Bond Calculator: Price, Yield and Rate Risk

A plain bond is a fixed set of promises: a coupon every six months and the face value at maturity. Those promises do not change after the bond is issued. What changes is the return buyers demand for them. The price of a bond is simply every remaining payment discounted at today's yield, so when the yield buyers demand goes up, the same payments are worth less today, and the price falls. When yields fall, the price rises. The SEC puts it in one line: when interest rates go up, prices of fixed-rate bonds fall (SEC investor bulletin).

Illustrative example: the people and figures below are hypothetical, not real clients. Numbers are calculated from the stated assumptions.

Take the calculator's illustrative bond: ten years to maturity, a 4.5% coupon paid twice a year, and a yield to maturity of 5%. This is a round number chosen to show the mechanics, not a market quote. Priced at 5%, the bond is worth 96.10 per 100 of face, or $9,610 for $10,000 of face value. It sells below face because its coupon is lower than the yield buyers want; a buyer gets the difference back as a gain when the bond matures at 100. If the yield were exactly 4.5%, the coupon would be all the return a buyer needs and the price would be 100.00, or par. If the yield rose a point to 6%, the price would drop to 88.84; if it fell a point to 4%, the price would rise to 104.09.

A bond priced above 100 is a premium bond, one below 100 a discount bond. Neither is better. A premium bond pays a coupon above today's yields and gives part of it back as a loss to face by maturity; a discount bond pays less now and makes it up at the end. The yield to maturity puts both on the same footing.

Four different numbers get called a bond's “yield”, and a broker's screen may show all of them.

• Coupon rate: the interest the bond pays as a percentage of face value, fixed at issue. The example pays 4.5%, so $450 a year on $10,000 of face, whatever you paid for it.
• Current yield: a year's coupons divided by today's price. At 96.10, the example's current yield is 4.68%. It ignores the gain or loss to face at maturity, so it flatters premium bonds and understates discount bonds.
• Yield to maturity (YTM): the single annual rate that makes every coupon and the final repayment, discounted back, equal the price. It counts the pull to face value, and it assumes you hold to maturity, every payment arrives, and coupons are reinvested at the same rate (FINRA on bond yield and return). It is the number to compare bonds by.
• Yield to call (YTC): the same calculation, but ending on the first date the issuer may repay early, at the call price. For a callable bond trading above its call price, the lower of YTM and YTC, the “yield to worst”, is the honest figure.

A callable example: a ten-year bond with a 6% coupon, bought at 105, callable at 100 in three years. Its current yield is 5.71% and its yield to maturity is 5.35%. But if rates stay where they are, the issuer will likely refinance and call it, and a buyer who paid 105 gets 100 back after three years. The yield to call is only 4.21%. Turn on the call feature in the calculator to see this for your own bond.

Duration is the measure of a bond's interest-rate sensitivity. FINRA's rule of thumb is that a bond's price changes by roughly its duration, in percent, for each percentage point that rates move, in the opposite direction (FINRA: interest rate changes and duration). There are two versions worth knowing.

Macaulay and modified duration

Macaulay durationis the average time until you receive the bond's cash, with each payment weighted by its present value. For the example bond it is 8.12 years, shorter than the ten-year maturity because the coupons arrive along the way. A zero-coupon bond pays nothing until maturity, so its Macaulay duration equals its maturity exactly.

Modified duration is Macaulay duration divided by one plus the yield per period, here 8.12 ÷ (1 + 5%/2) = 7.92. It is the percentage price change for a one-point change in yield. So a one-point rise should cut the example bond's price by about 7.92%. Repricing the bond exactly at 6% gives a loss of 7.56%, a little less. A one-point fall, by the same estimate, adds 7.92%; exactly, it adds 8.31%. On $10,000 of face, that is a swing from about $726 lost to $799 gained.

What makes duration longer or shorter

Longer maturities mean longer durations; higher coupons mean shorter ones, because more of the money comes back early. The table shows both effects at a 5% yield.

A two-year bond loses about 1.9% if yields rise a point; a thirty-year bond loses about 14.1%. Notice too that the thirty-year coupon bond's duration is far below thirty years: with decades of coupons paid along the way, the average wait for the money is much shorter than the final date. For a bond fund, the duration is on the fund's fact sheet, usually under “portfolio data” or a similar heading.

