# Duration and Convexity

Bond prices change inversely with interest rates, and, hence, there is interest rate risk with bonds. One method of measuring interest rate risk 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 is equal to the sum of the present value of each coupon payment plus the present value of the principal payment. That the present value of a future payment depends on the interest rate is what causes bond prices to vary with the interest rate, as well.

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

## Duration

Another method to measure interest rate risk, 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. Consequently, duration is also called the average maturity or the effective maturity. The longer the duration, the longer is the average maturity, and, therefore, the greater the sensitivity to interest rate changes. 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. 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.

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 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 the two: 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 risk, 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 risk. Therefore, Frederick Macaulay reasoned that a better measure of interest rate risk 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. Hence, duration is the effective maturity of a bond, which is why it is measured in years. Not only can the Macaulay duration measure the effective maturity of a bond, it can also be used to calculate the average maturity of a portfolio of fixed-income securities.

Consequently, duration has several simple properties:

• duration is proportional to the maturity of the bond, since the principal repayment is the largest cash flow of the bond and it is received at maturity;
• duration is inversely related to the coupon rate, since there will be a larger difference between the present values for the earlier payments over the lesser value for the principal repayment;
• duration decreases with increasing payment frequency, since half of the present value of the cash flows is received earlier than with less frequent payments, which is why coupon bonds always have a shorter duration than zeros with the same maturity.

The Macaulay duration is calculated by 1st calculating the weighted average of the present value (PV) of each cash flow at time t by the following formula:

 wt = CFt / (1 + y)tBond Price = Present Value of Cash FlowBond Price t =time in years wt = weighted average of cash flow at time t CFt = Cash flow at time t y = yield to maturity

For a interest rate that is continuously compounded, the weighted average is equal to the following:

wt = CFt /eyt

Then these weighted averages are summed:

 Macaulay Duration = T ∑ t=1 t × wt T = number of cash flow periods. t =time in years wt = weighted average of cash flow at time t CFt = Cash flow at time t y = yield to maturity

#### Example 1: Calculating Duration

A three-year bond has a par value of \$100 with a coupon rate of 5% and a current continuously compounded yield of 6%. The duration can be calculated as follows:

Bond Characteristics Value Time (Years) Cash Flow PV Weight Interest Rate(continuously compounded) 6% Coupon Rate 5% Par Value \$100 Current Bond Price \$97.05 0.5 \$2.50 \$2.43 0.025 0.012 1.0 \$2.50 \$2.35 0.024 0.024 1.5 \$2.50 \$2.28 0.024 0.035 2.0 \$2.50 \$2.22 0.023 0.046 2.5 \$2.50 \$2.15 0.022 0.055 3.0 \$102.50 \$85.62 0.882 2.647 Totals: \$115 \$97.05 1.000 2.820 = Duration

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

 ΔBB = -Δy × T ∑ t=1 CFt × teyt = -Δy × D

Multiply both sides by B:

 ΔB = -Δy × B × T ∑ t=1 CFt × teyt = -Δy × B × D

So if interest rates increased by 0.1%, then the change in the bond price in Example 1 can be calculated thus:

Example 2: New Bond Price = \$97.050.1% × 2.820 × \$97.05 ≈ \$96.776319

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

Time (Years) PV Weight Time × Weight Interest Rate 6.1% Coupon Rate 5.0% Par Value \$100 New Bond Price \$96.78 0.5 \$2.42 0.025 0.013 1.0 \$2.35 0.024 0.024 1.5 \$2.28 0.024 0.035 2.0 \$2.21 0.023 0.046 2.5 \$2.15 0.022 0.055 3.0 \$85.36 0.882 2.646 Totals: \$96.78 1.000 2.819

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. In fact, when rounded, the values are equal. Note that in the above example, if the yield had changed by 1% instead of 0.1%, then the bond price can simply be multiplied by the duration converted to a percentage, since 1% × 2.820 = .0282 = 2.82%.

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 the following formula:

 Duration = P- – P+ 2 × P0(Δy) P0 = Bond price. P- = Bond price when interest rate is incremented. P+ = Bond price when interest rate is decremented. Δy = change in interest rate in decimal form.

