Bond yield calculator

How do you calculate bond yield?

Two yields answer two different questions. The current yield is income only: annual coupon / price. A 5% coupon on a bond priced at 95 gives 5 / 95 = 5.26%. The yield to maturity (YTM) counts everything still to come: it is the one rate at which the present value of every remaining coupon plus the redemption payment equals today's price. In symbols, price = C / (1 + y)^1 + C / (1 + y)^2 + ... + (C + R) / (1 + y)^n, with C the coupon per period, R the redemption value and n the number of periods.

The calculator opens on that 5% bond, paying 2.50 every six months, priced at 95 per 100 of face value and repaid at 100 in five years. Its YTM is 6.18% a year. The extra 0.91 points over the current yield are the 5-point gain to par, earned gradually across the five years. The coupon rate stays at 5% throughout, because it is fixed when the bond is issued and only the price moves.

Worked example: yield to maturity by trial and error

With more than one cash flow there is no formula that isolates y, so YTM is found by trying rates until the price matches. For the default bond, with ten half-yearly coupons of 2.50 and 100 at the end:

The answer lies between 6% and 6.5%. A straight line between those two points gives 6% + (95.73 - 95.00) / (95.73 - 93.68) x 0.5% = 6.178%, close to the exact solution of 6.1776%. The calculator runs the search for you to many decimals. On a holding of $1,000 face value, the bond costs $950, pays $50 a year in two coupons of $25 and returns $1,000 at maturity: a gain of $300 over five years if held to the end.

A worked example in pounds: a bond with a 4% coupon paid half-yearly and seven years left is priced at £92.50 per £100. Its current yield is 4.32%, its YTM 5.30% and its effective annual yield 5.37%. £10,000 of face value costs £9,250 and pays £400 a year, in two coupons of £200.

How do you work out a bond's price from its yield?

Run the same sum the other way: discount each coupon and the redemption payment at the yield you require, and add. On the Price from yield tab, the default bond at a 6% yield is worth 95.73 per 100, so $1,000 of face value costs $957.35. At 4% the same bond is worth 104.49, above par, because its 5% coupon now pays more than the market asks. At exactly 5% it is worth 100.00.

Price and yield always move in opposite directions, and the further away maturity is, the further price moves for the same change in yield. At 6%, the five-year bond is worth 95.73 while a ten-year bond with the same 5% coupon is worth 92.56. The chart under the result draws the whole price-yield curve for your bond; it bends, which is why a change in yield of one point up and one point down does not move the price by the same amount.

How do you calculate the cost of redeemable debt for an exam?

For a company, the yield on its own traded debt is its cost of debt. A written answer treats the bond as a project: pay the ex-interest market value at year 0, receive interest each year, receive the redemption value at the end, and find the IRR. Interest is tax-deductible, so for the post-tax cost the yearly interest is taken after tax. With the defaults on the third tab, a 5% bond at 95, redeemed at par in five years, 25% tax, and trial rates of 4% and 8%:

Post-tax cost of debt = 4% + 3.895 / (3.895 + 11.926) x (8% - 4%) = 4.98%. The exact rate is 4.90%: a straight line across a curved NPV profile overstates it slightly, and the gap shrinks as the two rates close in. Pre-tax, with the full 5 of interest, the NPVs are 9.460 and -6.935, giving 6.31% by interpolation and 6.19% exactly. That exact pre-tax figure is the bond's annual yield to maturity, the same number the first tab shows with coupons set to once a year.

Worked example in pounds: loan notes redeemable at a premium

A company has 8% loan notes quoted at £104 per £100 nominal, ex interest, redeemable at £105 in six years. Tax is 25%, so each £100 of notes costs £8 x 0.75 = £6.00 of interest after tax a year.

• At 5%: -104 + 6.00 x 5.076 + 105 x 0.746 = -104 + 30.456 + 78.330 = 4.786.
• At 8%: -104 + 6.00 x 4.623 + 105 x 0.630 = -104 + 27.738 + 66.150 = -10.112.
• Post-tax cost = 5% + 4.786 / (4.786 + 10.112) x 3% = 5.96%, against 5.91% exactly.

Widen the second rate to 10% and the estimate becomes 6.02%, a little further from the exact figure. Before tax the notes cost 7.84% by interpolation and 7.82% exactly. If the company has £2,000,000 of nominal value in issue, their market value for a cost of capital calculation is £2,000,000 x 104 / 100 = £2,080,000. Enter these numbers on the third tab with the currency set to GBP and the working prints line by line.