Duration treats the price-yield relationship as a straight line. It is actually a curve that bends upward, and convexity measures the bend. For an ordinary bond, convexity is positive, which works in the holder's favor: when yields fall, the price rises more than duration predicts, and when yields rise, it falls less.

The example bond's convexity is 75.3. For a two-point move, the gap becomes visible. Duration alone predicts a loss of 15.8% if yields rise two points; the bond actually loses 14.4%. If yields fall two points, duration predicts a gain of 15.8% and the bond actually gains 17.5%. Adding the convexity term, half the convexity times the yield change squared, brings the estimates to -14.3% and +17.3%, close to the exact figures. The chart in the calculator draws both lines; the space between them is convexity.

A price drop on an individual bond is a loss only if you sell. Hold it to maturity and, as long as the issuer pays, you get every coupon and the full face value, exactly as promised when you bought it. What you lose when rates rise is the chance to have earned the higher rate; that is an opportunity cost, not a cash one.

Suppose you buy the 4.5% ten-year bond at par, and the next day yields on comparable bonds rise to 5.5%and stay there. The table shows the bond's market price as it ages. It drops at once, then climbs back to 100 as maturity approaches, the “pull to par”.

A bond fund works differently. Most funds keep their maturity in a target range by selling bonds as they age and buying longer ones, so the fund's duration stays roughly the same year after year instead of shrinking toward zero. There is no maturity date on which the fund is guaranteed to be back at its starting price. In exchange, a fund reinvests at the new, higher yields as it rolls its holdings, so over a period roughly equal to its duration, the higher income tends to make up for the price drop. Funds also give you diversification across hundreds of issuers and daily liquidity. Which one suits you depends on whether you need a known sum on a known date. The index funds vs. ETFs guide covers the fund wrappers themselves.

Every bond's yield is a promise only as good as the issuer. Credit rating agencies grade that risk. Bonds rated below Baa3 by Moody's, or below BBB by S&P and Fitch, are speculative grade, also called high-yield or junk bonds; they pay more because default is more likely (FINRA on high-yield bonds). A rating is an opinion, and it can be cut while you own the bond, which lowers the price even if rates have not moved.

For money a retiree needs to spend, the usual answer is to take credit risk sparingly: Treasuries, which carry the full faith and credit of the US government, highly rated munis and corporates, or diversified funds, rather than a handful of lower-rated individual bonds whose higher yield is payment for a risk you may not be able to absorb. The calculator's yield to maturity assumes every payment arrives; for a weak issuer, it is a ceiling, not an expectation.

Many corporate and municipal bonds are callable: the issuer may repay early, usually at face value, after a set date. Issuers call when rates have fallen, because they can borrow again for less. That is the worst time for you to get the money back, since you must reinvest it at the lower rates (FINRA on bond risks). A callable bond pays a little more than an otherwise identical non-callable one to compensate.

Before buying a callable bond above its call price, compare the yield to call with the yield to maturity and plan on the lower one. In the example above, paying 105 for a 6% bond callable at 100 in three years turns an apparent 5.35% into 4.21% if it is called, which is the likelier outcome if rates stay where they are. Treasury notes and bonds issued today are not callable.

Interest on most bonds issued by states, cities and their agencies is exempt from federal income tax (SEC investor bulletin on municipal bonds; IRS Publication 550). Munis therefore pay lower yields, and the useful question is whether the lower yield leaves you more after tax. The tax-equivalent yield answers it: the taxable yield you would need to match the muni.

For an in-state muni compared with a fully taxable bond, the formula is the muni yield divided by one minus your federal and state rates. For an out-of-state muni, your state usually taxes the interest, so the muni keeps only one minus the state rate of its yield before you divide. In plain terms: most states exempt their own munis and tax everyone else's, and a few states tax some of their own. If you live in a state without an income tax, in-state and out-of-state make no difference.

The table uses an illustrative 3.25% muni yield and a 5% state rate, with the 2026 federal brackets. It shows the taxable yield each muni has to beat.

