Effective Duration Calculator
Enter the face value, coupon rate, coupon frequency, years to maturity, yield to maturity and a small yield change to find the effective duration, the bond price and the coupons.
Updated
Effective Duration Calculator
Our Effective Duration Calculator shows how sensitive the price of a bond is to a change in interest rates. Enter the face value, the annual coupon rate, how often coupons are paid, the years to maturity, the yield to maturity and a small yield change. It reprices the bond with the yield moved up and down, and gives the effective duration together with the bond price, the modified and Macaulay durations and the coupons. It is made for investors, students and analysts who want a quick measure of interest rate risk.
Quick answer
A 1,200 bond with a 4% annual coupon, 5 years to maturity and a 4.4% yield is priced at 1,178.87 and has an effective duration of about 4.43 years.
- Formula: effective duration = (P− − P+) ÷ (2 × P × Δy).
- Meaning: the approximate percentage change in price for a 1% change in yield.
- Rule of thumb: a duration of 4.43 means a 1% rise in yield cuts the price by about 4.43%.
- Higher duration: more interest rate risk. Long maturities and low coupons raise it.
What Is Effective Duration?
Effective duration measures how much the price of a bond changes when interest rates change. It answers one question: if yields move by a small amount, by what percentage does the price of the bond go up or down?
It is found by experiment instead of by a closed formula. The bond is priced at its current yield, priced again with the yield a little higher, and once more with the yield a little lower. The gap between the two moved prices, compared with the starting price and the size of the move, is the effective duration.
The result is quoted in years. A bond with an effective duration of 4.43 loses about 4.43% of its price when yields rise by 1%, and gains about 4.43% when yields fall by 1%. Bond prices and yields always move in opposite directions.
Effective Duration Formula
÷ (2 × P × Δy)
- ED
- Effective duration, in years.
- P
- Bond price at the current yield.
- P−
- Bond price when the yield falls by Δy.
- P+
- Bond price when the yield rises by Δy.
- Δy
- The yield change as a decimal: 0.1% is 0.001.
Each of the three prices is the present value of the remaining coupons and the face value, discounted at the yield per coupon period. The Bond Price Calculator shows that step in full.
The top of the fraction is the full price swing from the lower yield to the higher one. That swing covers two yield moves, which is why the bottom is multiplied by 2.
How to Calculate Effective Duration
A bond has a face value of 1,200, a 4% coupon paid once a year, 5 years to maturity and a yield to maturity of 4.4%. Use a yield change of 0.1%:
P+ = 1,173.660140 (at 4.5%)
P− = 1,184.107615 (at 4.3%)
P− − P+ = 10.447475
2 × P × Δy = 2 × 1,178.869262
× 0.001 = 2.357739
ED = 10.447475 ÷ 2.357739
= 4.4311 years
- Find the coupon. 1,200 × 4% = 48 a year, paid in one payment of 48.
- Price the bond at its yield. Discount the five coupons and the face value at 4.4%: 1,178.87.
- Move the yield up and down. At 4.5% the price is 1,173.66. At 4.3% it is 1,184.11.
- Subtract the two moved prices. 1,184.107615 − 1,173.660140 = 10.447475.
- Divide by 2 × price × Δy. 10.447475 ÷ 2.357739 = 4.4311.
The bond price is below the 1,200 face value because the 4.4% yield is higher than the 4% coupon rate. The calculator keeps the unrounded prices all the way through.
What the Result Means for the Price
Multiply the effective duration by a yield move to estimate the percentage change in price:
≈ −ED × change in yield
For the example bond (price 1,178.87, effective duration 4.4311), a 1% rise in yield gives −4.4311 × 1% = −4.43%, or about 52.24 off the price:
| Yield move | Estimate | Actual |
|---|---|---|
| −2% | 1,283.34 | 1,289.46 |
| −1% | 1,231.11 | 1,232.60 |
| −0.5% | 1,204.99 | 1,205.36 |
| +0.5% | 1,152.75 | 1,153.11 |
| +1% | 1,126.63 | 1,128.06 |
| +2% | 1,074.39 | 1,079.99 |
The estimate is close for small moves and drifts for large ones, and the actual price is always a little higher than the estimate. That gap is convexity, which you can measure with the Bond Convexity Calculator.