Annual, semi-annual or quarterly coupons: which yield is quoted?

Coupon frequency changes both the cash flows and the way the yield is stated. This calculator quotes YTM as a nominal annual rate: the yield per period times the number of periods in a year. The effective annual yield compounds it instead. For the default bond:

Conventional UK gilts and US Treasury notes pay interest half-yearly. The UK Debt Management Office says conventional gilts pay their coupon in two equal semi-annual payments, six months apart, with the coupon stated as cash per £100 nominal each year and prices quoted per £100 face value. TreasuryDirect says US Treasury notes pay a fixed rate of interest every six months, for terms of 2, 3, 5, 7 or 10 years. For a bond that pays once a year, take a euro example: a 3% annual-coupon bond with eight years left at €97.20 per €100 has a current yield of 3.09% and a YTM of 3.41%. The exam tab on this page uses annual interest throughout.

How do you calculate yield to maturity in Excel?

Microsoft documents three functions that fit. YIELD(settlement, maturity, rate, pr, redemption, frequency, [basis]) returns the yield on a security that pays periodic interest, with pr and redemption per $100 of face value and frequency 1, 2 or 4. Basis defaults to 0, the US (NASD) 30/360 day count. With more than one coupon period left it is solved by the Newton method through 100 iterations, and it returns #NUM! when, among other cases, settlement is on or after maturity. PRICE(settlement, maturity, rate, yld, redemption, frequency, [basis]) runs the other way and returns the price per $100 of face value.

Without dates, RATE(nper, pmt, pv, [fv]) solves the same equation over whole periods. For the default bond, =RATE(10, 2.5, -95, 100) returns the half-yearly rate of 3.09%, and doubling it gives the 6.18% nominal YTM. For the exam example, =RATE(5, 3.75, -95, 100) gives the exact post-tax cost of 4.90%. Microsoft notes that RATE starts from a guess of 10%, returns #NUM! if it has not converged within 0.0000001 after 20 iterations, and needs nper and the guess in the same units, so a half-yearly bond needs half-yearly periods.

How does each input move the yield?

Price. With the other defaults held, a price of 90 gives a YTM of 7.43%, 100 gives 5.00% and 105 gives 3.89%. Years to maturity. A discount to par is spread over the years left, so at a price of 95 the YTM is 7.75% with two years to go, 6.18% with five, 5.66% with ten and 5.34% with thirty, closing in on the current yield. Coupon. Raising the coupon to 6% at the same price of 95 lifts the YTM to 7.21%. Redemption value. Repaying 105 instead of 100 lifts the default YTM to 7.06%, because the gain at maturity is larger.

Tax moves only the issuer's cost. On the exam tab, the default bond costs 6.19% exactly at 0% tax, 5.21% at 19% and 4.90% at 25%. The interpolation rates move only the estimate, never the exact answer; rates either side of the true cost and close together give the smallest error.

Common bond yield mistakes

• Treating the coupon rate as the yield. The coupon is fixed on face value; the yield depends on the price paid.
• Quoting the current yield as the return. It ignores any gain or loss to redemption, which can be most of the return on a deep-discount bond.
• Mixing periods. Ten half-yearly coupons of 2.50 are not five yearly coupons of 5; discount each at the matching periodic rate.
• Using a cum-interest price. In a cost of debt answer, take the interest about to be paid off the quoted price first.
• Taking tax off the redemption payment. Only the interest is tax-deductible; the capital repayment goes in at its full value.
• Leaving out the premium on redemption. Repayment at 105 is part of the return, and of the cost.
• Interpolating with both rates on one side of the answer. The calculator flags it when both NPVs share a sign.

What the bond yield calculator leaves out, and a glossary

It assumes you buy on a coupon date, so there is no accrued interest and no part period; a trade between coupon dates needs a day count, which spreadsheet functions handle. It treats every payment as certain, so it says nothing about the chance of default, and it ignores call features, inflation linking and the investor's own tax. YTM also assumes each coupon is reinvested at the same yield. Once you have a cost of debt, the WACC calculator weights it with the cost of equity, which the CAPM calculator works out from beta.

Face value (par, nominal): the amount the coupon is calculated on, usually the amount repaid. Coupon: the interest paid each year as a percentage of face value. Current yield: annual coupon / price. Yield to maturity: the discount rate at which the remaining cash flows are worth the price. Ex interest: a price that excludes the interest about to be paid. Redemption value: the amount repaid at maturity per 100 of face value. Cost of debt (kd): the yield to maturity seen from the issuer's side, usually taken after tax.

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