Yield to maturity, internal rate of return, and net present value are one idea in three forms: discounting. This guide works a bond and a project end to end, shows where the measures agree and where they disagree, and gives the steps in plain language.

One question, three measures

Three measures dominate bond and project analysis, and beginners often treat them as rivals. They are not rivals. They are three ways of asking the same question about a stream of future cash flows: net present value (NPV) asks what the stream is worth today at a stated discount rate, in dollars; internal rate of return (IRR) asks what discount rate makes the stream worth exactly what it costs, as a percentage; and yield to maturity (YTM) is the IRR of a bond — the discount rate that makes the bond’s coupons and principal worth exactly its price. NPV needs a rate and gives a value. IRR needs a value and gives a rate. YTM is IRR with a bond’s cash flows. Once that relationship is clear, the three measures stop competing and start checking each other: an NPV computed at the IRR is always zero, by definition, and a YTM above the coupon rate always means the bond trades below its face value.

All three rest on one idea: a dollar in the future is worth less than a dollar today, by a factor that compounds with time and with the rate. Discounting is the operation that converts future dollars to present dollars, and every formula in this guide is discounting applied to a particular stream. The financial calculator’s cash-flow functions do the discounting mechanically — the user lists the flows, states the rate, and reads the answer — but the machine cannot supply the judgement: which rate to discount at, which flows belong in the list, and whether the answer is even meaningful for the stream in front of you. This guide works two complete examples, a bond and a project, so both the mechanics and the judgement are visible.

Yield to maturity, worked on a bond

Take a bond with a face value of $1,000, a coupon rate of 5% a year paid half-yearly, and 10 years left to maturity. The holder receives $25 every six months for 20 periods, plus the $1,000 principal at the end. If the bond trades at its face value of $1,000, the yield to maturity is exactly the coupon rate: 5% a year, or 2.5% per half-year, because each $25 coupon is exactly the interest on $1,000 for half a year at that rate. If the bond trades below face value, the yield must be higher, because the buyer pays less for the same stream and also collects the $1,000 at maturity — a gain on top of the coupons. If it trades above face value, the yield must be lower, because the buyer overpays for the same stream and gives back part of the premium at maturity.

Price the bond at $950 and solve for the yield: the half-yearly rate that makes the present value of twenty $25 coupons plus $1,000 in twenty periods equal $950 is 2.8308% per half-year, which is a nominal annual yield of 5.66% compounded half-yearly. The steps a calculator takes are exactly that search: guess a rate, discount the stream, compare with the price, adjust, and repeat until the discounted stream equals the price to the cent. The table shows the price at three yields, so the inverse relationship is unmistakable — price down, yield up — and so the common question “how do I find YTM on a financial calculator” has a plain answer: enter the bond’s cash flows (price as the outflow, coupons and principal as inflows), and solve for the rate that equates them, which is the cash-flow IRR function applied to a bond’s stream.

Two warnings belong with the yield. First, yield to maturity assumes the coupons are reinvested at the yield itself for the whole life of the bond — an assumption that is rarely true in practice, since rates move, and one that makes YTM an “if everything stays put” measure rather than a realised return. Second, YTM assumes the issuer pays: it is a promised yield, not a delivered one, and a bond that defaults delivers far less than its YTM whatever the calculation says. Both warnings apply equally to the IRR of a project, which assumes intermediate cash flows are reinvested at the IRR — an assumption that flatters projects with high IRRs, because the cash rarely finds another home earning the same rate.

Net present value, worked on a project

Now a project. It costs $10,000 today and returns $3,000 at the end of each of the next four years, plus $4,000 at the end of year five. The discount rate — the hurdle the project must clear, usually the cost of the capital funding it — is 8% a year. Net present value discounts each inflow at 8% and subtracts the cost: $3,000 discounted one year is $2,777.78, two years $2,572.02, three years $2,381.50, four years $2,205.09, and the $4,000 in year five is $2,722.33. The discounted inflows total $12,658.71, the cost is $10,000, and the NPV is $2,658.71 — positive, so the project creates value at an 8% hurdle, by that amount, in today’s dollars. The sign is the verdict: NPV above zero means accept, NPV below zero means reject, and NPV of exactly zero means the project earns exactly its hurdle rate and no more.

The mechanics on a calculator follow the same shape as the bond, with the cash flows listed rather than level. Enter the initial outlay as a negative flow at time zero, the five inflows as positive flows at their dates, state the 8% discount rate, and read the NPV. In a spreadsheet the same calculation is =NPV(8%, 3000, 3000, 3000, 3000, 4000) – 10000, because the spreadsheet NPV function discounts its first value by one period and the initial outlay must stay outside it. The most common NPV error is not arithmetic but inclusion: cash flows that belong to the project must all be listed — working capital, tax, the terminal value — and cash flows that do not belong, like sunk costs already spent or financing flows already reflected in the discount rate, must be left out. A perfect discounting of the wrong list is still a wrong answer.

Internal rate of return, on the same project

The IRR of the same project is the discount rate that makes its NPV exactly zero: the rate at which the discounted inflows of $12,658.71 at 8% shrink to exactly $10,000. Solving gives 17.23% a year. The interpretation is direct: the project earns 17.23% a year on the money tied up in it, and it clears any hurdle below that rate — at 8% it clears comfortably, which is why the NPV at 8% is positive, and at a hurdle above 17.23% the NPV would turn negative and the verdict would flip. IRR and NPV agree whenever the cash flows change sign only once, as they do here: money out once, money in thereafter. That agreement is the normal case for simple projects, and it is why the two measures are so often quoted together — NPV in dollars for the size of the gain, IRR in percent for the rate of the gain.

