Duration and Convexity

Although bonds are generally safer investments than stocks, bonds do have risks. A common risk is interest rate risk. Interest rates change because the supply and demand of loanable funds changes, because inflation or expected inflation changes, or the central bank changes the target interest rate, which affects all other interest rates. For instance, in the United States, the Federal Reserve influences interest rates by setting the federal funds rate, which is the rate charged in interbank lending, and which affects all other interest rates in the United States. These factors make bond prices volatile, but certain bonds are more volatile than others, meaning they have greater interest rate risk.

Duration measures this risk, and convexity measures how duration itself changes with interest rates. So, duration estimates bond price changes with small changes in interest rates, while duration plus convexity estimates bond price changes with larger changes in interest rates.

This article explains duration and convexity, and it presents several formulas for calculating each, but a bond investor generally does not need to know this, since most bond listings list the duration. Nonetheless, it is still helpful to understand these concepts.

The primary point to remember is that the longer the duration, the greater the interest rate risk and volatility, and vice versa. Bonds with the longest duration will have the greatest volatility.

Bond prices change inversely with interest rates, so bonds have interest rate risk. However, interest rate risk is greatest when interest rates are near 0 and least when interest rates are high. Therefore, it is more accurate to describe the interest rate sensitivity of bonds. One method of measuring interest rate sensitivity due to changes in market interest rates is by the full valuation approach, which simply calculates what bond prices will be if the interest rate changed by specific amounts. The full valuation approach is based on the fact that the price of a bond = the sum of the present value of each coupon payment + the present value of the principal repayment. Present value depends on the market interest rate, so bond prices depend on the market interest rate.

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

Duration

Another method to measure interest rate sensitivity, 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. The longer the duration, the greater the sensitivity to interest rate changes. Mathematically, duration is the 1st derivative of the price-yield curve, which is a line tangent to the curve at the current price-yield point.

Graphically, the duration of a bond can be envisioned as the fulcrum under a seesaw, placed so as to balance the weights of the present values of the coupon payments and the principal payment.

Graph of the present value of cash flows for a 30-year Treasury bond with a 5% coupon, 5% market rate, and a duration of 15.84 years.

The effective duration compares the interest rate sensitivities of different securities, especially for securities where the cash flow may change because of the yield change, such as for a callable bond. Securities with the same duration have the same interest rate sensitivity. For instance, since zero-coupon bonds only pay the face value at maturity, the duration of a zero = 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 small change in its yield to maturity (YTM). It should not be surprising that there is a relationship between the change in bond price and the change in duration when the yield changes, since both the bond and duration depend on the present values of the bond's cash flows. In fact, a very simple relationship exists between them: when the YTM changes by 1%, the bond price changes by the duration converted to a percentage. So, for instance, the price of a bond with a 10-year duration would change by 10% for a 1% change in the interest rate.

Macaulay Duration

Before 1938, it was well known that the maturity of a bond affected its interest rate sensitivity, but it was also known that bonds with the same maturity could differ widely in price changes with changes to yield. On the other hand, zero-coupon bonds always exhibited the same interest rate sensitivity. Therefore, Frederick Macaulay reasoned that a better measure of interest rate sensitivity is to consider a coupon bond as a series of zero-coupon bonds, where each payment is a zero-coupon bond weighted by the present value of the payment divided by the bond price.

Consequently, duration has several simple properties:

If a bond is callable or has prepayment risk, such as with mortgage bonds, then the maturity of the bond, and therefore its duration, may be less than what the stated maturity would suggest. In these cases, duration is calculated by using the effective maturity (a.k.a. average maturity), which is the shortest maturity if the bond was called or the average maturity of a group of bonds with prepayment risk that have similar characteristics, such as similar mortgage bonds.

The Macaulay duration is calculated by this formula:

Macaulay Duration Formula
Macaulay Duration = 1 × PVCF1 + 2 × PVCF2 + ... + n × PVCFn
k × p

Using sigma notation:

Macaulay Duration Formula
Macaulay Duration = T

t=1
t × PVCFt

k × p

Example 1: Calculating Macaulay Duration

A 3-year bond has a par value of $100 with a coupon rate of 6% and a current yield of 6%. The duration can be calculated as follows:

Interest Rate 6.0%
Coupon Rate 6.0%
Par Value $100
Current Bond Price $100.00
Number of Payments
per Year
2
Time Period Cash Flow Discount PV Time Period × Present Value of Cash Flow
1 $3.00 0.9709 $2.91 $2.91
2 $3.00 0.9426 $2.83 $5.66
3 $3.00 0.9151 $2.75 $8.24
4 $3.00 0.8885 $2.67 $10.66
5 $3.00 0.8626 $2.59 $12.94
6 $103.00 0.8375 $86.26 $517.57

