Yield to Maturity

Yield to maturity is the foundational yield measure of fixed income analysis — most bond pricing calculations and yield spread comparisons use yield to maturity as a primary analytical input — and is directly and extensively tested on the Series 65 examination in the context of bond pricing, the price-yield relationship, and the comparison among the multiple yield measures.

The Three Assumptions of Yield to Maturity

The precision of yield to maturity as a return measure depends critically on three assumptions that are explicitly embedded in its definition and that investors must understand to correctly interpret the YTM figure.

The first assumption is that the investor holds the bond to maturity — if the bond is sold before the stated maturity date, the actual realised return will generally differ from the YTM, because the sale price is set by the market yield prevailing at the date of sale rather than by the YTM at purchase. Only if the market yield at the sale date is unchanged from the YTM at purchase, and the coupons received have been reinvested at that yield, does the realised return equal the YTM. YTM is a hold-to-maturity return measure — it is the return only for investors who genuinely intend to hold the bond until it matures and receives par.

The second assumption is that all coupon payments are received as scheduled — there is no default by the issuer. For investment-grade corporate bonds and Treasury securities this assumption is generally reasonable. For high-yield bonds with meaningful default probability, the YTM overstates the expected return because it does not account for the possibility that some coupons will not be received or that the par value will not be repaid in full.

The third assumption — and the most significant practical limitation of YTM — is that all coupon payments received are reinvested at the same rate as the yield to maturity itself throughout the bond's remaining life. In practice, the reinvestment rate will change as market interest rates change over the bond's life — when interest rates fall below the YTM, reinvested coupons earn less than assumed, and the realised return falls below the YTM. When interest rates rise above the YTM, reinvested coupons earn more than assumed, and the realised return exceeds the YTM. Only zero coupon bonds — which pay no interim coupon and therefore have no reinvestment rate assumption — actually deliver the YTM free of any reinvestment assumption if held to maturity and paid in full by the issuer.

The Price-Yield Relationship — The Foundational Bond Theorem

The relationship between a bond's market price and its yield to maturity is the most fundamental theorem in all of fixed income analysis — an inverse relationship whose mathematical certainty derives directly from the discounted cash flow framework through which bond prices are determined.

When interest rates rise — meaning the market requires a higher yield to maturity on bonds of a given credit quality and maturity — the fixed future cash flows of existing bonds become less valuable because they must be discounted at a higher rate. The bond's price must fall until its yield to maturity — the discount rate that equates its future cash flows to its price — equals the new higher market yield. The mathematical consequence is that bond prices fall when interest rates rise.

When interest rates fall — meaning the market is satisfied with a lower yield to maturity — existing bonds with above-market coupons become more valuable. The bond's price must rise until its yield to maturity equals the new lower market yield. Bond prices rise when interest rates fall.

This inverse relationship — bond prices move opposite to interest rates — is the most critical single fact in fixed income investing and is tested on virtually every fixed income question in the Series 65 curriculum. Every bond question about price movements in response to interest rate changes is answered by this inverse relationship.

The magnitude of the price change for a given yield change depends on the bond's duration — the present-value-weighted average time to receipt of all cash flows. Longer duration bonds have greater price sensitivity to interest rate changes — a thirty-year bond changes in price far more than a two-year bond for the same one percentage point change in yield. Duration is covered in the Duration entry of this dictionary.

The YTM and the Three-Way Yield Relationship

As established in the Yield entry of this dictionary, yield to maturity participates in a precise three-way relationship with nominal yield and current yield depending on whether the bond trades at par, at a discount, or at a premium.

At par — when market price equals face value — all three yields are equal. YTM equals current yield equals nominal yield. The coupon rate precisely equals the market required return — no capital gain or loss occurs at maturity.

At a discount — when market price is below face value — YTM exceeds current yield which exceeds nominal yield. The below-market coupon income is supplemented by a capital gain at maturity — the bond purchased below par will appreciate to par — adding to the total annualised return above what the current yield alone would suggest.

At a premium — when market price is above face value — YTM is below current yield which is below nominal yield. The above-market coupon income is reduced by a capital loss at maturity — the bond purchased above par will decline to par — reducing the total annualised return below what the current yield alone would suggest.

