Below is a Long-Run Yield Curve tool. It charts the gap between the 10-year US Treasury yield and a blended short-term rate – the classic "long minus short" spread – for every month back to 1871. When the line drops below zero the curve is inverted, the signal that has led most US recessions.
Normally, you'd expect that lending to the government for ten years pays more than lending for a few months – you want to be paid for tying your money up for a longer period of time. When that flips and short rates climb above long rates, the market is effectively betting that rates (and growth) will be lower down the road. Since World War I that inversion has been mostly a reliable – if early and sensitive – recession warning.
There are two big caveats here worth keeping in mind:
And the other, more theoretical problem going forward – the indicator is well-known now. There's a name for the risk: Goodhart's Law, which holds that "when a measure becomes a target, it ceases to be a good measure" – and fittingly, it came out of monetary economics.
Once everyone is watching an indicator, it's ripe for weird second-order effects, or even manipulation, from participants positioning around it. So, even though there is good theoretical grounding for why this one works: past performance is not a guarantee of future results.
This is the long-history, two-point view. For the daily, full-curve picture from 1990 to today – every maturity from 1 month to 30 years, in rotatable 3D – see the 3D Treasury Yield Curve.
On this page is a Stocks vs Bonds Historical Returns Calculator. It puts the S&P 500, Dow Jones Industrial Average, NASDAQ Composite, the 10-year US Treasury, short-term cash (3-month T-bill), and a classic 60/40 portfolio on the same chart, so you can see how each one performed over history.
The default view grows $10,000 in each asset from 1972, where the NASDAQ data starts. Everything recomputes in real time as you change a setting.
In Calculation Options:
Three views are toggleable across the top:
Each asset starts from a monthly-average price (or yield) series, then is mapped to total-return level the way an investor would experience it. The five series:
Below is an interactive 3D treasury yield curve chart showing U.S. Treasury yields from 1977 to the present. Watch rates rise, fall, and even invert over decades of economic history.
Drag to rotate, scroll to zoom. Use the timeline slider or play button to animate through history.
The tool loads with the last 10 years of data by default. Here's where to explore:
On desktop, drag to rotate the 3D view and scroll to zoom. Hover over the surface to see exact yields for any date and maturity.
Click Advanced Options to access additional settings:
Drag, zoom, and pan to find the best angle:
Below the 3D chart, a traditional 2D yield curve shows the current date's rates across all maturities. The 10Y-2Y spread indicator on the right shows whether the curve is normal (positive, blue) or inverted (negative, red).
The yield curve plots interest rates on U.S. Treasury securities across different maturities, from 1-month T-bills to 30-year bonds. Normally, longer-term bonds pay higher rates (an upward-sloping yield curve) because investors demand more compensation for locking up their money longer.
When short-term rates exceed long-term rates, the curve inverts. An inverted yield curve has preceded every U.S. recession since the 1970s; the signal isn’t perfect, with a notable mid‑1960s inversion that didn’t lead to a recession. Its success predicting recessions makes it one of the most watched indicators in economics.
(And yes, an inversion predicting a recession somehow included the recession that coincided with COVID-19 going worldwide in 2020.)
This 3D visualization adds a time dimension, letting you see how the entire curve shape evolves: not just today's snapshot, but decades of shifting rates, inversions, and recoveries.
You might notice that our tool highlights the 10-year minus 2-year spread, not the 10-year minus 3-month spread or some other combination. Here's why:
The yield curve's recession-predicting power was first documented by Campbell Harvey in his 1986 dissertation at the University of Chicago (as an aside, his dissertation committee included three future Nobel laureates: Eugene Fama, Lars Hansen, and Merton Miller!). Harvey originally used the 5-year minus 90-day spread. Later, Arturo Estrella and Frederic Mishkin at the NY Fed popularized the 10Y-3M spread in their influential 1996 paper, and that's what the Fed's official recession probability model still uses today.