Totals: $118 $100.00 $557.97

Macaulay Duration Example
Duration = Sum of (Time Period × Present Value of Cash Flows)
(Number of Payments per Year × Bond Price)
= $557.97 / (2 × $100)
= 2.79

Because the bond price = the total present value of all bond payments, the bond price will change inversely to changes in yield, which can be calculated approximately by this equation:

ΔP
P
= −Δy × T

t=1
CFt × t
eyt
= −Δy × D

Multiply both sides by P:

ΔP = −Δy × P × T

t=1
CFt × t
eyt
= −Δy × P × D

New Bond Price = P + ΔP

So if interest rates increased by 1%, then the change in the bond price in Example 1 can be calculated thus:

Example: New Bond Price

Compare this calculation with the bond price as given by the sum of the present value of its payments:

Interest Rate 7.0%
Coupon Rate 6.0%
Par Value $97.34
Current Bond Price $100.00
Number of Payments
per Year
2
Time Period Cash Flow Discount PV
1 $3.00 0.9662 $2.90
2 $3.00 0.9335 $2.80
3 $3.00 0.9019 $2.71
4 $3.00 0.8714 $2.61
5 $3.00 0.8420 $2.53
6 $103.00 0.8135 $83.79

Totals: $118 $97.34

As you can see, bond prices as calculated using Macaulay duration is very close to the price calculated with the present values of the cash flows when the interest rate change is small. Note that in the above example, when the yield changes by 1%, then the bond price can simply be multiplied by the duration converted to a percentage, since 1% × 2.79 = .0279 = 2.79%.

The duration adjustment is a close approximation for small changes in interest rates. However, duration changes as well, which is measured by the bond's convexity (discussed later). Because duration also changes, larger changes in interest rates will yield larger discrepancies between the actual bond price and the price calculated using duration. Duration can also be approximated by this formula:

Duration Approximation Formula
Duration = P − P+
2 × P0(Δy)

Modified Duration

Macaulay duration is a better measure of interest rate sensitivity than the term of a bond. So, interest rate sensitivity can easily be compared by comparing the durations of different bonds or portfolios. Modified duration gives a better measure of the sensitivity of changes in bond price with changes in yield. Specifically:

dP/P
dy
= DMac
1 + ytm/k
Modified Duration Formula
Modified Duration = DMac
1 + y/k

So equating the change in bond price calculated for Example 1 above to modified duration yields:

Modified Duration

As you can see, modified duration gives a better estimate of the new price than Macaulay duration, since it is closer to the price as determined by discounted cash flows. Like Macaulay duration, modified duration is valid only when the change in yield is small and the yield change 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 it will be called. Of course, interest rates usually change in small steps, so duration measures interest rate sensitivity effectively.

Duration and Modified Duration Formulas for Bonds using Microsoft Excel

Duration or Modified Duration

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

Modified duration

Note that modified duration is always slightly less than duration, since the modified duration is the duration divided by 1 + 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.

Note that the price-yield curve is convex, and that the modified duration is the slope of the tangent line to a particular market yield, and that the discrepancy between the price-yield curve and the modified duration increases with greater changes in the interest rate. It can easily be seen that modified duration changes as the yield changes because it is obvious that the slope of the line changes with different yields. The gap between the modified duration and the convex price-yield curve is the convexity adjustment, which — as can be easily seen — is greater on the upside than on the downside.

Diagram of modified duration and convexity and the price-yield curve of a bond.

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 average of the present value of the bond's cash flows, a simple calculation for duration is not valid if the change in yield could change cash flow. Valuation models must be used to calculate new prices for yield changes 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

Effective Duration

= ΔP/P
Δi

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:

Duration Formula for Coupon Bond
Coupon Bond Duration = 1 + ytm
ytm
(1 + ytm) + T (c − ytm)
c [(1 + ytm)T− 1] + ytm

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
Duration for Coupon Bond Selling for Face Value = 1 + ytm
ytm
[ 1 − 1
(1 + ytm)T
]

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

Fixed Annuity Duration Formula
Fixed Annuity Duration = 1 + ytm
ytm
T
(1 + ytm)T − 1

A perpetuity is a bond without a maturity date, so it 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

Perpetuity Duration

= 1 + ytm
ytm

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 sensitivity to the portfolio.

