Yield and Yield Spread Measures for Fixed-Rate Bonds

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

Fixed IncomeYield and Yield Spread Measures for Fixed-Rate Bonds
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The lesson

Runtime 9 minutes 45 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 calculate and compare current yield, yield to maturity and their fixed ordering against coupon rate, convert a nominal yield to an effective annual yield, calculate G-spread, I-spread and Z-spread, and calculate option-adjusted spread in the correct direction for a callable versus a putable bond.

Yield to maturity is the single discount rate that makes the present value of every one of a bond's future cash flows equal its current market price, effectively the bond's own internal rate of return. It is not automatically the return an investor actually earns. That equality only holds if every coupon received along the way is reinvested at that exact same YTM rate, and because real reinvestment rates move with the market over time, realized return almost never matches the original YTM except for a zero-coupon bond, which has no coupons to reinvest at all. Current yield, a simpler figure, is just the annual coupon divided by the current price, and it ignores the capital gain or loss a bond will realize by maturity entirely.

Coupon rate, current yield and YTM line up in a fixed order that depends only on whether a bond trades at a premium or a discount. For a discount bond, the order climbs: coupon rate is less than current yield, which is less than YTM, since YTM alone captures the capital gain the price earns as it is pulled up to par. For a premium bond the order falls in exactly the reverse direction: coupon rate exceeds current yield, which exceeds YTM, since YTM alone captures the capital loss as price falls back to par. For a par bond, all three are identical. A stated yield compounded more than once a year also understates its own true annual return: converting it to an effective annual yield, (1 + nominal yield / m)^m - 1, always produces a number above the nominal figure whenever compounding happens more than once a year.

Three spread measures compare a bond's yield to a benchmark, and they differ in how much of the yield curve's actual shape they respect. G-spread is simply a bond's YTM minus the yield of a government bond of matching maturity, a single number against a single benchmark. I-spread does the same thing against the swap curve instead of a government bond. Both use one blended benchmark rate and lose accuracy whenever the yield curve is steep. Z-spread is more careful: it adds one constant spread to every point on the government zero-coupon spot curve at once, so that discounting the bond's actual cash flows at spot-plus-Z reproduces its market price exactly, respecting the curve's full shape rather than collapsing it into one number.

Option-adjusted spread strips the value of an embedded option back out of the Z-spread, and the direction of that adjustment depends entirely on which side holds the option. For a callable bond, the issuer holds the option, so part of the Z-spread is really compensating the investor for having effectively sold that option away: OAS = Z-spread minus the option's cost, which makes OAS the smaller of the two figures. For a putable bond, the investor holds the option instead, so less spread compensation is actually needed: OAS = Z-spread plus the value the investor gains from holding the put, making OAS the larger figure this time. An option-free bond needs no adjustment at all, so its OAS equals its Z-spread exactly.

Worked in full

A callable corporate bond has a Z-spread of 210 basis points over the government spot curve. Its embedded call option is estimated to be worth 40 basis points to the issuer. What is the bond's option-adjusted spread, and what does it represent? Because the issuer holds the call option on this bond, OAS = Z-spread - option cost = 210 - 40 = 170 basis points. The 170 basis points is the pure credit and liquidity compensation an investor is actually earning, once the 40 basis points of compensation for having effectively sold the issuer a call option is stripped back out.

The same problem, one step removed

Same bond: Z-spread of 210 basis points, callable, with the call option valued at 40 basis points. Confirm which side holds the option, then apply the matching direction to find OAS yourself.

The trap

Subtracting a cost from a number that is supposed to describe compensation feels backwards, which is exactly why candidates add the option cost instead of subtracting it on a callable bond; OAS is always smaller than Z-spread for a callable bond and always larger for a putable one, never the reverse.

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.

A stated annual yield compounded more than once a year understates the true effective annual yield

Converting a nominal yield quoted on an m-times-per-year compounding basis to an effective annual yield requires (1 + nominal yield / m)^m - 1, which is always greater than the nominal yield itself whenever m exceeds one; a semi-annual bond-basis yield of 7.2% therefore corresponds to an effective annual yield above 7.2%, not equal to it.