The Chicago Fed's research found that academics prefer 10Y-3M while practitioners prefer 10Y-2Y, and both "produce qualitatively similar results." So why show 10Y-2Y?
If you want to track the Fed's preferred 10Y-3M spread, FRED has it here: T10Y3M. The Cleveland Fed also publishes a yield curve recession probability model.
3D charts have a bad reputation in data visualization (usually for good reasons – they're pretty terrible visualizations, most of the time). But yield curves are genuinely three-dimensional data: time × maturity × yield. A 2D chart can only show one dimension at a time (for example, the curve on a date... like, you know, the view under the 3D chart).
This visualization was inspired by The New York Times' 2015 piece by Amanda Cox and Gregor Aisch, which was highlighted by the Information is Beautiful Awards. As Gregor explained in a Data Stories podcast interview, the key to making 3D work is creating "a 3D chart that people can actually read" with optimal camera views and good narrative – my nod to that is the camera presets and historical jump points.
The yield data comes from the Federal Reserve Economic Data (FRED) database, updated daily. We pull all 11 constant maturity Treasury rates, and cross them with the official NBER recession indicator from FRED.
With the 3D surface, I interpolate between maturities for visual smoothness – exact yields (well, CMT yields, anyway) are shown in tooltips and the 2D chart below the 3D graph.
For broader market context:
The yield curve isn't going to tell you exactly when a recession will hit. But that's not the goal – watching it evolve in 3D should give you a better feel for the economic terrain than any single shock headline trying to grab your attention. Enjoy the 3D navigation!
This Treasury Return Calculator uses long-run 10-year Treasury data from the US Treasury Department and Robert Shiller to compute the total return of a 10 Year Treasury – coupon income plus the price change as yields move – for any period from 1871 until today.
(We have a similar calculator for the S&P 500, Dow Jones Industrial Average, NASDAQ, Gold, and Daily Inflation. Or, compare stocks, bonds, and short-term investments with the Stocks vs Bonds Historical Returns tool.)
Pick a starting and ending month and the calculator updates in real time. Here's what it shows:
The 10-year yield comes from Robert Shiller's compiled data, which runs back to 1871, with infill from the Treasury and FRED.
Each month's figure isn't a single day's price – it's the blended average of that month's daily 10-year constant-maturity yields. In other words these are "fake" numbers – a monthly average of the yield at which a new 10-year note could have been issued, not a price you could have transacted at on a given day. Because it's a monthly average rather than a month-end snapshot, returns here differ slightly from year-end-based sources – that is, returns are not based on an individual investment and sale date but a blended average.
By default, the calculator uses the par-bond method for turning constant-maturity yields into total returns (see Swinkels, 2019 for the canonical writeup). Each month you hold a freshly-issued par 10-year bond – coupon equal to the prevailing yield and priced at par – earn the coupon, and a month later reprice that now-slightly-older bond at the new yield using full present-value bond math. The monthly total return is the coupon you earned plus that price change; then you roll into a fresh par bond and repeat.
Because the bond is fully repriced every month, this captures convexity exactly – no duration or convexity coefficient to fit. It's the method behind most published Treasury total-return indices.
And yes, this model uses the same general engine behind our Bond Convexity and Bond Pricing calculators.
This calculator originally had a different performance methodology which built a bond ladder. The legacy algorithm would buy a 10-year note, collect its semiannual coupons, and sell it after three years (holding, effectively, a 7-year bond). The model would then roll any proceeds into a new 10-year note assuming bonds are bought at face value with no transaction fees and a 0% tax rate.
While I've maintained the model for comparability across time (via Show legacy return calculation), there are two big caveats.
The constant-maturity method has neither problem because it marks to market every month... which is why it's the default now.
It's worth internalizing: the 10 year Treasury total return is not its yield at purchase. Indexes like the one I've constructed here try to keep your maturity at roughly ten years. When rates rise, the price of existing bonds falls, and that capital loss can swamp the coupon.