Calculating YTM — The Approximation Formula

The exact yield to maturity can only be calculated through iterative numerical methods — trial and error using a financial calculator or spreadsheet — because there is in general no closed-form algebraic solution for the YTM of a coupon-bearing bond. For examination purposes, the approximation formula provides a reasonably accurate estimate.

The approximate YTM formula equals the annual coupon payment plus the quantity face value minus market price divided by years to maturity — all divided by the quantity face value plus market price divided by two.

Approximate YTM equals coupon plus the quantity par minus price divided by years — divided by the quantity par plus price divided by two.

A ten-year bond with a face value of one thousand dollars, a five percent annual coupon paying fifty dollars annually, currently trading at nine hundred dollars. Approximate YTM equals fifty plus the quantity one thousand minus nine hundred divided by ten — equalling fifty plus ten — equalling sixty — divided by the quantity one thousand plus nine hundred divided by two — equalling nine hundred and fifty — equalling six point three two percent. The bond's yield to maturity of approximately six point three two percent exceeds both the nominal yield of five percent and the current yield of approximately five point five six percent — confirming the discount bond relationship.

Exact Yield to Maturity and the Error in the Approximation

The approximation formula is an estimate, and its error can be measured against the exact figure. Solving the ten-year example above exactly — finding the single annual discount rate at which ten coupons of fifty dollars and the one thousand dollar repayment at maturity are together worth nine hundred dollars today — gives a yield to maturity of about six point three eight percent, against the approximation's six point three two percent. For this discount bond the approximation understates the true yield to maturity.

The direction of the error depends on which side of par the bond trades. Repeating the exact calculation for the same five percent ten-year bond at other prices gives about seven point nine eight percent exact against seven point seven eight percent approximate at a price of eight hundred dollars, and about three point seven eight percent exact against three point eight one percent approximate at a price of eleven hundred dollars. In these computations the approximation understates the yield of every bond priced below par and overstates the yield of every bond priced above par, and the gap widens as the price moves further from par — at nine hundred and fifty dollars the two figures are about five point six seven and five point six four percent, much closer together than at nine hundred.

A zero coupon bond needs no approximation at all. Because it makes a single payment, its price equation can be solved for the yield directly: the yield per period equals the face value divided by the price, raised to the power of one divided by the number of periods, minus one. A twenty-year zero priced at about three hundred and six dollars and fifty-six cents per one thousand dollars of face value, on the semi-annual convention of forty periods, gives three percent per half-year — a six percent yield to maturity — with no trial and error. The iterative search is needed only where there are several payments to discount.

The Reinvestment Risk Limitation and Zero Coupon Bonds

The reinvestment rate assumption is the most significant practical limitation of YTM as a return predictor — and the zero coupon bond is the instrument that eliminates this limitation entirely.

A coupon-bearing bond that matures in twenty years and pays six percent semi-annual coupons will only actually deliver a six percent YTM if each of the forty coupon payments received over twenty years is reinvested at exactly six percent for its remaining period. If interest rates fall to three percent after five years, the subsequent coupons are reinvested at three percent — producing a realised return below the six percent YTM. If rates rise to nine percent, the subsequent coupons are reinvested at nine percent — producing a realised return above the YTM.

A zero coupon bond with a twenty-year maturity and a six percent YTM purchased at the discounted price — approximately three hundred and seven dollars for a one thousand dollar face value on the semi-annual compounding convention — will deliver exactly its six percent yield to maturity if held to maturity and paid in full, regardless of what happens to interest rates during the twenty-year period, because there are no interim coupons to reinvest. Zero coupon bonds are the only fixed income instruments that lock in the yield to maturity without any reinvestment assumption, provided the issuer pays in full — making them ideal for liability-driven investing strategies where a specific dollar amount must be available at a specific future date.

How the Treasury Turns an Auction Yield into a Coupon

Treasury notes and bonds are sold at auction, and for a new issue Treasury's auction regulations fix the coupon from the auction result rather than before it. Under 31 CFR 356.20(b), the interest rate Treasury establishes for a new note or bond is the rate that produces the price closest to, but not above, par when evaluated at the yield of awards to successful competitive bidders. The coupon is therefore set at or just below the auction yield, so a new Treasury coupon issue is ordinarily sold at par or at a small discount, and its yield to maturity at issue is equal to or slightly above its coupon. Appendix B to Part 356 shows the pattern: an eight and three-quarter percent thirty-year bond evaluated at a yield of eight point eight four percent is priced at 99.057893 per 100 of par.

The rule has a floor. If an auction of a non-indexed or inflation-protected note or bond results in a yield lower than 0.125 percent, the interest rate is set at one-eighth of one percent and the successful bidders' award prices are calculated accordingly. In that case the coupon is above the auction yield, and the price that equates the bond's cash flows with that lower yield is above par — the one situation in which a new fixed-coupon Treasury note or bond is sold at auction at a premium.

A reopening works differently. The regulations define a reopening as the auction of an additional amount of an outstanding security, and the interest rate of that security was fixed when it was first issued. Bidding in a reopening is still in yield, but the auction now determines only the price: Appendix B prices reopened issues at whatever yield the reopening produces, above or below the coupon, so a ten and three-quarter percent bond reopened at a yield of ten point four seven percent is priced above par. The coupon a reopened security carries does not set the yield to maturity at which the additional amount is sold.

Yield to Maturity in Treasury's Pricing Formulas

Appendix B to 31 CFR Part 356 sets out the formulas Treasury uses to convert a yield into a price for its notes and bonds, and its definitions fix the quotation convention. The yield in those formulas is the nominal annual rate of return or yield to maturity, based on semiannual interest payments, and each future payment is discounted by one divided by the quantity one plus half the yield, once for each semiannual period. A quoted Treasury yield to maturity is therefore twice the rate earned per half-year rather than a rate compounded once a year — a six percent yield means three percent per half-year.

A note or bond is often issued, or reopened, part-way through an interest period, and the formulas deal with the part period directly. They use the ratio of r to s — the number of days from the issue date to the first interest payment, over the number of days in the full semiannual period ending on that payment date — and apply that fraction of one period's yield to the stub, while the remaining full semiannual periods are discounted in the ordinary way. A security bought part-way through a coupon period is priced for exactly the time that remains, not for a whole period.

Where a security carries accrued interest at settlement — a reopened issue, or a new issue whose interest accrues from the coupon date before its issue date — the formula equates the discounted cash flows with the price plus the accrued interest, and only then subtracts the accrued interest to leave the price. Appendix B's example of a nine and one-half percent ten-year note at a yield of nine point five four percent gives price plus accrued interest of 100.098321, accrued interest of 0.367403 and a price of 99.730918 per 100. The yield to maturity is the rate that equates the cash flows with the full amount the buyer pays, quoted price and accrued interest together, not with the quoted price alone.

Real Yields and Discount Margins at Treasury Auctions

Not every Treasury coupon security is bid in a nominal yield to maturity. Under 31 CFR 356.12(c), a competitive bid for a Treasury inflation-protected security must show the real yield bid, expressed with three decimals, and that real yield may be a positive number, a negative number or zero — whereas the yield bid on a non-indexed note or bond may be a positive number or zero.

Treasury converts a real yield into a price in two steps. Under Appendix B to Part 356, it first computes an unadjusted, or real, price per 100 by discounting the real coupon and the real principal at the real yield with the same semiannual formula used for nominal securities. It then multiplies that real price, and any accrued interest, by the index ratio for the settlement date to obtain the inflation-adjusted price and the settlement amount. The Appendix's example of a three and seven-eighths percent ten-year inflation-protected note shows it issued at a discount to a real yield of three point eight nine eight percent.

Floating rate notes have no fixed coupon, so no conventional yield to maturity can be bid for them. A competitive bid shows a discount margin instead. The note's interest rate is the index — the high discount rate on 13-week bills determined by Treasury's auctions of those bills — plus a spread that is fixed for the life of the note and that, for a new note, equals the high discount margin at its auction, subject to a minimum daily interest accrual rate of zero percent. Because the spread equals the discount margin at issue, a new floating rate note is ordinarily issued at par; Appendix B shows a premium arising where the discount margin at auction is negative and the zero accrual floor applies.

The Municipal Yield Formula Under MSRB Rule G-33

For municipal securities the arithmetic of yield to maturity is not left to each firm. MSRB Rule G-33 requires brokers, dealers and municipal securities dealers effecting transactions in municipal securities to compute yields and dollar prices in accordance with prescribed formulas, and it defines the yield as the yield on the investment if the security is held to redemption. For a security with periodic interest payments the rule draws a line at the last coupon period: one formula applies to a security with one coupon period or less to redemption, and a different formula, which computes and sums the present value of each interest payment between settlement and redemption together with the redemption value, applies to a security with more than one coupon period to redemption.

The formula for more than one coupon period was amended with effect from February 23, 2016. Before the amendment it presumed a semi-annual payment of interest. As amended it requires the actual interest payment frequency of the security to be factored in — monthly, quarterly or annually, for example — through a term defined as the number of interest payment periods per year standard for the security, so that a municipal bond paying interest once a year is no longer priced as though it paid twice a year.

The rule also sets standards of accuracy. All values used in computing accrued interest, yield and dollar price must be computed to not less than ten decimal places. Dollar prices are truncated to three decimal places before they are presented on the confirmation and used to compute extended principal. Yields are truncated to four decimal places and rounded to three, although for the confirmation display that Rule G-15(a) requires, a yield accurate to the nearest .05 percentage point is deemed satisfactory.

The Yield Shown on a Municipal Confirmation

MSRB Rule G-15(a) requires a confirmation of a municipal securities transaction with a customer to show both a dollar price and a yield, each derived from the other. Where the transaction was effected on the basis of yield, the yield at which it was effected is shown and a dollar price is computed from it, and that dollar price is used in computing the extended principal and final monies. Where the transaction was effected on the basis of a dollar price, the dollar price is shown and a yield must be computed and shown — unless the transaction was effected at par.

The computed yield is not a gross figure. The rule requires yield computations to take into account dollar price concessions granted to the customer, commissions charged to the customer and adjustable tender fees applicable to puttable securities. Incidental transaction fees and miscellaneous charges are not taken into account in the yield; they must instead be indicated separately on the confirmation. A confirmed yield on a commission trade therefore reflects the commission charged, not only the price at which the trade was executed.

Nor is the computed yield always a yield to the nominal maturity date. For securities that are prerefunded or called to a date before maturity, the date and dollar price of redemption set by the prerefunding are used in the computation in place of the maturity date and maturity value, and for securities subject to a mandatory tender the tender date and its dollar price of redemption are used instead. Whenever the computed yield is not based on the nominal maturity date, the date used must be identified and stated on the confirmation, and whenever it is not based on a redemption value of par, the dollar price used in the computation must be shown.

When a Municipal Confirmation Shows No Yield

A yield to maturity presupposes a schedule of payments that will be made, and Rule G-15(a) recognises the securities for which no such figure can honestly be stated. For securities that have defaulted in the payment of interest or principal, the rule is categorical: a yield shall not be shown. The second assumption embedded in yield to maturity — that every payment is received as scheduled — has already failed for such a security, and the confirmation does not present a yield computed as though it had not.

For municipal securities with a variable interest rate the future coupons are not known in advance, and the rule provides that a yield shall not be shown unless the transaction was effected on the basis of yield to put. The exception follows the way the trade was struck: where the parties priced the transaction to a put date, that yield is the basis of the trade and is shown; otherwise no yield appears for the variable rate security.

A third group is treated more leniently. For securities that prepay principal periodically, and for municipal collateralized mortgage obligations, a yield computation and display of yield is not required, but if a yield is displayed the confirmation must include a statement describing how the yield was computed. Because the timing of principal on such securities depends on prepayments, any yield shown rests on an assumption about that timing, and the required statement discloses the basis on which the figure was produced rather than leaving the customer to take it as a conventional yield to maturity.

Zero Coupon and Discounted Municipal Securities on the Confirmation

Rule G-15(a) adjusts the final monies section of the confirmation for securities that do not pay interest periodically. For zero coupon securities no figure for accrued interest shall be shown, because there is no current coupon accruing between payment dates; the whole return of such a security is the difference between the price paid and the amount received at redemption, which is what its yield to maturity measures.

Securities traded on a discounted basis are treated separately. For such securities a yield shall not be shown, no figure for accrued interest is shown unless the security is a discounted security traded on a yield-equivalent basis, and the total dollar amount of discount may be shown in place of the resulting dollar price and extended principal amount. Rule G-33 follows the same division in its formulas: the dollar price of a discounted security is computed by its own discount formula, while a discounted security traded on a yield-equivalent basis is priced with the formula for securities paying interest solely at redemption.

Day counting follows the same line. Rule G-33 requires its computations to be made on the basis of a thirty-day month and a three-hundred-sixty-day year, except that for securities paying interest solely at redemption the computations are made on the day count basis selected by the issuer of the securities. A dealer pricing a municipal note that pays all of its interest at maturity therefore takes the day count from the terms the issuer set rather than from the standard convention used for periodic-interest municipal bonds.

Yield to Maturity in the Tax Rules for Original Issue Discount

The income tax regulations on original issue discount define yield to maturity in their own terms. Under Treasury Regulation section 1.1272-1(b)(1)(i), the yield to maturity of a debt instrument is the discount rate that, when used in computing the present value of all principal and interest payments to be made under the instrument, produces an amount equal to the issue price of the instrument. The yield must be constant over the term of the instrument and, when expressed as a percentage, must be calculated to at least two decimal places. The tax yield is fixed by the issue price, not by the price a later buyer pays in the market.

The regulation measures the accrual of discount over accrual periods, and it gives the holder latitude in choosing them. Accrual periods may be of any length and may vary in length over the term of the instrument, provided that each accrual period is no longer than one year and each scheduled payment of principal or interest occurs either on the final day or on the first day of an accrual period. The regulation notes that the computation is generally simplest when the accrual periods correspond to the intervals between the payment dates the instrument provides.

Whatever accrual periods are chosen, the yield applied in each must be stated appropriately for the length of the period. The regulation's examples state a yield of eight percent compounded semiannually, and they give a formula for converting a yield based on compounding a given number of times a year into the equivalent yield based on compounding a different number of times a year. The same instrument can therefore be accrued over quarterly, semiannual or annual periods without changing its economic yield, only the way that yield is stated.

The Savings Bond Doubling Guarantee as a Yield

United States Series EE savings bonds issued on or after May 1, 2005 offer a fixed rate of interest that the Treasury establishes for the life of the bond, including the extended maturity period, unless before that period begins the Secretary announces a different fixed rate for it or amends its terms. Interest accrues monthly and compounds semiannually under 31 CFR 351.35. Such a bond reaches original maturity twenty years after its issue date and final maturity thirty years after it, when it ceases to earn interest.

At original maturity, the redemption value of a book-entry Series EE bond shall not be less than double its purchase price. Doubling over twenty years is a minimum compound return: two raised to the power of one-twentieth, minus one, is about three point five three percent a year compounded annually, or about three point five percent compounded semiannually. A holder who keeps the bond to its twentieth anniversary therefore earns at least that yield, whatever fixed rate the bond carries, and when the fixed rate is lower the guaranteed value at original maturity — not the fixed rate — sets the holder's return to that date.

Like a yield to maturity, the guarantee is earned only by holding to the date at which it is measured. The doubling floor applies at original maturity; a bond redeemed before then is worth only what its fixed rate has accrued, less any early-redemption penalty. The regulation adds a penalty for the earliest redemptions: a bond redeemed less than five years after its issue date has its earning period reduced by three months, although its redemption value is not reduced below the issue price, and the penalty does not apply to a bond redeemed five years or more after issue.

Examination Relevance and Key Takeaways

Yield to maturity is tested on the Series 65 examination as the primary bond yield measure — in the context of the price-yield relationship, the three-way yield comparison, the YTM approximation calculation, the three embedded assumptions, and the comparison with current yield and yield to call.

The key points to retain are these.

Yield to maturity is the total annualised return on a bond assuming purchase at the current market price, all coupons received as scheduled, coupons reinvested at the YTM rate, and the bond held to maturity. YTM is the bond's internal rate of return — the discount rate making the present value of all future cash flows equal to the current market price — incorporating coupon income, capital gain or loss, and the time value of money simultaneously. The three YTM assumptions are hold to maturity, no default, and coupon reinvestment at the YTM rate — the reinvestment assumption is the most significant practical limitation.

The foundational inverse price-yield relationship — bond prices fall when interest rates rise and bond prices rise when interest rates fall — is the most critical fact in fixed income analysis. The three-way relationship — at discount YTM exceeds current yield exceeds nominal yield; at par all three are equal; at premium nominal yield exceeds current yield exceeds YTM. The approximate YTM formula equals coupon plus the quantity par minus price divided by years — divided by the quantity par plus price divided by two. Only zero coupon bonds lock in the YTM free of reinvestment risk, absent default — because there are no interim coupons to reinvest, realised return equals YTM regardless of interest rate movements during the holding period.