Effective, Modified and Macaulay Duration
The calculator shows three duration figures. They are related but answer different questions:
- Macaulay
- The average time until the cash flows arrive, weighted by their present value. A time measure.
- Modified
- Macaulay duration ÷ (1 + yield per period). The price sensitivity from a formula.
- Effective
- The price sensitivity measured by repricing the bond after moving the yield up and down.
For the example bond the Macaulay duration is 4.6261 years, the modified duration is 4.6261 ÷ 1.044 = 4.4311 and the effective duration is also 4.4311. For a plain bond with fixed coupons, effective and modified duration agree almost exactly, and the small difference comes from the size of the yield change.
The two part ways for bonds with embedded options, such as callable or putable bonds and mortgage backed securities. Their cash flows change when rates move, so modified duration no longer works and effective duration is the measure to use. Pricing those bonds needs an option model and a shift in the benchmark yield curve, which this calculator does not do: it reprices a plain bond by moving its own yield.
Effective Duration Examples
Each row uses a yield change of 0.1%:
| Bond | Result |
|---|---|
| 1,200 face, 4% annual 5 years, yield 4.4% | Price 1,178.87 Duration 4.4311 |
| 1,200 face, 4% semi-annual 5 years, yield 4.4% | Price 1,178.67 Duration 4.4780 |
| 1,000 face, 3% quarterly 2 years, yield 4% | Price 980.87 Duration 1.9288 |
| 1,000 face, 6% semi-annual 10 years, yield 5% | Price 1,077.95 Duration 7.5726 |
| 1,000 face, zero coupon 10 years, yield 4% | Price 675.56 Duration 9.6156 |
| 1,000 face, 7% semi-annual 15 years, yield 7% | Price 1,000.00 Duration 9.1963 |
| 1,000 face, 5% semi-annual 30 years, yield 6% | Price 861.62 Duration 14.3411 |
The face value does not change the duration, only the price. The amounts carry no currency symbol, so the calculator works in any currency.
What Changes Effective Duration?
Maturity. The longer the bond, the higher the duration, but duration grows more slowly than maturity because the coupons arrive along the way. A 4% annual bond at a 4.4% yield:
Coupon rate. Lower coupons mean a higher duration, because more of the value arrives at maturity. A 5 year annual bond at a 4.4% yield:
| Coupon | Duration |
|---|---|
| 0% | 4.79 |
| 2% | 4.59 |
| 4% | 4.43 |
| 6% | 4.30 |
| 8% | 4.18 |
Yield. Higher yields mean a lower duration. The same 4% annual, 5 year bond has a duration of 4.61 at a 1% yield, 4.43 at 4.4% and 4.15 at 10%.
Coupon frequency. In this calculator the yield is compounded at the coupon frequency, and the combined effect is a slightly higher figure for more frequent coupons: 4.4311 with annual coupons, 4.4780 semi-annual, 4.5020 quarterly and 4.5181 monthly.
Effective Duration of a Portfolio
The duration of a bond portfolio is the average of the durations of its bonds, weighted by market value:
= Σ (weight × duration)
Suppose 60,000 is held in the 5 year example bond (duration 4.4311) and 40,000 in a 10 year bond with the same coupon and yield (duration 8.0512):
0.4 × 8.0512 = 3.2205
Portfolio duration = 5.88
A 1% rise in yields across the board would cut the value of this 100,000 portfolio by about 5.88%, or roughly 5,880. Shifting money towards the shorter bond lowers that figure, which is how investors reduce interest rate risk when they expect rates to rise.
The Six Inputs
- Face value
- The par value repaid when the bond matures.
- Coupon rate
- Yearly interest as a percentage of the face value. 4% on 1,200 pays 48 a year. Enter 0 for a zero coupon bond.
- Coupons paid
- Payment frequency, from annually to daily. Most government and corporate bonds pay semi-annually or annually.
- Years
- The remaining life of the bond.
