Practice: Fixed-Income Bond Valuation: Prices and Yields
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Fixed IncomeFixed-Income Bond Valuation: Prices and Yields
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Question 1Exam level
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:
How sure are you?
Correct: 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.
B. $1,000.00 is the face value with no adjustment for yield. Since YTM (8%) exceeds the coupon (6%), the bond must price below par, not at par.
C. $917.35 is below the correct discounted value. PV of coupons ($154.63) plus PV of par ($793.83) totals $948.46, not $917.35.
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:
How sure are you?
Correct: A. Semi-annual coupon = (0.05/2) x $1,000 = $25. Semi-annual YTM = 4%/2 = 2%. N = 4 x 2 = 8 periods. PV of coupons = $25 x [1-(1.02)^-8]/0.02 = $25 x 7.3255 = $183.14. PV of par = $1,000/(1.02)^8 = $853.49. Price = $183.14 + $853.49 = $1,036.63 (approximately $1,036.30 depending on rounding). Since coupon rate (5%) > YTM (4%), bond trades at a premium.
B. Choosing $963.70 might tempt you if you mistakenly calculated the price using an annual instead of a semi-annual YTM, leading to an incorrect discounting of cash flows and a price that reflects a discount rather than the premium the bond should trade at given its higher coupon rate compared to its YTM.
C. You might be tempted to choose $1,027.15 if you incorrectly calculated the present value using an annual instead of a semi-annual YTM, which violates the bond pricing rule requiring consistent periodicity between coupon payments and yield to maturity.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.