Models like this one approximate how you would have done if you – in some magical, frictionless way – constantly turned over 10-year treasuries just to purchase newly issued ones. Treat the output as a decent approximation of how an investor would have fared, not the ultimate arbiter of returns. And always remember that, just like with stocks, the time of day, the weather, general sentiment, daily inflation, and countless other factors move the real price of a security at any point in time, including the moment you'd actually have bought it.
To Robert Shiller, of course, for posting his data publicly. To Laurens Swinkels, whose 2019 paper documents the constant-maturity return method cleanly. And to Jim at Free By 50, who assisted with some of the original assumptions.
Is this a useful tool? Anything else you'd like to see added to the Treasury Return Calculator?
On this page is a bond convexity calculator. It will compute a bond's convexity as the second derivative of the bond's price in relation to the interest rate. Optionally, it will show the price and yield relationship estimate from duration and convexity.
You can input either the market yield or yield to maturity, or the bond's current price and coupon and par, and the tool will compute the bond duration and convexity.
Our convexity tool will run the math starting from either the bond's market price or if you know the current yield to maturity. (Choose whichever is easier for you.)
If you prefer to start from the bond's current market price, ensure You Know Market Price is depressed.
Optionally, if you click the "Draw Price vs. Yield Graph", the tool will show the estimates change in price if the market yield moves.
The above graph shows the relationship for price and yield using the default values in the tool. Note the following outputs:
If you are on desktop, you can hover your cursor for a point estimate of price. On mobile or tablet, if you click you will see a tool-tip.
If you click the "hamburger" menu in the graph's upper right corner, you can download the price sensitivity graph in svg or png format. You can also download the backing data in csv format.
Ensure the "You Know Yield to Maturity" button is depressed if you'd prefer to enter the bond's par value and yield to maturity to compute convexity.
See the above section for Graph Output where you know bond price - behind the scenes it's the same function. The only difference is:
Bond convexity is a measure of the curve's degree when you plot a bond's price (on the y-axis) against market yield (on the x-axis). As the market yield changes, a bond's price does not move linearly – convexity is a measure of the bond price's sensitivity to interest rate changes.
It's built off the convexity work of Hon-Fei Lai, and started to gain popularity after Stanley Diller's 1984 paper Parametric analysis of fixed income securities.
(...a paper which I can't find online – please get in touch if you can)
Bond duration is also a measure of a bond's sensitivity to interest rate changes. Modified duration is the estimate of the price change of the bond for a 1% move in interest rates.
However, the duration is only a linear approximation. Specifically, the duration is the first derivative of the bond's price as it relates to interest rate changes. Convexity is the second derivative.
Drawn on a graph with bond price and yield, duration is tangent to convexity at the current price and interest rate.
At 'small' changes in interest rates, duration is a fine estimate of a bond's price change. For larger changes, using convexity will better approximate the real-world behavior of the bond.
The bond convexity formula (written as a series) is:
(\frac{coupon}{price}*
(\frac{1*(1+1)}{(1+ytm)^{1+2}}+\frac{2*(2+1)}{(1+ytm)^{2+2}}+...\\~\\+
\frac{(n-1)*((n-1)+1)}{(1+ytm)^{(n-1)+2}}+\frac{n*(n+1)}{(1+ytm)^{n+2}}) )+\\~\\
\frac{face\ value}{price}*\frac{n*(n+1)}{(1+ytm)^{n+2}}
And note that if the bond pays out multiple coupons per year, you can either:
Where:
There are a lot of factors, but it's reasonably straightforward. Next, let's manually compute the convexity of a made-up bond and walk through the calculation.
Sometimes you need a guess at convexity instead of working through the full formula. Don't worry, there's a way to reasonably estimate a bond's convexity with fewer terms.
