i}{\frac {d}{dy}}{\Bigl (}1+{\frac {y}{m}}{\Bigr )}^{-mt_{i}}=-{\frac {1}{1+y/m}}\sum _{i=1}^{n}t_{i}\,PV_{i}\,.$$

Hence the modified duration is $$D_{\text{mod}}(y)\equiv -{\frac {1}{P(y)}}\,{\frac {dP}{dy}}={\frac {\sum _{i}t_{i}\,PV_{i}}{P(y)}}\cdot {\frac {1}{1+y/m}}={\frac {D_{\text{Mac}}}{1+y/m}}\,,$$ where the Macaulay duration is the present-value-weighted average time to cash flow $$D_{\text{Mac}}=\sum _{i=1}^{n}t_{i}\,w_{i}={\frac {\sum _{i}t_{i}\,PV_{i}}{\sum _{i}PV_{i}}}\,.$$ For a small change $\Delta y$, the first-order approximation is $${\frac {\Delta P}{P}}\approx -\,D_{\text{mod}}\,\Delta y\,.$$ These relations assume fixed cash flows and a small parallel move in the quoted yield.

Continuous compounding

If pricing uses a continuously compounded rate $r$, then $$P(r)=\sum _{i=1}^{n}C_{i}\,e^{-rt_{i}},\qquad {\frac {dP}{dr}}=-\sum _{i=1}^{n}t_{i}\,C_{i}\,e^{-rt_{i}}=-\sum _{i=1}^{n}t_{i}\,PV_{i}\,,$$ so $$-\,P^{-1}{\frac {dP}{dr}}=D_{\text{Mac}}\,.$$ Thus modified and Macaulay duration coincide under continuous compounding.

Term-structure version (Fisher–Weil)

When the term structure is not flat, discount each cash flow at its own zero-coupon rate $z(t)$. For a parallel shift $a$ to the zero curve, $$P(a)=\sum _{i=1}^{n}C_{i}\exp \!{\Bigl (}-\!\int _{0}^{t_{i}}[z(u)+a]\,du{\Bigr )}=\sum _{i=1}^{n}C_{i}\,e^{-at_{i}}\,e^{-\int _{0}^{t_{i}}z(u)\,du}\,.$$ Differentiating at $a=0$ yields $$-\,P^{-1}\,{\frac {\partial P}{\partial a}}{\Big |}_{a=0}={\frac {\sum _{i}t_{i}\,PV_{i}}{\sum _{i}PV_{i}}}=D_{\text{FW}}\,,$$ the Fisher–Weil duration, which preserves present-value weighting with a full term structure.

Money duration and DV01

The dollar sensitivity is $${\frac {\partial P}{\partial y}}=-\,P\,D_{\text{mod}}\,,$$ so the value change for a one-basis-point move is $$\mathrm {DV01} =P\,D_{\text{mod}}\times 10^{-4}\,.$$

Macaulay duration

Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average time until cash flows are received, where each payment is weighted by its present value. The denominator is the sum of the weights, which is precisely the price of the bond. Consider some set of fixed cash flows. The present value of these cash flows is:

$V\;=\;\sum _{i=1}^{n}PV_{i}$

The Macaulay duration is defined as:

(1)     ${\text{MacD}}\;=\;{\frac {\sum _{i=1}^{n}t_{i}\,PV_{i}}{\sum _{i=1}^{n}PV_{i}}}\;=\;{\frac {\sum _{i=1}^{n}t_{i}\,PV_{i}}{V}}\;=\;\sum _{i=1}^{n}t_{i}\,{\frac {PV_{i}}{V}}$

where:

• $i$ indexes the cash flows,
• $PV_{i}$ is the present value of the $i$th cash payment from an asset,
• $t_{i}$ is the time in years until the $i$th payment will be received,
• $V$ is the present value of all future cash payments from the asset.

Here the weights are $w_{i}=PV_{i}/V$, which sum to 1. Hence ${\text{MacD}}$ is the present-value-weighted average of the times $t_{i}$.