The duration for a bond portfolio = the weighted average of the duration for each type of bond in the portfolio:

Portfolio Duration = w1D1 + w2D2 + … + wKDK

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

Investment Tip: Minimize Duration Risk

When yields are low, investors, who are risk-averse but who want to earn a higher yield, often buy bonds with longer durations, since longer-term bonds pay higher interest rates. But even the yields of longer-term bonds are only marginally higher than short-term bonds, because insurance companies and pension funds, who are major buyers of bonds, are restricted to investment grade bonds, so they bid up those prices, forcing the remaining bond buyers to bid up the price of junk bonds, thereby diminishing their yield even though they have higher risk. Indeed, interest rates may even turn negative. In June 2016, the 10-year German bond, called the bund, sported negative interest rates several times, when the price of the bond actually exceeded its principal.

Interest rates cycle endlessly from high to low to high, so buying long-duration bonds when yields are low increases the likelihood bond prices will be lower if the bonds are sold before maturity. Sometimes called duration risk, it is more commonly called interest rate risk. Duration risk would be especially large in buying bonds with negative interest rates. On the other hand, if long-term bonds are held to maturity, then you may incur an opportunity cost, earning low yields when interest rates are higher.

Therefore, especially when yields are extremely low, as they were starting in 2008 and continuing even into 2016, it is best to buy bonds with the shortest durations, especially when the difference in interest rates between long-duration portfolios and short-duration portfolios is less than the historical average.

On the other hand, buying long-duration bonds make sense when interest rates are high, to not only earn the high interest, but also to realize capital appreciation when interest rates decline.

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.

A diagram of the convexity of 2 representative bond portfolios, showing the general relationship between the percentage change in the value of bond portfolios to a change in yield..
Diagram of the convexity of 2 bond portfolios.

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. Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is greater on the upside than on the downside, so the absolute change in price for a given change in yield will be slightly greater when yields decline rather than increase. Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

Some additional properties of convexity include:

Convexity is calculated by this equation:

Convexity Formula
Convexity = 1
P × (1 + y)2
T

t=1
[ CFt
(1 + y)t
(t2 + t) ]

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

Change in Bond Prices Using Duration + Convexity Adjustment
ΔP
P
= −Dm × Δy + (Δy)2
2
× Convexity

Convexity can also be estimated with a simpler formula:

1. Convexity Approximation Formula
Convexity = P+ + P − 2P0
2 × P0(Δy)2

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
Convexity Adjustment = Convexity × 100 × (Δy)2

Hence:

Bond Price Change Formula
Bond Price Change = Duration × Yield Change + Convexity Adjustment

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

2. Convexity Approximation Formula
Convexity = P+ + P − 2P0
P0(Δy)2
  • P0 = initial bond price
  • P+ = bond price when interest rate is decremented
  • P = bond price when interest rate is incremented
  • Δy = 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 × 100 × (Δy)2

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 online calculators 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, like 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.

Basis Point Value (BPV) Measures the Change in Cash Price of a Bond When Yield Changes by 1 Basis Point

Bond managers often want to know how much the market value of a bond portfolio changes when interest rates change by 1 basis point. This can be calculated using the basis point value (BPV) [aka price value of a basis point (PVBP), dollar value of a 01 (DV01)], which also measures the volatility of bond prices to interest rates, calculated as the absolute value of the change in price when the interest rate changes by 1 basis point (0.01%). At bond trading desks, trading exposure is often set in terms of the BPV.

BPV = |initial price − price if yield changes by 1 basis point|

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

Because BPV depends on modified duration and on the convexity of the bond price/yield, BPV is larger at lower interest rates, and the difference in BPV between an upward shift and downward shift in interest rates will be larger for longer maturities.

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, although this difference is larger for longer maturities. Since modified 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 price change predicted by modified duration. Recall that:

Basis Point Value Formula
Change in Market Price = Yield Change Percentage × Modified Duration
100
× Bond Price

So the price change per basis point change in market yield is:

Basis Point Value Formula
BPV = .01 × Modified Duration
100
× Bond Price

Examples: Calculating the Price Change of a Basis Point Change in Yield for a Given Duration

Given:

Case #1:

Case #2:

Yield Volatility (Interest Rate Volatility)

Duration gives an estimate of the interest rate sensitivity 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 bond yields can also change because of changes in the credit default risk as evidenced by changes in the credit ratings of the issuer or because of detrimental changes to the economy that may increase the credit default risk of many businesses.

For instance, U.S. Treasuries 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 bond prices, blunting or augmenting what duration would predict.

For instance, during the 2007 - 2009 subprime mortgage crisis, many bonds were riskier 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 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 held for the benefit of others in a fiduciary capacity be invested only in investment grade securities. So when investment ratings decline for many securities to below investment grade, managers of funds held in trust must sell the riskier securities and buy securities likely to retain an investment grade rating or be free of default risk, such as 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.