The bond convexity approximation formula is:
Bond\ Convexity\approx\frac{Price_{+1\%}+Price_{-1\%}-(2*Price)}{2*(Price*\Delta yield^2)}Where:
But – stick with the better convexity formula if you have time to calculate it (or come back and visit this page!).
In the bond duration example, we computed the duration for a made up bond. Let's use the same example and compute convexity:
(\frac{50}{960.27}*
(\frac{1*(1+1)}{(1+.065)^{1+2}}+\frac{2*(2+1)}{(1+.065)^{2+2}}+\frac{3*(3+1)}{(1+.065)^{3+2}}))+\\~\\
\frac{1000}{960.27}*\frac{3*(3+1)}{(1+.065)^{3+2}}=\\~\\
(\frac{50}{960.27}*
(\frac{2}{(1.065)^3}+\frac{6}{(1.065)^4}+\frac{12}{(1.065)^5}))+\\~\\
\frac{1000}{960.27}*\frac{12}{(1.065)^{5}}=\\~\\
(0.05206868901 * 15.0782067672 ) + (1.04137378029 * 8.75857003825)=\\~\\
Bond\ Convexity \approx{9.906}
For this bond, the Bond Convexity is roughly 9.906. Next, let's look at how you can use that information.
Now you know how to compute a bond's duration as well as its convexity. Now that you have those numbers, you can use them to predict a bond's price after a given interest rate movement.
The formula for estimated price change for a given interest rate move is:
\frac{price*convexity*(\Delta yield)^2}{2}-(price*modified\ duration*\Delta yield)Where:
Let's continue working with the same made-up bond from above.
How much would the price change for a market rate change from 6.5% to 8%? How about from 6.5% to 5%?
We'll use the following computed values (and the same fantasy bond):
\frac{960.27*9.906*(.015)^2}{2}-(960.27*2.682*.015) = \\~\\
1.07014889475 - 38.6316621\approx -37.561
If interest rates rise 1.5% to 8%, we'd expect the bond price to fall $37.561 to around $920.59.
(You can verify it's close in the bond pricing calculator – which estimates $922.69.)
\frac{960.27*9.906*(-.015)^2}{2}-(960.27*2.682*-.015) = \\~\\
1.07014889475-(-38.6316621)\approx 39.702
If interest rates fall 1.5% to 5%, we'd expect the bond price to rise $39.702 to around $999.97.
(We don't need the bond price calculator here – we essentially reverse engineered the par value of $1000. Pretty close!)
Using the bond convexity calculator above, we can create a line chart showing the price estimates using convexity and duration:
In the duration calculator, I explained that a zero coupon bond's duration is equal to its years to maturity. However, it does have a modified (dollar) duration and convexity.
The formula for convexity of a zero coupon bond is:
zero\ coupon\ convexity=\frac{ttm^2+\frac{ttm}{2}}{(1+\frac{rate}{2})^{(2+(2*ttm))}}Where:
Convexity and duration both help you estimate your interest rate risk for bonds in your portfolio. As bonds with different characteristics will have different values for duration and convexity, they're important numbers to know so you know your exposure to market yield changes.
Are they a panacea? No.
As with many models, they assume a responsible bond issuer and continued payments through maturity. You'll nearly always see different prices in practice than those implied by bond price models. That is: convexity and duration are just decent estimates.
But – while they may be wrong, they're often useful. They give us a quick check on interest rate sensitivity at a glance and help construct portfolios hardened for different theoretical scenarios.
Also, if you see any issues in the tool let me know. Lotta ins, lotta' outs, lotta' what have yous in this one. (I tested many scenarios, but can't guarantee it works for every corner case!).
On this page is a bond duration calculator. It will compute the mean bond duration measured in years (the Macaulay duration), and the bond's price sensitivity to interest rate changes (the modified duration).
You can input either the market yield or yield to maturity, or the bond's price, and the tool will compute the associated durations.
This bond duration tool can calculate the Macaulay duration and modified duration based on either the market price of the bond or the yield to maturity (or the market interest rate) of the bond.