For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely $t_{1}$ (the time to the first payment) and the time of the final cash flow. The Macaulay duration will equal the final maturity if and only if there is only a single payment at maturity. In symbols, if the cash-flow times are $(t_{1},\ldots ,t_{n})$, then

$t_{1}\;\leq \;{\text{MacD}}\;\leq \;t_{n},$

with the inequalities being strict unless it has a single cash flow. In terms of standard bonds (for which cash flows are fixed and positive), this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond.

Macaulay duration has the diagrammatic interpretation shown in figure 1.

Fig. 1: Macaulay duration

This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.

For most practical calculations, Macaulay duration is computed by discounting cash flows at the yield to maturity $y$:

(2)     $V\;=\;\sum _{i=1}^{n}PV_{i}\;=\;\sum _{i=1}^{n}CF_{i}\,e^{-y\,t_{i}}$

(3)     ${\text{MacD}}\;=\;\sum _{i=1}^{n}t_{i}\,{\frac {CF_{i}\,e^{-y\,t_{i}}}{V}}$

where:

• $i$ indexes the cash flows,
• $PV_{i}$ is the present value of the $i$th cash payment from an asset,
• $CF_{i}$ is the cash flow of the $i$th payment from an asset,
• $y$ is the yield to maturity (continuously compounded) for an asset,
• $t_{i}$ is the time in years until the $i$th payment will be received,
• $V$ is the present value of all cash payments from the asset until maturity.

Macaulay gave two alternative measures:

• Expression (1) is the general weighted-average form. When the present values $PV_{i}$ are computed using zero-coupon (spot) discount factors, it is termed the Fisher–Weil duration.
• Expression (3) uses the bond’s yield to maturity $y$ (assumed flat across maturities) to compute the discount factors.

The key difference between the two durations is that the Fisher–Weil duration allows for the possibility of a sloping yield curve, whereas the second form is based on a constant value of the yield $y$, not varying by term to payment. With the use of computers, both forms may be calculated but expression (3), assuming a constant yield, is more widely used because of the application to modified duration.

Duration versus Weighted Average Life (WAL)

Similarities in both values and definitions of Macaulay duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two. For example, a 5-year fixed-rate interest-only bond would have a Weighted Average Life of 5, and a Macaulay duration that should be very close. Mortgages behave similarly. The differences between the two are as follows:

1. Macaulay duration only measures fixed period cash flows, Weighted Average Life factors in all principal cash flows whether they be in fixed or floating. Thus for Fixed Period Hybrid ARM mortgages, for modeling purposes, the entire fixed period ends on the date of the last fixed payment or the month prior to reset.
2. Macaulay duration discounts all cash flows at the corresponding cost of capital. Weighted Average Life does not discount.
3. Macaulay duration uses both principal and interest when weighting cash flows. Weighted Average Life only uses principal.

Modified duration

In contrast to Macaulay duration, modified duration (sometimes abbreviated MD) is a price sensitivity measure, defined as the percentage derivative of price with respect to yield (the logarithmic derivative of bond price with respect to yield). Modified duration applies when a bond or other asset is considered as a function of yield. In this case one can measure the logarithmic derivative with respect to yield:

${\text{ModD}}(y)\equiv -{\frac {1}{V}}\cdot {\frac {\partial V}{\partial y}}=-{\frac {\partial \ln(V)}{\partial y}}$

When the yield is expressed continuously compounded, Macaulay duration and modified duration are numerically equal. To see this, if we take the derivative of price or present value, expression (2), with respect to the continuously compounded yield $y$ we see that:

${\frac {\partial V}{\partial y}}=-\sum _{i=1}^{n}t_{i}\cdot CF_{i}\cdot e^{-y\cdot t_{i}}=-{\text{MacD}}\cdot V,$

In other words, for yields expressed continuously compounded, ${\text{ModD}}={\text{MacD}}.$

where:

• $i$ indexes the cash flows,
• $t_{i}$ is the time in years until the $i$th payment will be received,
• $V$ is the present value of all cash payments from the asset.