### Modified Duration

Duration is measured in years, so it does not directly measure the change in bond prices with respect to changes in yield. Nonetheless, interest rate risk can easily be compared by comparing the durations of different bonds or portfolios. Modified duration, on the other hand, does measure the sensitivity of changes in bond price with changes in yield. Specifically:

 dP/Pdy = – DMac1 + y/k DMac = Macaulay Duration dP/P = small change in bond price dy = small change in yield y = yield to maturity k = number of payments per year
 Modified Duration = DMac 1 + y/k DMac = Macaulay Duration dP/P = small change in bond price dy = small change in yield y = yield to maturity k = number of payments per year

So equating the change in bond price calculated for Example 2 above to modified duration yields:

dP/P ÷ dy = –.27 ÷ 0.1 = –2.7 = –2.82 ÷ (1 + 6%/2) = –2.82 ÷ 1.03 = Macaulay Duration ÷ (1 + y/k)

In other words:

Bond Price Change = Yield Change × Modified Duration × Bond Price

So for the example above:

Bond Price Change = 0.1 × –2.7 × \$97.05 = –\$0.26.2035 ≈ \$0.26

The above calculation differs by less than a penny from the actual difference of \$.27 as calculated using the present value of the 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 that it will be called. Of course, interest rates usually only change in small steps, so duration measures interest rate risk effectively.

 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. 1 = Annual 2 = Semiannual 4 = Quarterly Basis = Day count basis. 0 = 30/360 (U.S. basis). This is the default if the basis is omitted. 1 = actual/actual (actual number of days in month/year). 2 = actual/360 3 = actual/365 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 = – ΔP/PΔi Δi = interest rate differential ΔP = Bond price at i + Δi – bond price at i - 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 is the following:

 Coupon Bond Duration = 1 + yy – (1 + y) + T (c – y)c [(1 + y)T– 1] + y 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 for Coupon Bond Selling for Face Value = 1 + yy [ 1 – 1 (1 + y)T ] 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 = 1 + yy – T (1 + y)T – 1 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 = 1 + y y Δi = interest rate differential ΔP = Bond price at i + Δi – bond price at i - Δi.

## 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 × Duration of Issue

## Investment Tip: Minimize Duration Risk

When yields are low, investors, who are risk-averse but who want to earn a higher yield, will 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, known as the bund, sported negative interest rates several times, when the price of the bond actually exceeded its principal.

Interest rates vary continually from high to low to high in an endless cycle, so buying long-duration bonds when yields are low increases the likelihood that bond prices will be lower if the bonds are sold before maturity. This is sometimes called duration risk, although it is more commonly known as 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, since you not only earn the high interest, but you may also realize capital appreciation if you sell when interest rates are lower.

## 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. Convexity is always positive for vanilla bonds. Furthermore, the price-yield curve flattens out at higher interest rates, so convexity is usually 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 the following:

• Convexity increases as yield to maturity decreases, and vice versa.
• Convexity decreases at higher yields because the price-yield curve flattens at higher yields, so modified duration is more accurate, requiring smaller convexity adjustments. This is also why convexity is more positive on the upside than on the downside.
• Among bonds with the same YTM and term length, lower coupon bonds have a higher convexity, with zero-coupon bonds having the highest convexity.
• This results because lower coupons or no coupons have the highest interest rate volatility, so modified duration requires a larger convexity adjustment to reflect the higher change in price for a given change in interest rates.

Convexity is calculated by the following equation:

 Convexity = 1P × (1 + y)2 T ∑ t=1 [ CFt(1 + y)t (t2 + t) ] 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:

 ΔPP = -Dm × Δy + (Δy)22 × Convexity Δy = yield change ΔP = Bond price change

Convexity can also be estimated with a simpler formula:

 Convexity = P+ + P- - 2P02 × P0(Δy)2 P0 = Bond price. P- = Bond price when interest rate is incremented. P+ = Bond price when interest rate is decremented. Δy = 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:

 Convexity Adjustment = Convexity × 100 × (Δy)2 Δy = change in interest rate in decimal form.

Hence:

 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 calculators on the Internet calculate convexity according to the following formula:

 Convexity = P+ + P- - 2P0 P0(Δy)2 P0 = Bond price. P- = Bond price when interest rate is incremented. P+ = Bond price when interest rate is decremented. 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:

 Convexity Adjustment = Convexity/2 × 100 × (Δy)2 Δy = 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, 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 will often want to know how much the market value of a bond portfolio will change 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:

 Change in Market Price = Yield Change Percentage × Modified Duration 100 × Bond Price

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

 BPV = 0.01 × Modified Duration 100 × Bond Price

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

Given:

• Modified Duration = 7.45% = 7.45/100 = .0745

Case #1:

• Market Price of Bond = \$1,000
• BPV = .0745 × .01 × 1,000 = 0.75
• So if the yield fell by 1 basis point, the bond price would rise to \$1000 + 0.75 = \$1000.75

Case #2:

• Market Price = \$900
• BPV = 0.0745 × .01 × 900 = 0.67
• So if the yield rose by 1 basis point, the bond price would decline to \$900 – 0.67 = \$899.33

## 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 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 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 bond prices, blunting or augmenting what duration would predict.

For instance, during the recent subprime mortgage crisis, many bonds were perceived to be 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 that are held for the benefit of others in a fiduciary capacity be invested only 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 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.