Since you'll have one or the other, choose the easier path to compute the duration.
If you have all of the details of the bond and know the market price, click the blue "You Know Market Price" button.
If you have all of the details of the bond and know the market yield or the bond's yield to maturity, use the "You Know Yield to Maturity" option.
The Macaulay duration of a bond is the weighted average payout of the bond, measured in years.
Practically, a longer Macaulay duration shows at a glance (and relative to another bond) a bond's interest rate risk. Longer duration bonds are more volatile – they are more sensitive to interest rate changes.
It was first introduced by Frederick Macaulay.
The Macaulay duration formula (written as a series) is:
\frac{
1*\frac{Payment_1}{(1+yield)^1} + 2*\frac{Payment_2}{(1+yield)^2} +...+
(n-1)*\frac{Payment_{n-1}}{(1+yield)^{n-1}} + n*\frac{Payment_n+Par\ Value}{(1+yield)^n}
}
{Current\ Price}Where:
From the series, you can see that a zero coupon bond has a duration equal to it's time to maturity – it only pays out at maturity.
Let's compute the Macaulay duration for a bond with the following stats:
\frac{
1*\frac{50}{(1+.065)^1}+2*\frac{50}{(1+.065)^2}+3*\frac{50 + 1000}{(1+.065)^3}
}{960.27} =
\\~\\
\frac{46.948 + 88.166 + 2607.72}{960.27} =
\\~\\
2742.834/960.27 = 2.856\ years
For this bond, the Macaulay duration is 2.856 years, heavily weighted towards maturity (3 years).
The modified duration of a bond is a measure of the sensitivity of a bond's market price to a change in interest rates. It's the percentage change of a bond's price based on a one percentage point move in market interest rates.
Bond prices move in an inverse direction from interest rates.
For a one percent increase in interest rates, the bond's market price will decrease by the percentage shown by the modified duration. For a one percentage point decrease in interest rates, the bond price will increase by the percentage shown by the modified duration.
The modified duration formula is:
\frac{Macaulay\ Duration}{1+\frac{YTM}{Annual\ Payments}}Where:
Let's extend the above example (from the Macaulay section) for a bond with the following characteristics:
\frac{2.856}{1+\frac{.065}{1}}=\\~\\2.856/1.065 =\\~\\2.682Remember, the modified duration is a measure of sensitivity to interest rate changes at a point in time. Here's the relationship:
Bond duration is a linear estimate of a bond's price sensitivity to changes in market yield. It's the first derivative of price with respect to market yield. However – the relationship between yield and price isn't linear, it's a curve.
Bond convexity is the second derivative, and a measure of the "curvedness" of the relationship. Here's how the price estimate looks for the example bond in this post:
The difference is slight – for small changes in yield – but it is real. Using convexity gives you a better measure.
Duration helps you understand, at a glance, how sensitive your bond portfolio is to interest rate changes.
Shorter duration bonds will be relatively price stable; they will pay out most of their promised cash flow in the near future. Longer duration bonds are less stable; long duration bonds have all the risk of taking longer to pay out their funds, including a shift in the market's demanded yield.
So, to insulate yourself from interest rate risk pick shorter duration bonds. If you want to take on more interest rate risk, pick longer. (You can also compute the Macaulay and modified duration of an entire portfolio by summing cash flow).
For other bond calculators, check out the following:
On this page is a bond yield to put calculator. It automatically calculates the annual yield earned on a puttable bond assuming you put it back to the issuer at the first possible time. Importantly, it assumes all payments and coupons are on time (no defaults).
Also, find the approximate yield to put formula. Like with Yield to Maturity (YTM), Yield to Put is calculated iteratively.
The calculation for Yield to Put is very similar to Yield to Maturity – and equal to the Yield to Call calculation (just with the incentives flipped).