Periodically compounded

In financial markets, yields are usually expressed periodically compounded (say annually or semi-annually) instead of continuously compounded. Then expression (2) becomes:

$V(y_{k})=\sum _{i=1}^{n}PV_{i}=\sum _{i=1}^{n}{\frac {CF_{i}}{(1+y_{k}/k)^{k\cdot t_{i}}}}$

${\text{MacD}}=\sum _{i=1}^{n}{\frac {t_{i}}{V(y_{k})}}\cdot {\frac {CF_{i}}{(1+y_{k}/k)^{k\cdot t_{i}}}}$

To find modified duration, when we take the derivative of the value $V$ with respect to the periodically compounded yield we find

${\frac {\partial V}{\partial y_{k}}}=-{\frac {1}{(1+y_{k}/k)}}\cdot \sum _{i=1}^{n}t_{i}\cdot {\frac {CF_{i}}{(1+y_{k}/k)^{k\cdot t_{i}}}}=-{\frac {{\text{MacD}}\cdot V(y_{k})}{(1+y_{k}/k)}}$

Rearranging (dividing both sides by -V ) gives:

${\frac {\text{MacD}}{(1+y_{k}/k)}}=-{\frac {1}{V(y_{k})}}\cdot {\frac {\partial V}{\partial y_{k}}}\equiv {\text{ModD}}$

which is the well-known relationship between modified duration and Macaulay duration:

${\text{ModD}}={\frac {\text{MacD}}{(1+y_{k}/k)}}$

where:

• $i$ indexes the cash flows,
• $k$ is the compounding frequency per year (1 for annual, 2 for semi-annual, 12 for monthly, 52 for weekly, etc.),
• $CF_{i}$ is the cash flow of the $i$th payment from an asset,
• $t_{i}$ is the time in years until the $i$th payment will be received (e.g. a two-year semi-annual would be represented by a $t_{i}$ index of 0.5, 1.0, 1.5, and 2.0),
• $y_{k}$ is the yield to maturity for an asset, periodically compounded
• $V$ is the present value of all cash payments from the asset.

This gives the well-known relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment (measured in units of time such as years) while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield.

Units

Macaulay duration has units of time (years).

Modified duration is dimensionless (a semi-elasticity). For a small change in the annual yield $\Delta y$ (in decimal form),

${\frac {\Delta V}{V}}\;\approx \;-{\text{ModD}}\;\Delta y.$

For a 100-basis-point change ($\Delta y=0.01$), the approximate percentage price change is ${\text{ModD}}\times 1\%$.

Non-fixed cash flows

Modified duration can be extended to instruments with non-fixed cash flows, while Macaulay duration applies only to fixed cash flow instruments. Modified duration is defined as the logarithmic derivative of price with respect to yield, and such a definition will apply to instruments that depend on yields, whether or not the cash flows are fixed.

Finite yield changes

Modified duration is defined above as a derivative (as the term relates to calculus) and so is based on infinitesimal changes. Modified duration is also useful as a measure of the sensitivity of a bond's market price to finite interest rate (i.e., yield) movements. For a small change in yield, $\Delta y$,

${\text{ModD}}\approx -{\frac {1}{V}}{\frac {\Delta V}{\Delta y}}\Rightarrow \Delta V\approx -V\cdot {\text{ModD}}\cdot \Delta y$

Thus modified duration is approximately equal to the percentage change in price for a given finite change in yield. So a 15-year bond with a modified duration of roughly 7 would fall approximately 7% in value for a 100-basis-point rise in yield in value if the interest rate increased by one percentage point (say from 7% to 8%).

Fisher–Weil duration

Fisher–Weil duration is a refinement of Macaulay’s duration which takes into account the term structure of interest rates. Fisher–Weil duration calculates the present values of the relevant cashflows (more strictly) by using the zero coupon yield for each respective maturity.