- Yield
- The yield to maturity as a yearly percentage. Find it with the Bond YTM Calculator if you only know the price.
- Yield change
- The size of the up and down move in percentage points. 0.1 means 10 basis points each way.
What the Calculator Assumes
- No embedded options. Fixed coupons and the face value repaid at maturity. The cash flows stay the same when the yield moves.
- Whole coupon periods. Years × payments per year is rounded to a whole number of payments, with no accrued interest.
- One yield for every payment. The yield to maturity is compounded at the coupon frequency.
- A parallel move. The yield goes up and down by the same amount. Duration says nothing about a change in the shape of the yield curve.
- Small moves. Duration is a straight line estimate, so it is less exact for large changes in yield.
How to Use the Effective Duration Calculator
- Enter the face value and the coupon rate. Type the face value of the bond, such as 1,200, and its annual coupon rate as a percentage, such as 4.
- Choose the coupon frequency. Select annually, semi-annually, quarterly, monthly, weekly or daily.
- Enter the years to maturity and the yield. Type the remaining life of the bond in years, such as 5, and the yield to maturity as a percentage, such as 4.4.
- Enter the yield change. Type the size of the up and down yield move in percentage points, such as 0.1.
- Read and copy the result. The effective duration appears as you type, with the bond price, the prices after the yield moves, the modified and Macaulay durations and the coupons. Press Copy to copy the duration, or Clear or Esc to start again.
Example: a face value of 1,000, a 6% coupon paid semi-annually, 10 years to maturity, a 5% yield and a yield change of 0.1. The result is an effective duration of 7.5726 years and a bond price of 1,077.95.
Who Can Use This Calculator?
Frequently Asked Questions (FAQs)
What is effective duration?
Effective duration is a measure of how sensitive the price of a bond is to a change in interest rates. It is the approximate percentage change in price for a 1% change in yield, found by repricing the bond with the yield moved up and down.
How do you calculate effective duration?
Price the bond at its yield (P), at the yield minus a small change (P−) and at the yield plus the same change (P+). Effective duration = (P− − P+) ÷ (2 × P × Δy), with Δy written as a decimal.
What is the effective duration of a 5 year 4% bond at a 4.4% yield?
About 4.43 years. With annual coupons and a face value of 1,200 the price is 1,178.87, and with a yield change of 0.1% the moved prices are 1,184.11 and 1,173.66, which gives 4.4311.
How is effective duration different from years to maturity?
Years to maturity only says when the face value is repaid. Effective duration says how far the price moves when yields change. Two bonds that both mature in 5 years can have different durations: a 5 year zero coupon bond at a 4.4% yield has a duration of 4.79, and one with an 8% coupon has 4.18.
What is the difference between effective and modified duration?
Modified duration comes from a formula that assumes the cash flows never change. Effective duration is measured by repricing the bond, so it also works when cash flows depend on rates. For a plain fixed coupon bond the two are almost equal: both are 4.4311 for the example bond.
What yield change should I use?
A small one, such as 0.1 (10 basis points). For a plain bond the choice hardly matters: the example bond gives 4.4311 with 0.01 or 0.1, 4.4312 with 0.19 and 4.4338 with 1.
Does coupon frequency change effective duration?
Slightly. With the same coupon rate, maturity and yield, the example bond has an effective duration of 4.4311 with annual coupons, 4.4780 with semi-annual coupons and 4.5181 with monthly coupons.
Can I use it for callable bonds?
Only as a rough guide. The calculator prices a plain bond with fixed cash flows. A callable or putable bond needs an option pricing model, because its expected cash flows change when rates move.
Is the Effective Duration Calculator free?
Yes. It is completely free, needs no sign-up, and can be used as many times as you like.
Sources
- Wikipedia. Duration (finance).
- AnalystPrep. Effective Duration and Interest Rate Risk.
- WallStreetMojo. Effective Duration: Formula and Calculation.
- OpenStax. Principles of Finance, 10.4 Risks of Interest Rates and Default.
- Investor.gov (U.S. SEC). Bonds.
- Wikipedia. Bond valuation.
Try our Effective Duration Calculator to see how far a bond price moves when interest rates change.