The calculator assumes you will put the bond back to the issuer at your first chance – although, of course, you only want to do that if you would make money on the trade (and assuming all payments are made). And just like how callable bonds usually have a call price higher than par value, puttable bonds generally have a put value below face value.
However, that doesn't mean we can't estimate and come close. The formula for the approximate yield to put on a bond is:
\frac{(Annual\ Interest)+((Price\ to\ Put-Current\ Price)/(Years\ to\ Put))}{(Price\ to\ Put+Current\ Price ) / 2}Let's solve the default yield to put calculation inside the tool:
\frac{(100)+((970-920)/5)}{(970+920 ) / 2}=\\~\\\frac{100+10}{945}=11.640\%If you actually run the scenario in the tool, you'll find that yield to put is actually 11.700% – what's going on there?
As mentioned, the above formula is just an estimate. A real calculation would sum the present value of all future cash flows in almost the same way you calculate yield to maturity.
Here is the summation:
Price = Coupon\ Payment/(1+rate)^{-1} + Coupon\ Payment/(1+rate)^{-2} +\\ ... + Coupon\ Payment/(1+rate)^{-n}+Price\ to\ Put/(1+rate)^{-n}Internally, the tool adapts the method from Github user ndongo. See the yield to maturity calculator's methodology discussion for more.
As mentioned, bonds with options (calls and puts) introduce a wrinkle into price calculations. Depending on who is able to exercise, it gives that party an option to either call back the bond or put it back to the issuer for a set price.
In this case, a bond with a put option means at some time in the future you can put it back to the issuer if the price is right (and assuming all other incentives are aligned).
Read more in the yield to call discussion (just flip the incentives!).
Use the Yield to Put as you would use other measures of bond valuation: a factor in your decision to buy or avoid. Bonds with put options usually have the deck stacked against them in terms of face value vs. put price, but sometimes when bonds trade at a discount the put option can be worth it for you to exercise.
Here are some other bond calculators:
Here you'll find a tax equivalent yield calculator, sometimes called a TEY calculator. When presented with investments that are free from taxation at the state, federal, and/or local level, you can use your tax rate to determine the equivalent taxable yield using this tool.
*If you live in a location with local tax (for example, New York City) and a security is exempt from tax there as well, add it here. Additionally, note that due to the maximum state tax exemption of $10,000, the calculator adds marginal tax at the state and local level to the federal marginal tax. If you have tax cap space, omit this field.
Tax equivalent yield is the yield you would need on a taxable investment to match the return you receive on a tax-advantaged investment.
Since most investments are taxable, you have to pay for any dividends, capital gains, coupon payments, and interest you receive at the end of the year. By applying tax "in reverse" such as with this tool, you normalize payouts on tax free investments to other potential investments.
It's impossible to list every tax-advantaged or tax-free investment in the United States, especially since some are only tax-free under certain conditions (for example, sometimes the alternative minimum tax (AMT) cancels tax-advantaged treatment).
Always consult with your tax advisor if you have any doubts – but here are examples of nominally tax-advantaged investments:
Note that for most securities, the interest might be tax-advantaged, while any capital gains (or, in the case of TIPS, inflation adjustments) are probably not. Again, check with your tax advisor with any questions.
The tax current yield formula is:
\frac{yield}{1-marg\_tax}Where:
Let's say you live in California and are evaluating a municipal bond which pays 1.2% and is exempt from taxes at the Federal and State level.
Your Federal marginal tax rate is 37% and you would owe an additional 3.8% Medicare surcharge on taxable interest, for a total of 40.8%. Your California marginal rate is 12.3%, for a total marginal tax rate of 53.1%.
What would yield more after tax – a bank savings account at 2.15% or the municipal bond?
\frac{yield}{1-marg\_tax} = \\~\\\frac{.012}{1-.531}=\\~\\0.025586 = 2.56\%In this case, the Municipal Bond would provide as much income as a savings account paying 2.56%, handily defeating the 2.15% account.