Key rate duration

Key rate durations (also called partial DV01s or partial durations) are a natural extension of the total modified duration to measuring sensitivity to shifts of different parts of the yield curve. Key rate durations might be defined, for example, with respect to zero-coupon rates with maturity '1M', '3M', '6M', '1Y', '2Y', '3Y', '5Y', '7Y', '10Y', '15Y', '20Y', '25Y', '30Y'. Thomas Ho (1992) introduced the term key rate duration. Reitano covered multifactor yield curve models as early as 1991 and has revisited the topic in a recent review.

Key rate durations require valuing an instrument off a yield curve and thus require building a yield curve. Ho's original methodology was based on valuing instruments off a zero or spot yield curve and used linear interpolation between "key rates", but the idea is applicable to yield curves based on forward rates, par rates, and so forth. Many technical issues arise for key rate durations (partial DV01s) that do not arise for the standard total modified duration because of the dependence of the key rate durations on the specific type of the yield curve used to value the instruments (see Coleman, 2011 ).

Bond formulas

For a standard bond with fixed, semi-annual payments the bond duration closed-form formula is:

${\text{Dur}}={\frac {1}{P}}\left(C{\frac {(1+ai)(1+i)^{m}-(1+i)-(m-1+a)i}{i^{2}(1+i)^{(m-1+a)}}}+{\frac {FV(m-1+a)}{(1+i)^{(m-1+a)}}}\right)$

• FV = par value
• C = coupon payment per period (half-year)
• i = discount rate per period (half-year)
• a = fraction of a period remaining until next coupon payment
• m = number of full coupon periods until maturity
• P = bond price (present value of cash flows discounted with rate i)

For a bond with coupon frequency $k$ but an integer number of periods (so that there is no fractional payment period), the formula simplifies to:

$MacD=\left[{\frac {(1+y/k)}{y/k}}-{\frac {100(1+y/k)+m(c/k-100y/k)}{(c/k)[(1+y/k)^{m}-1]+100y/k}}\right]/k$

where

• y = Yield (per year, in percent),
• c = Coupon (per year, in decimal form),
• m = Number of coupon periods.

Example 1

Consider a 2-year bond with face value of $100, a 20% semi-annual coupon, and a yield of 4% semi-annually compounded. The total PV will be:

$V=\sum _{i=1}^{n}PV_{i}=\sum _{i=1}^{n}{\frac {CF_{i}}{(1+y/k)^{k\cdot t_{i}}}}=\sum _{i=1}^{4}{\frac {10}{(1+.04/2)^{i}}}+{\frac {100}{(1+.04/2)^{4}}}$

$=9.804+9.612+9.423+9.238+92.385=130.462$

The Macaulay duration is then

${\text{MacD}}=0.5\cdot {\frac {9.804}{130.462}}+1.0\cdot {\frac {9.612}{130.462}}+1.5\cdot {\frac {9.423}{130.462}}+2.0\cdot {\frac {9.238}{130.462}}+2.0\cdot {\frac {92.385}{130.462}}=1.777\,{\text{years}}$.

Noting that ${\frac {y}{k}}={\frac {.04}{2}}=.02,{\frac {c}{k}}={\frac {20}{2}}=10$, the simple formula above gives:

${\text{MacD}}=\left[{\frac {(1.02)}{0.02}}-{\frac {100(1.02)+4(10-2)}{10[(1.02)^{4}-1]+2}}\right]/2=1.777\,{\text{years}}$

The modified duration, measured as percentage change in price per one percentage point change in yield, is:

${\text{ModD}}={\frac {\text{MacD}}{(1+y/k)}}={\frac {1.777}{(1+.04/2)}}=1.742$ (% change in price per 1 percentage point change in yield)

The DV01, measured as dollar change in price for a $100 nominal bond for a one percentage point change in yield, is

${\text{DV01}}={\frac {{\text{ModD}}\cdot 130.462}{100}}=2.27$ ($ per 1 percentage point change in yield)

where the division by 100 is because modified duration is the percentage change.