IRR has a practical advantage and a practical trap. The advantage is that it needs no hurdle rate: it reports the project’s own rate, and the decision-maker compares it with whatever hurdle applies, which makes it the natural measure when the hurdle is disputed or unknown. The trap is that IRR is a rate, not a value: a small project with a 40% IRR creates less wealth than a large project with a 15% IRR, and ranking projects by IRR alone systematically favours the small and the quick. IRR also assumes reinvestment at the IRR itself, as YTM does, which overstates the realised return of high-IRR projects whose intermediate cash cannot be reinvested so profitably. Where the decision is accept-or-reject for one project, IRR is sufficient. Where the decision is which of several projects to fund with limited capital, NPV in dollars is the measure that maximises wealth, and IRR is the supporting evidence.

When NPV and IRR disagree, trust NPV

The measures disagree in two well-defined cases, and both are worth recognising before they arrive. The first is a stream that changes sign more than once — money out, money in, money out again, as with a project that needs a costly decommissioning at the end, or a mine that must be restored. Such a stream can have two or more IRRs, or none at all, because the NPV curve crosses zero several times as the discount rate rises; the calculator may report one of them, or fail to converge, and none of the reported figures is “the” return. The second is mutually exclusive projects of different scale or timing: the smaller, faster project usually shows the higher IRR, while the larger, slower project shows the higher NPV, and choosing by IRR leaves money on the table. In both cases the remedy is the same: compute NPV at the stated discount rate, which is always defined and always single-valued, and treat IRR as a secondary check. If the IRR function returns an error or a wildly implausible rate, the first suspicion should be the cash-flow list — a missing outflow, a sign error, a flow dated wrong — because the function is almost never the problem; the list usually is.

The steps, in plain language

On any financial calculator, the cash-flow workflow has the same four steps, whatever the keys are labelled. First, clear the cash-flow list, so no stale flow from the previous problem contaminates this one — the most common source of wrong answers in practice. Second, enter the flows in date order, with outflows negative and inflows positive, including the initial outlay as the first flow; for a bond, the price is the outflow and the coupons plus principal are the inflows. Third, for NPV, state the discount rate and read the net present value; for IRR or YTM, solve for the rate that makes the NPV zero, starting from a sensible guess such as 10% if the calculator asks for one. Fourth, sanity-check the answer against the table logic: a bond below par must yield above its coupon rate, a project with NPV positive at the hurdle must have an IRR above the hurdle, and an answer that violates either check has a sign or timing error in the list. The calculator does the discounting; the checks are the user’s, and they are what separate a computed number from a trusted one.

Frequently asked questions

What is yield to maturity (YTM)?

Yield to maturity is the discount rate that makes a bond’s future coupons and principal worth exactly its current price. It is the internal rate of return of the bond. On the worked example — a 1,000 dollar bond paying 25 dollars every six months for 10 years, priced at 950 dollars — the YTM is 5.66 percent a year compounded half-yearly, above the 5 percent coupon rate because the bond trades below face value. YTM assumes coupons are reinvested at the yield and the issuer pays in full; neither is guaranteed.

How do I find YTM on a financial calculator?

Enter the bond’s cash flows and solve for the rate. Clear the cash-flow list, enter the price as a negative outflow, enter the coupons as positive inflows with the principal added to the final coupon, then use the IRR function to solve for the rate that makes the net present value zero. That rate, annualised for the coupon frequency, is the YTM. Check it: a bond priced below face value must yield above its coupon rate, and a bond above face value must yield below it.

What is net present value (NPV)?

Net present value is a stream of future cash flows discounted to today at a stated rate, minus the initial cost. On the worked project — 10,000 dollars out, 3,000 dollars a year for four years plus 4,000 dollars in year five, discounted at 8 percent — the discounted inflows total 12,658.71 dollars and the NPV is 2,658.71 dollars. A positive NPV means the project creates value above its hurdle rate; a negative NPV means it destroys value.

How do I calculate IRR on a financial calculator?

List the cash flows in date order with outflows negative and inflows positive, including the initial outlay first, then use the IRR function to solve for the discount rate that makes the NPV zero. On the worked project the IRR is 17.23 percent a year. If the calculator asks for a starting guess, 10 percent is a sensible default. If it returns an error, check the cash-flow list first — a sign error or a missing flow is the usual cause, especially with streams that change sign more than once.

NPV versus IRR: which should I trust when they disagree?

Trust NPV at the stated discount rate. IRR and NPV agree for simple streams that change sign once, but streams that change sign more than once can have multiple IRRs or none, and mutually exclusive projects of different scale can rank oppositely by the two measures. NPV is always single-valued and measures wealth created in dollars, which is what the decision is about. Use IRR as supporting evidence — the rate of return — and NPV as the verdict.

What is the difference between YTM and the coupon rate?

The coupon rate is the interest the bond pays on its face value — 5 percent on 1,000 dollars is 50 dollars a year. YTM is the total return implied by the price paid, including the gain or loss at maturity. A bond bought at face value has a YTM equal to its coupon rate. A bond bought below face value, at 950 dollars, has a YTM of 5.66 percent, above the coupon rate, because the buyer also gains 50 dollars of principal at maturity. A bond bought above face value yields below its coupon rate for the mirror-image reason.