In the 24% bracket with a 5% state tax, the 3.25% in-state muni is worth the same as a 4.58% taxable bond; in the 35% bracket, a 5.42% taxable bond. The net investment income tax of 3.8%, which applies to taxable interest once modified AGI passes $250,000 on a joint return or $200,000 single, pushes the bar higher still, because muni interest is exempt from it. That is why munis are mainly a higher-bracket holding, and why they rarely belong inside an IRA, where the exemption is wasted.

One more wrinkle: interest on certain private activity bonds is a preference item for the alternative minimum tax, and Form 1099-INT reports it separately (IRS Publication 550). Few retirees owe AMT, but ask before buying a large position. The guide to taxes in retirement puts these pieces together.

Interest on Treasury bills, notes and bonds is subject to federal income tax but exempt from all state and local income taxes (IRS Publication 550). In a state with an income tax, that makes a Treasury worth more than a corporate bond or CD with the same yield. At an illustrative 4.75% yield, in the 24% bracket with a 5% state tax, a CD or corporate bond keeps 3.37% after tax and a Treasury keeps 3.61%. Choose “Treasury” in the calculator's muni tab to compare one with a muni on that basis.

Accrued interest, clean and dirty prices

Bonds trade between coupon dates, and the seller has earned part of the next coupon. Quotes are “clean”, without that interest; you pay the clean price plus accrued interest, the “dirty” or full price. Treasuries count the days actually elapsed (actual/actual); corporates and munis use a 30-day-month, 360-day-year convention (30/360). For example, a 4% Treasury note maturing 2030-11-15 and bought for settlement on 2026-08-01 is 78 days into a 184-day coupon period, so the buyer owes the seller 0.8478 per 100 of accrued interest, $424 on $50,000 of face. At a 4% yield the full price is 100.843 and the quoted price 99.995 (TreasuryDirect on pricing). The accrued interest you pay comes back with the next coupon; for tax purposes it reduces your basis rather than counting as income (Pub 550). The calculator handles this when you choose “Between coupon dates”.

TIPS

Treasury Inflation-Protected Securities are sold in 5-, 10- and 30-year terms. Their principal rises and falls with inflation, the fixed coupon is paid on the adjusted principal, and at maturity you get the adjusted principal or the original amount, whichever is larger (TreasuryDirect). The yearly increase in principal is taxable as it accrues even though you receive it only at maturity, so TIPS usually sit best in an IRA. This calculator prices fixed-coupon bonds; it does not model TIPS. The inflation calculator shows what rising prices do to a fixed payment.

Bond ladders

A ladder is a set of individual bonds or CDs maturing in successive years, say one a year for ten years. Each rung pays face value when it comes due, so you can plan spending against known sums and avoid selling in a bad market; and because something matures every year, you reinvest at whatever rates prevail rather than betting on one rate. Each rung is held to maturity, so the price swings in the table above matter only if you sell early. The CD ladder calculator builds a ladder rung by rung, and the retirement withdrawal calculator shows how much spending a bond allocation has to cover.

The price you pay can change how the bond is taxed. These are the rules in IRS Publication 550, in outline; your 1099s and your CPA handle the arithmetic.

• Original issue discount (OID). A bond issued below face, including every zero-coupon bond, generally has OID, which you include in income as it accrues each year, whether or not you receive any cash. A discount smaller than one-quarter of 1% of face times the full years to maturity is treated as zero.
• Market discount. If you buy an existing bond below face in the market, the gain up to the accrued market discount is taxed as ordinary interest income when you sell or redeem it, unless you elect to include it each year. The same quarter-of-a-percent-per-year de minimis rule applies. For the ten-year example bond the threshold is 2.50 points; buying it at 96.10, a discount of 3.90 points, would make it a market discount bond. On a muni bought after April 30, 1993, market discount is taxable ordinary income too, even though the coupons are tax-exempt.
• Premium. On a taxable bond bought above face, you can elect to amortize the premium, reducing the taxable interest each year. On a tax-exempt bond, you must amortize it, and the amortized amount is not deductible; it simply reduces your basis.

Bonds are usually the part of a portfolio meant to be spent: they fund the next several years of withdrawals so stocks can be left alone in a downturn. Duration is the dial. Short durations protect the price you could sell at; matched maturities, as in a ladder, protect the dollars you will have on a date. How much to hold in bonds at all is covered in asset allocation by age, and the asset allocation worksheet helps you set and record a target.