Example 2

Consider a bond with a $1000 face value, 5% coupon rate and 6.5% annual yield, with maturity in 5 years. The steps to compute duration are the following:

1. Estimate the bond value The coupons will be $50 in years 1, 2, 3 and 4. Then, in year 5, the bond will pay coupon and principal, for a total of $1050. Discounting to present value at 6.5%, the bond value is $937.66. The detail is the following:

Year 1: $50 / (1 + 6.5%) ^ 1 = 46.95

Year 2: $50 / (1 + 6.5%) ^ 2 = 44.08

Year 3: $50 / (1 + 6.5%) ^ 3 = 41.39

Year 4: $50 / (1 + 6.5%) ^ 4 = 38.87

Year 5: $1050 / (1 + 6.5%) ^ 5 = 766.37

2. Multiply the time each cash flow is received, times its present value

Year 1: 1 * $46.95 = 46.95

Year 2: 2 * $44.08 = 88.17

Year 3: 3 * $41.39 = 124.18

Year 4: 4 * $38.87 = 155.46

Year 5: 5 * 766.37 = 3831.87

TOTAL: 4246.63

3. Compare the total from step 2 with the bond value (step 1)

Macaulay duration: 4246.63 / 937.66 = 4.53

Money duration

The money duration, or basis point value or Bloomberg Risk, also called dollar duration or DV01 in the United States, is defined as negative of the derivative of the value with respect to yield:

$D_{\$}=DV01=-{\frac {\partial V}{\partial y}}.$

so that it is the product of the modified duration and the price (value):

$D_{\$}=DV01=BPV=V\cdot ModD/100$ ($ per 1 percentage point change in yield)

or

$D_{\$}=DV01=V\cdot ModD/10000$ ($ per 1 basis point change in yield)

The DV01 is analogous to the delta in derivative pricing (one of the "Greeks") – it is the ratio of a price change in output (dollars) to unit change in input (a basis point of yield). Dollar duration or DV01 is the change in price in dollars, not in percentage. It gives the dollar variation in a bond's value per unit change in the yield. It is often measured per 1 basis point - DV01 is short for "dollar value of an 01" (or 1 basis point). The name BPV (basis point value) or Bloomberg "Risk" is also used, often applied to the dollar change for a $100 notional for 100bp change in yields - giving the same units as duration. PV01 (present value of an 01) is sometimes used, although PV01 more accurately refers to the value of a one dollar or one basis point annuity. (For a par bond and a flat yield curve the DV01, derivative of price w.r.t. yield, and PV01, value of a one-dollar annuity, will actually have the same value.) DV01 or dollar duration can be used for instruments with zero up-front value such as interest rate swaps where percentage changes and modified duration are less useful.

Application to value-at-risk (VaR)

Dollar duration $D_{\$}$ is commonly used for value-at-risk (VaR) calculation. To illustrate applications to portfolio risk management, consider a portfolio of securities dependent on the interest rates $r_{1},\ldots ,r_{n}$ as risk factors, and let

$V=V(r_{1},\ldots ,r_{n})\,$

denote the value of such portfolio. Then the exposure vector ${\boldsymbol {\omega }}=(\omega _{1},\ldots ,\omega _{n})$ has components

$\omega _{i}=-D_{\$,i}:={\frac {\partial V}{\partial r_{i}}}.$

Accordingly, the change in value of the portfolio can be approximated as

$\Delta V=\sum _{i=1}^{n}\omega _{i}\,\Delta r_{i}+\sum _{1\leq i,j\leq n}O(\Delta r_{i}\,\Delta r_{j}),$

that is, a component that is linear in the interest rate changes plus an error term which is at least quadratic. This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms. Typically cubic or higher terms are truncated. Quadratic terms, when included, can be expressed in terms of (multi-variate) bond convexity. One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases (e.g., Gaussian distribution assuming a linear approximation), even analytically. The formula can also be used to calculate the DV01 of the portfolio (cf. below) and it can be generalized to include risk factors beyond interest rates.