(Note, however, there are other considerations than yield – liquidity, default risk vs. FDIC insurance, etc. Always consult your advisor with any questions.)
Taxes are very complex. Since there are so many security types and tax systems even in the United States, it's important to compare like to like when evaluating your asset allocation. Normalizing every security to the taxable-equivalent is good practice for your own portfolio.
Note that it's not the only way to "increase" your tax equivalent yield, however. By investing in tax-advantaged account types – such as IRAs, HSAs, 401(k)s, and 529s – you can defer or eliminate tax even on taxable accounts.
For other retirement account posts, plus financial basics and bond calculators, please see:
On this page is a bond yield calculator to calculate the current yield of a bond. Enter the bond's trading price, face or par value, time to maturity, and coupon or stated interest rate to compute a current yield.
The tool will also compute yield to maturity, but see the YTM calculator for a better explanation plus the yield to maturity formula.
The current yield of a bond is the annual payout of a bond divided by its current trading price. That is, you sum up all coupon payments over one year and divide by what a bond is paying today.
A bond's yield to maturity is the annual percentage gain you'll make on a bond if you hold it until maturity (assuming it doesn't miss payments). It's expressed in an annual percentage, just like the current yield. However, YTM is not current yield – yield to maturity is the discount rate which would set all bond cash flows to the current price of the bond.
You can find more information (including an estimated formula to calculate YTM) on the yield to maturity calculator page.
The bond current yield formula is:
\frac{ACF}{P}Where:
Let's work through an example and compute the current yield for an example bond. We'll use the example in the tool's defaults.
\frac{ACF}{P} = \\~\\ \frac{100}{920} = 10.87\%So, a bond trading at $920 with a face value of $1000 and a 10% interest rate has a 10.87% current yield, higher than the one stated by the bond.
Current yield gives you a quick read of how a bond compares in the market. It is inferior to yield to maturity, although YTM does come with the risk that a bond may stop paying out (while your next year of payments is more certain). In almost all cases you should compute both, though.
For other financial basics and bond calculators, please see:
On this page is a zero coupon bond calculator, to calculate the market price or fair value of a zero coupon bond of known time to maturity, par or face value, and interest rate.
A zero coupon bond is a bond which doesn't pay any periodic payments. Instead it has only a face value (value at maturity) and a present value (current value). The entire face value of the bond is paid out at maturity.
It is also known as a deep discount bond.
Zero coupon bonds have a duration equal to their time until maturity, unlike bonds which pay coupons.
Duration of a bond is a length of time representing how sensitive a bond is to changes in interest rates. Since zero coupon bonds have an equal duration and maturity, interest rate changes have more effect on zero coupon bonds than regular bonds maturity at the same time. (Whether that's good or bad is up to you!)
Zero coupon bonds are particularly sensitive to interest rates, so they are also sensitive to inflation risks. Inflation both erodes the value of the dollars the bond will eventually pay.
In the United States, you need to impute the interest for some zero coupon bonds to pay taxes in the current year (possibly also for state or local taxes). One tax workaround is to purchase zero coupon bonds in tax-free accounts such as IRAs, or to purchase zero coupon municipal bonds with no tax obligations. Consult your tax advisor for a full breakdown.
The zero coupon bond price formula is:
\frac{P}{(1+r)^t}where:
Let's walk through an example zero coupon bond pricing calculation for the default inputs in the tool.
Substituting into the formula:
\frac{P}{(1+r)^t} = \\~\\ \frac{1000}{(1+.1)^{10}} = \\~\\ \frac{1000}{2.5937424601} = \\~\\ \$385.54So a 10 year zero coupon bond paying 10% interest with a $1000 face value would cost you $385.54 today.
In the opposite direction, you can compute the yield to maturity of a zero coupon bond with a regular YTM calculator.
Zero coupon bonds are yet another interesting security in the fixed income world. For other bond calculators, check out the following: