Fixed-Income Bond Valuation: Prices and Yields

Fixed Income. Worth 11 to 14 percent of the exam. One session: the lesson, the rules, the method, then the questions.

Fixed IncomeFixed-Income Bond Valuation: Prices and Yields
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The lesson

Runtime 10 minutes 37 seconds, measured from the published video.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to price a bond from its coupon, maturity and required yield, read premium, par or discount pricing straight off the coupon-versus-yield comparison, work out accrued interest and the full price a buyer actually pays between coupon dates, and explain how matrix pricing estimates a yield for a thinly traded bond.

A bond's coupon rate is fixed at issuance and never changes; it is the label on the box. What actually moves the bond's price every day is the yield to maturity, the market's current required return, and the market reprices the bond until its total return, coupon income plus any capital gain or loss by maturity, equals that required yield. Bond price is simply the present value of every promised cash flow, each coupon plus the final par redemption, discounted at that required yield. For a bond paying semi-annually, three adjustments happen together before any present-value step: the number of periods doubles, the coupon per period halves, and the yield per period halves, relative to the annual figures a question might quote.

The relationship between coupon rate and yield tells you a bond's pricing category without any calculation at all. A coupon rate above the yield to maturity means the bond prices above par, a premium; a coupon rate below the yield means it prices below par, a discount; equal rates mean it prices exactly at par. This works in reverse too: a price above $1,000 par, by itself, proves the coupon rate exceeds the yield, with no further arithmetic required. Price and yield always move in opposite directions. A higher discount rate reduces the present value of the same fixed cash flows. Every bond's price is also pulled toward par as maturity approaches, regardless of whether it started as a premium or a discount bond.

When a bond trades between coupon dates, the price quoted on a screen, the flat or clean price, is not what the buyer actually pays. The buyer pays the full, or dirty, price: the flat price plus accrued interest owed to the seller for the portion of the current coupon period that has already elapsed. Accrued interest equals the coupon for the period multiplied by the fraction of days elapsed since the last coupon date. Quoting the flat price is what keeps market screens from showing an artificial price drop on every coupon date; the full price is the honest cash amount that actually changes hands at settlement.

Holding every other feature constant, a longer maturity makes a bond more sensitive to a given change in yield. A lower coupon rate makes it more sensitive too. More of the bond's total value sits further out in time and is therefore hit harder by discounting. When two bonds differ on both dimensions at once, the one that is both longer and lower-coupon dominates in sensitivity over one that is only longer or only lower-coupon.

Matrix pricing exists for a bond that lacks a reliable market price of its own, typically one that trades infrequently or has not yet been actively quoted. An analyst estimates its required yield by interpolating between the yields of actively traded, comparable-credit-quality bonds whose maturities bracket the target bond's own maturity, then prices the bond in the ordinary present-value way using that estimated yield.

Bond price falls as yield rises price yield par yield
Price and yield move in opposite directions. As the required yield rises, the price you would pay today for the same fixed coupons falls, and the curve bends, it does not fall in a straight line.

Worked in full

A bond has a 5 percent annual coupon rate, paid semi-annually, 4 years to maturity, and a required yield to maturity of 7 percent. What is the bond's price? Semi-annual adjustments: N = 4 x 2 = 8, I/Y = 7% / 2 = 3.5%, PMT = 5% x $1,000 / 2 = $25, FV = $1,000. Price = $25 x [1 - (1.035)^-8] / 0.035 + $1,000 x (1.035)^-8 = $931.26. Since the coupon rate, 5 percent, is below the yield, 7 percent, this is a discount bond, and $931.26 sits below the $1,000 par value, exactly as the price-versus-par shortcut predicts.

The same problem, one step removed

Same bond: 5 percent annual coupon paid semi-annually, 4 years to maturity, 7 percent yield to maturity. Make the three semi-annual adjustments yourself, then discount the coupons and the par value to find the price.

The trap

Pricing a semi-annual coupon bond with annual inputs instead of the three semi-annual adjustments produces a plausible-looking but wrong answer; the exam's default for USD bonds is semi-annual, and skipping the double-N, halve-rate, halve-coupon conversion is the most repeated setup error on this module.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

Bond price is the present value of every promised cash flow, coupons and the final par redemption, discounted at the required yield

A bond's price equals the sum of each coupon payment's present value plus the present value of the par amount returned at maturity, all discounted at the market's required yield to maturity for that period; for a semi-annual coupon bond, the number of periods doubles, the coupon per period halves, and the yield per period halves relative to the annual figures before this present-value calculation is run.

Pricing between coupon dates requires the full price, built from the flat price plus accrued interest

When a bond is purchased between coupon payment dates, the buyer pays the full (dirty) price, which equals the flat (clean) price, the quoted market price, plus accrued interest owed to the seller for the portion of the current coupon period already elapsed; accrued interest is the coupon for the period multiplied by the fraction of days elapsed since the last coupon date, using the day-count convention the market specifies.

The relationship between coupon rate and yield to maturity sets whether a bond trades at a premium, at par, or at a discount

Coupon rate greater than yield to maturity means the bond prices above par, a premium; coupon rate equal to yield to maturity means the bond prices exactly at par; coupon rate less than yield to maturity means the bond prices below par, a discount. This relationship can be read directly from price versus par without any calculation: a price above $1,000 par by itself proves the coupon rate exceeds the yield, and the reverse for a discount.

Price and yield always move in opposite directions, and every bond's price converges to par as maturity approaches

Raising the required yield always lowers a bond's price and lowering the yield always raises it, because a higher discount rate reduces the present value of the same fixed cash flows. Independent of that relationship, as a bond approaches maturity its price is pulled toward par (pull to par), a premium bond's price declines toward par and a discount bond's price rises toward par, purely because fewer and fewer future cash flows remain to be discounted, until at maturity the only remaining cash flow is the par redemption itself.

Longer maturity and lower coupon both increase a bond's sensitivity to yield changes

Holding other features constant, a bond with a longer maturity is more price-sensitive to a given change in yield than a shorter one, and a bond with a lower coupon rate is more sensitive than one with a higher coupon rate, because more of its total value sits further in the future and is therefore more heavily affected by discounting. When comparing two bonds on both dimensions at once, the longer-maturity, lower-coupon bond dominates in sensitivity over the shorter-maturity, higher-coupon one.

Matrix pricing estimates a yield, and therefore a price, for a bond that lacks a reliable market price of its own

When a bond trades infrequently, or is new to market and not yet actively quoted, an analyst can estimate its required yield by interpolating between the yields of actively traded, comparable-credit-quality bonds with maturities that bracket the target bond's own maturity; the estimated yield is then used to price the bond in the ordinary present-value way. This technique is a practical substitute for a missing observed price, not a formula that changes how the price itself is calculated once the yield is estimated.

The trick

Price versus par tells you the coupon-versus-yield relationship for free

Above par means coupon rate exceeds yield; below par means yield exceeds coupon rate; at par means they are equal. No calculation is needed, only a comparison to $1,000.

Semi-annual bonds: double N, halve the coupon, halve the yield

The three adjustments needed whenever a bond pays semi-annually rather than annually, applied together, before any present-value step.

Longer maturity AND lower coupon both push sensitivity the same direction

When comparing bonds, check both dimensions; a bond that is longer AND lower-coupon dominates in sensitivity over one that is only longer or only lower-coupon.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. Identify whether the coupon is annual or semi-annual, and adjust the number of periods, the coupon per period, and the yield per period accordingly before any calculation.
  2. Discount every coupon and the par redemption at the appropriate periodic yield, then sum the present values for the bond price.
  3. If the bond trades between coupon dates, add accrued interest (coupon for the period times the fraction of the period elapsed) to the flat price to get the full price the buyer actually pays.
  4. Use the coupon-versus-yield comparison, not a fresh calculation, to answer premium/par/discount questions whenever price and par are both given.
  5. If a bond lacks a reliable observed yield, estimate it by interpolating between comparable, actively traded bonds bracketing its maturity (matrix pricing), then price it in the ordinary way using that estimated yield.
  6. [BA II Plus: enter N (periods), I/Y (periodic yield), PMT (periodic coupon), FV (par value), then CPT PV for price; for semi-annual bonds enter N as years x 2, I/Y as annual yield / 2, and PMT as annual coupon / 2]

Two worked examples, then you are on your own

The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.

Worked in full

A 6% annual coupon bond with a face value of $1,000 has 3 years to maturity. If the required yield to maturity is 8%, the bond's price is closest to:

Answer A. Bond price = PV of coupons + PV of par. Coupon = 0.06 x $1,000 = $60 per year. PV of coupons = $60 x [1 - (1+0.08)^-3] / 0.08 = $60 x 2.5771 = $154.63. PV of par = $1,000 / (1.08)^3 = $793.83. Price = $154.63 + $793.83 = $948.46. Since coupon rate (6%) < YTM (8%), the bond trades at a discount, confirming price < $1,000.

Your turn, setup given

A 5% semi-annual coupon bond with $1,000 face value has 4 years to maturity. The bond's YTM is 4% (annual). The bond's price is closest to:

Identify whether the coupon is annual or semi-annual, and adjust the number of periods, the coupon per period, and the yield per period accordingly before any calculation.

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Exam level

A zero-coupon bond with a face value of $1,000 matures in 5 years. If the YTM is 6% (semi-annual compounding), the bond's price is closest to:

How sure are you?

Correct: A. For a zero-coupon bond with semi-annual compounding: Price = $1,000 / (1 + 0.06/2)^(5x2) = $1,000 / (1.03)^10 = $1,000 / 1.34392 = $744.09. Answer A ($747.26) uses annual compounding: $1,000/(1.06)^5 = $747.26. The exam typically uses semi-annual for consistency with coupon bonds. On the actual exam, read carefully whether compounding is specified as annual or semi-annual.
B. Choosing B ($712.99) might tempt you if you incorrectly apply an annual compounding rate of 12% instead of 6%, violating the given semi-annual compounding rule and leading to an underestimation of the bond's price.
C. Choosing $862.61 might tempt you if you mistakenly used a lower yield to maturity or incorrectly applied a different compounding frequency, but the correct calculation requires using the given 6% YTM with semi-annual compounding to accurately determine the bond's price.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 2Exam level

Which of the following statements about the price-yield relationship for a non-callable bond is most accurate?

How sure are you?

Correct: B. The price-yield relationship for a non-callable bond is convex (curved, not linear). This means that for equal basis point increases and decreases in yield, the price increase (when yield falls) is greater than the price decrease (when yield rises). This is the positive convexity property that makes bonds attractive to investors. A is wrong (inverse relationship). B is wrong (the relationship is curved, not linear. Though duration provides a linear approximation). D is wrong (a par bond will move off par when yields change).
A. You might be thinking that duration provides a good linear approximation for small yield changes, but this overlooks the inherent convexity of the price-yield relationship, which means that even for small changes, the relationship is not truly linear as convexity ensures price increases more when yields fall than it decreases when yields rise.
C. Choosing C might seem logical if you think that a bond at par remains stable, but this ignores the fundamental principle that bond prices move inversely to yield changes, meaning a par bond will move off par as yields fluctuate.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 3Exam level

A bond with a coupon rate of 7% is priced at $1,050. The bond's YTM must be closest to:

How sure are you?

Correct: B. When a bond's price is above par ($1,050 > $1,000), the bond is trading at a premium. A premium bond always has a YTM less than the coupon rate. Intuition: if you pay more than face value, your effective return (YTM) must be less than the stated coupon rate, because you also suffer a capital loss as the bond is pulled to par at maturity. Premium bond: coupon rate > YTM. Discount bond: coupon rate < YTM. Par bond: coupon rate = YTM.
A. Choosing A might seem logical if you assume the bond price directly reflects the coupon rate, but this ignores the premium pricing effect; when a bond trades above par, its YTM must be lower than the coupon rate, not equal to it.
C. Choosing C might tempt you if you think the bond's maturity is crucial for calculating YTM, but the relationship between price and YTM is clear here: a premium bond always has a YTM less than the coupon rate, regardless of maturity.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 4Exam level

A bond's full price is $1,020 and the accrued interest is $15. The bond's flat price (also called the clean price) is closest to:

How sure are you?

Correct: A. Full price (dirty price) = Flat price (clean price) + Accrued interest. Therefore: Flat price = Full price - Accrued interest = $1,020 - $15 = $1,005. The flat price is what is quoted in the market (Bloomberg, financial press). The full price is what the buyer actually pays. Accrued interest compensates the seller for the coupon earned but not yet paid since the last coupon date.
B. Choosing $1,035 might tempt you if you mistakenly add the accrued interest to the full price, but this violates the rule that the flat price is derived by subtracting accrued interest from the full price, not adding it.
C. Choosing $15 might tempt you if you confuse the flat price with the accrued interest, but the flat price represents the bond's price excluding accrued interest, not the interest itself, making $15 far too low.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 5Exam level

A semi-annual coupon bond has a face value of $1,000, a coupon rate of 8%, and 10 years to maturity. If the bond's current price is $950, the bond is most likely described as:

How sure are you?

Correct: A. Since price ($950) < face value ($1,000), this is a discount bond. For a discount bond, the YTM must be greater than the coupon rate (8%). The investor buys below par and receives par at maturity, creating a capital gain that supplements the coupon return. The total return (YTM) therefore exceeds the coupon rate. The YTM calculation (using calculator: N=20, PV=-950, PMT=40, FV=1000) confirms YTM per period > 4%, annualized > 8%.
B. Choosing B might be tempting if you assume the bond price equals its face value, but a par bond would have a price of $1,000 with a YTM equal to the coupon rate, which is not the case here since the bond is priced at a discount, indicating a YTM greater than 8%.
C. You might be thinking that a lower YTM could still result in a discount bond, but this violates the principle that for a bond priced below par, the YTM must be higher than the coupon rate to compensate for the lower purchase price and ensure a return above the coupon payments alone.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 6Exam level

An investor purchases a bond between coupon payment dates. The bond has a semi-annual coupon of $40, and 45 days have passed since the last coupon date out of a 180-day coupon period (30/360 day count). The accrued interest per $1,000 face value is closest to:

How sure are you?

Correct: A. Accrued interest = Coupon x (Days since last coupon / Days in coupon period). Using 30/360 convention: AI = $40 x (45/180) = $40 x 0.25 = $10.00. The buyer pays the seller $10 of accrued interest as part of the full (dirty) price, compensating the seller for holding the bond for 45 of the 180-day coupon period. The flat (clean) price does not include this $10.
B. Choosing $40.00 might seem logical if you mistakenly believe it represents the full coupon payment due at the next payment date, but this ignores the partial period for which interest has accrued; the correct calculation only accounts for the 45 days since the last coupon date, not the full coupon amount.
C. Choosing $5.00 might tempt you if you mistakenly halve the days since the last coupon payment, using 22.5 instead of 45, which violates the correct application of the 30/360 day count convention.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 7Exam level

Which of the following bonds will most likely have the greatest price sensitivity to a given change in yield?

How sure are you?

Correct: B. Price sensitivity to yield changes (duration) increases with longer maturity and lower coupon rate. Longer maturity = more cash flows far in the future, heavily discounted, making price more sensitive to yield changes. Lower coupon = less cash flow received early, so more weight in the distant par payment. Bond C has the longest maturity AND the lowest coupon, giving it the highest duration and greatest price sensitivity. This is a concept question about the determinants of duration. No calculation needed.
A. You might be tempted by choice A because a higher coupon rate could suggest greater stability, but remember that higher coupon rates actually reduce price sensitivity due to more frequent cash flows, contrasting with the correct answer which has a lower coupon rate and thus higher price sensitivity for the same maturity.
C. You might be tempted by choice C because a lower coupon rate generally increases price sensitivity, but choice C has a shorter maturity than choice B, reducing its overall price sensitivity despite the low coupon rate.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 8Exam level

A 3-year, 6% annual coupon bond with face value $1,000 currently yields 6%. If the yield immediately rises to 7%, the bond's new price is closest to:

How sure are you?

Correct: A. At YTM = 6%, the bond trades at par ($1,000) since coupon rate = YTM. When YTM rises to 7%: PV of coupons = $60 x [1-(1.07)^-3]/0.07 = $60 x 2.6243 = $157.46. PV of par = $1,000/(1.07)^3 = $816.30. New price = $157.46 + $816.30 = $973.76. The bond falls from $1,000 to $973.76. A price drop of $26.24 for a 1% yield increase. This illustrates the inverse price-yield relationship.
B. Choice B ($1,000) is the trap. Candidates who remember the bond was at par don't recalculate after the yield change. The yield changed; the price MUST change. Choice C ($1,026.24) is what happens if yield falls to 5% (not rises to 7%). Confusing the direction.
C. Choosing $947.51 might tempt you if you incorrectly assume a larger price drop for a 1% increase in yield, but the inverse relationship between bond price and yield does not imply a linear drop, and the magnitude of price change is smaller at higher yields, making this choice too low compared to the actual price drop.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 9Exam level

As a bond approaches maturity, its price will approach par value, regardless of whether it is a premium or discount bond. This phenomenon is most likely called:

How sure are you?

Correct: A. Pull to par (also called price convergence) describes the phenomenon whereby a bond's price converges toward its face value (par) as it approaches maturity, assuming no change in credit quality or required yield. A premium bond's price declines toward par (capital loss). A discount bond's price rises toward par (capital gain). At maturity, the bond's price must equal par because the only remaining cash flow is the face value redemption. This is a definitional question but also appears in YTM calculation context.
B. Choice A (duration drift) sounds plausible as a technical term. Choice C (yield compression) also sounds reasonable. These are distractor terms. Pull to par is the official CFA curriculum term and the correct answer.
C. You might be tempted by convexity convergence because it deals with how bond prices change with yield, but convexity describes the curvature in the price-yield relationship, not the linear approach to par value as maturity nears, which is what pull to par describes.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 10Exam level

A bond with a face value of $1,000 has 5 years to maturity and pays semi-annual coupons of $35. The bond's YTM is 8% (annual). Using a financial calculator, which inputs are correct? The value is closest to:

How sure are you?

Correct: A. For a semi-annual coupon bond: (1) N = years x 2 = 5 x 2 = 10 periods. (2) I/Y = annual YTM / 2 = 8% / 2 = 4% per period. (3) PMT = semi-annual coupon = $35 (already stated as semi-annual. Do not double it). (4) FV = $1,000. The coupon rate is $35 x 2 / $1,000 = 7% annually, and YTM is 8%, so the bond should trade at a discount. A sanity check on the output.
B. You might be tempted to choose B because it keeps the number of periods correct at 10, but it incorrectly uses the annual YTM of 8% instead of the semi-annual rate of 4%, and doubles the coupon payment to $70, which violates the semi-annual payment rule.
C. You might be tempted to choose C because it correctly sets the number of years to 5 and the semi-annual YTM to 4%, but it incorrectly doubles the semi-annual coupon payment to 70, which violates the rule that the PMT should match the actual semi-annual payment of 35.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 11Exam level

A bond's YTM equals its coupon rate. Which of the following must most likely be true?

How sure are you?

Correct: B. When YTM = coupon rate, the discount rate exactly equals the return on each coupon payment, so the present value of all cash flows exactly equals the face value. Price = Par. This is the definition of a par bond. Premium: coupon rate > YTM. Discount: coupon rate < YTM. Par: coupon rate = YTM.
A. Choosing A might tempt you if you confuse a lower price with a higher YTM, but remember, a bond trading below par has a YTM higher than its coupon rate, not equal to it as required here.
C. You might be tempted to think that if YTM equals the coupon rate, the bond behaves simply, leading to duration equaling maturity, but duration equals maturity only for zero-coupon bonds, not for coupon-paying bonds trading at par.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 12Above the exam

A 3-year, 6% annual-pay coupon bond with $1,000 face value is priced using the following spot rates: 1-year spot = 4%, 2-year spot = 5%, 3-year spot = 6%. Combining the arbitrage-free (spot-rate) valuation approach with the single discount rate a flat-YTM approach would use instead, the bond's price using the CORRECT spot-rate approach is most likely to be:

How sure are you?

Correct: B. Under the arbitrage-free approach, each cash flow is discounted at the spot rate matching ITS OWN maturity: the year-1 coupon at 4%, the year-2 coupon at 5%, and the year-3 coupon-plus-principal at 6%. Since the year-1 and year-2 spot rates (4% and 5%) are both BELOW the flat 6% YTM a single-rate approach would use, those earlier cash flows are discounted less heavily under the spot-rate approach, making their present value HIGHER than under a flat 6% rate; the year-3 cash flow is discounted the same either way. The overall spot-rate price is therefore higher than the flat-YTM price.
A. The 3-year spot rate matching the coupon rate does not make the two approaches equivalent; the EARLIER cash flows (years 1 and 2) are discounted at their OWN, different (lower) spot rates under the correct approach, which a flat 6% YTM calculation would not capture.
C. Using the term structure of spot rates does not mechanically always reduce price relative to a flat rate; the direction of the difference depends on whether the yield curve is upward- or downward-sloping relative to the flat rate used, and here the upward-sloping curve (rates below 6% at shorter maturities) pushes the spot-rate price HIGHER, not lower.

Unit: fixed-income-bond-valuation-prices-and-yields

Question 13Above the exam

A bond is priced at a premium to par. An analyst incorrectly states that 'as this bond approaches maturity, its price will rise steadily toward the premium price it is trading at today.' Combining the concept of the constant-yield price trajectory with how a premium bond's price actually behaves over time (assuming yields do not change), this statement is most likely:

How sure are you?

Correct: B. Assuming the market discount rate (yield) does not change, a bond's price moves along its constant-yield price trajectory toward par value as it approaches maturity. A PREMIUM bond's price starts above par and DECLINES over time toward par (amortizing the premium); a DISCOUNT bond's price starts below par and RISES over time toward par (accreting the discount). The analyst's claim that a premium bond's price will rise is backwards.
A. Bond prices do not always rise as maturity approaches; the direction depends on whether the bond trades at a premium or a discount to par. Only discount bonds see their price rise toward par over time (holding yield constant); premium bonds see their price fall toward par.
C. The constant-yield price trajectory applies to option-free bonds generally, and the premium/discount amortization logic described is a basic bond math result, not something limited to bonds with embedded options; embedded options introduce additional complications but are not required for this basic premium-versus-discount price trajectory to hold.

Unit: fixed-income-bond-valuation-prices-and-yields