Risk – duration as interest rate sensitivity

The primary use of duration (modified duration) is to measure interest rate sensitivity or exposure. Thinking of risk in terms of interest rates or yields is very useful because it helps to normalize across otherwise disparate instruments. Consider, for example, the following four instruments, each with 10-year final maturity:

Description	Coupon ($ per year)	Initial Price (per $100 notional)	Final Principal Re-Payment	Yield	Macaulay Duration (years)	Modified Duration (% per 100bp yld ch)	BPV or DV01 ($ per 100bp yld ch)
5% semi-annual coupon bond	$5	$100	$100	5%	7.99yrs	7.79%	$7.79
5% semi-annual annuity	$5	$38.9729	$0	5%	4.84yrs	4.72%	$1.84
zero-coupon bond	$0	$61.0271	$100	5%	10yrs	9.76%	$5.95
5% fixed-floating swap, Receive fixed	$5	$0	$0	5%	NA	NA	$7.79

All four have a 10-year maturity, but the sensitivity to interest rates, and thus the risk, will be different: the zero-coupon has the highest sensitivity and the annuity the lowest.

Consider first a $100 investment in each, which makes sense for the three bonds (the coupon bond, the annuity, the zero-coupon bond - it does not make sense for the interest rate swap for which there is no initial investment). Modified duration is a useful measure to compare interest rate sensitivity across the three. The zero-coupon bond will have the highest sensitivity, changing at a rate of 9.76% per 100bp change in yield. This means that if yields go up from 5% to 5.01% (a rise of 1bp) the price should fall by roughly 0.0976% or a change in price from $61.0271 per $100 notional to roughly $60.968. The original $100 invested will fall to roughly $99.90. The annuity has the lowest sensitivity, roughly half that of the zero-coupon bond, with a modified duration of 4.72%.

Alternatively, we could consider $100 notional of each of the instruments. In this case the BPV or DV01 (dollar value of an 01 or dollar duration) is the more natural measure. The BPV in the table is the dollar change in price for $100 notional for 100bp change in yields. The BPV will make sense for the interest rate swap (for which modified duration is not defined) as well as the three bonds.

Modified duration measures the size of the interest rate sensitivity. Sometimes we can be misled into thinking that it measures which part of the yield curve the instrument is sensitive to. After all, the modified duration (% change in price) is almost the same number as the Macaulay duration (a kind of weighted average years to maturity). For example, the annuity above has a Macaulay duration of 4.8 years, and we might think that it is sensitive to the 5-year yield. But it has cash flows out to 10 years and thus will be sensitive to 10-year yields. If we want to measure sensitivity to parts of the yield curve, we need to consider key rate durations.

For bonds with fixed cash flows a price change can come from two sources:

1. The passage of time (convergence towards par). This is of course totally predictable, and hence not a risk.
2. A change in the yield. This can be due to a change in the benchmark yield, and/or change in the yield spread.

The yield-price relationship is inverse, and the modified duration provides a very useful measure of the price sensitivity to yields. As a first derivative it provides a linear approximation. For large yield changes, convexity can be added to provide a quadratic or second-order approximation. Alternatively, and often more usefully, convexity can be used to measure how the modified duration changes as yields change. Similar risk measures (first and second order) used in the options markets are the delta and gamma.

Modified duration and DV01 as measures of interest rate sensitivity are also useful because they can be applied to instruments and securities with varying or contingent cash flows, such as options.

Embedded options and effective duration

See also: Effective convexity

For bonds that have embedded options, such as putable and callable bonds, modified duration will not correctly approximate the price move for a change in yield to maturity.

Consider a bond with an embedded put option. As an example, a $1,000 bond that can be redeemed by the holder at par at any time before the bond's maturity (i.e. an American put option). No matter how high interest rates become, the price of the bond will never go below $1,000 (ignoring counterparty risk). This bond's price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cash flows.

To price such bonds, one must use option pricing to determine the value of the bond, and then one can compute its delta (and hence its lambda), which is the duration. The effective duration is a discrete approximation to this latter, and will require an option pricing model.

${\text{Effective duration}}={\frac {V_{-\Delta y}-V_{+\Delta y}}{2(V_{0})\Delta y}}$

where Δ y is the amount that yield changes, and $V_{-\Delta y}$ and $V_{+\Delta y}$ are the values that the bond will take if the yield falls by y or rises by y, respectively. (A "parallel shift"; note that this value may vary depending on the value used for Δ y.)

These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model (finance) § Interest rate derivatives.

Spread duration

Spread duration is the sensitivity of a bond's market price to a change in option-adjusted spread (OAS). Thus the index, or underlying yield curve, remains unchanged. Floating rate assets that are benchmarked to an index (such as 1-month or 3-month LIBOR) and reset periodically will have an effective duration near zero but a spread duration comparable to an otherwise identical fixed rate bond.

Average duration

The sensitivity of a portfolio of bonds such as a bond mutual fund to changes in interest rates can also be important. The average duration of the bonds in the portfolio is often reported. The duration of a portfolio equals the weighted average maturity of all of the cash flows in the portfolio. If each bond has the same yield to maturity, this equals the weighted average of the portfolio's bond's durations, with weights proportional to the bond prices. Otherwise the weighted average of the bond's durations is just a good approximation, but it can still be used to infer how the value of the portfolio would change in response to changes in interest rates.

Convexity

Main article: Bond convexity

Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative.

Convexity also gives an idea of the spread of future cashflows. (Just as the duration gives the discounted mean term, so convexity can be used to calculate the discounted standard deviation, say, of return.)

Note that convexity can be positive or negative. A bond with positive convexity will not have any call features - i.e. the issuer must redeem the bond at maturity - which means that as rates fall, both its duration and price will rise.

On the other hand, a bond with call features - i.e. where the issuer can redeem the bond early - is deemed to have negative convexity as rates approach the option strike, which is to say its duration will fall as rates fall, and hence its price will rise less quickly. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality. Similar to the above, in these cases, it may be more correct to calculate an effective convexity.

Mortgage-backed securities (pass-through mortgage principal prepayments) with US-style 15- or 30-year fixed-rate mortgages as collateral are examples of callable bonds.

Sherman ratio

The "Sherman ratio" is the yield offered per unit of bond duration, named after DoubleLine Capital's chief investment officer, Jeffrey Sherman. It has been called the "Bond Market's Scariest Gauge", and hit an all-time low of 0.1968 for the Bloomberg Barclays US Corporate Bond Index on Dec 31, 2020. The ratio is simply the yield offered (as a percentage), divided by the bond duration (in years).

See also

• Bond convexity
• Bond valuation
• CS01
• Day count convention
• Duration gap
• Fixed-income attribution § First principles versus perturbational attribution
• Immunization (finance)
• List of finance topics
• Stock duration

Further reading

• Fabozzi, Frank J. (1999), "The basics of duration and convexity", Duration, Convexity, and Other Bond Risk Measures, Frank J. Fabozzi Series, vol. 58, John Wiley and Sons, ISBN 9781883249632
• Mayle, Jan (1994), Standard Securities Calculation Methods: Fixed Income Securities Formulas for Analytic Measures, vol. 2 (1st ed.), Securities Industry and Financial Markets Association, ISBN 1-882936-01-9. The standard reference for conventions applicable to US securities.

• Rojas Arzú, Jorge; Roca, Florencia (December 2018), Risk Management and Derivatives Explained, Amazon Kindle Direct Publishing, pp. 43–44, ISBN 9781791814342

External links

• Risk Encyclopedia for a good explanation on the multiple definitions of duration and their origins.
• Step-by-step video tutorial
• Investopedia’s duration explanation