Current yield, coupon rate, and yield to maturity line up in a fixed, predictable order that depends only on premium or discount status

For a discount bond, coupon rate is less than current yield, which is less than yield to maturity, since YTM alone captures the capital gain earned as price rises to par. For a premium bond the order reverses entirely, coupon rate exceeds current yield, which exceeds yield to maturity, since YTM alone captures the capital loss as price falls to par. For a par bond all three are equal. This ordering can be read directly once premium or discount status is known, without any additional calculation.

Yield to maturity assumes every coupon is reinvested at the YTM itself, an assumption that rarely holds in practice

YTM is the internal rate of return of the bond's cash flows at its current price, which mathematically requires that every coupon received along the way be reinvested at that same YTM rate; because actual reinvestment rates move with the market over time, the return an investor actually realizes almost never equals the original YTM, except for a zero-coupon bond, which has no coupons to reinvest and therefore no reinvestment risk.

G-spread and I-spread each use a single benchmark yield; Z-spread respects the full shape of the yield curve

G-spread is simply a bond's yield to maturity minus the yield of a government bond of matching maturity; I-spread is the bond's yield minus the swap rate of matching maturity. Both use one single benchmark number and become less accurate when the yield curve is steep. Z-spread instead adds one constant spread to every point on the government zero-coupon spot curve simultaneously, so that the present value of the bond's actual cash flow schedule, each cash flow discounted at its own appropriate maturity's spot rate plus that constant, equals the bond's price; this makes Z-spread the more accurate measure whenever the curve is not flat.

Option-adjusted spread strips the embedded-option value out of the Z-spread, and the direction of the adjustment depends on who holds the option

For a callable bond, the issuer holds the option, so part of the Z-spread compensates investors for having effectively sold that option: OAS = Z-spread minus the option's cost, making OAS the smaller of the two. For a putable bond, the investor holds the option, so less spread compensation is needed: OAS = Z-spread plus the value the investor gains from holding the put, making OAS the larger of the two. For an option-free bond, there is no option to adjust for, so OAS equals Z-spread exactly. OAS is the correct measure for comparing bonds that carry different embedded-option structures on an apples-to-apples basis; Z-spread and G-spread are not, since they still embed the option's value.

The trick

Discount bond: yields climb from coupon to YTM. Premium bond: yields fall from coupon to YTM

Discount: coupon < current yield < YTM. Premium: coupon > current yield > YTM. The direction flips entirely between the two cases; memorize both, not just one and its reverse.

Callable: OAS is smaller than Z-spread. Putable: OAS is larger than Z-spread

The issuer's call option costs the investor value, so it is subtracted out. The investor's put option adds value to the investor, so it is added back. Option-free bonds need no adjustment at all: OAS = Z-spread.

G for Government, I for Interest-rate swap, Z for Zero-coupon curve

G-spread benchmarks against a government bond's yield; I-spread benchmarks against the swap curve; Z-spread benchmarks against the full zero-coupon government curve, one point per cash flow date.

The method

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

  1. For a nominal-to-effective yield conversion, apply (1 + nominal/m)^m - 1 using the stated compounding frequency m.
  2. For a yield-ordering question, first determine premium or discount status from price versus par, then apply the corresponding fixed ordering of coupon rate, current yield, and YTM.
  3. For a spread question, identify which benchmark is used, single government yield (G-spread), swap rate (I-spread), or the full zero-coupon curve (Z-spread), before comparing spreads across bonds.
  4. For an OAS question, first identify whether the bond is callable, putable, or option-free, then apply the matching direction: subtract the option cost for callable, add it for putable, no adjustment for option-free.
  5. [BA II Plus: for YTM, enter N, PV (as a negative, the price paid), PMT (periodic coupon), FV (par), then CPT I/Y; multiply by the number of periods per year for the nominal annual yield]

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 bond has a face value of $1,000, a coupon rate of 8% paid semi-annually, 5 years to maturity, and a current price of $920. Which of the following best describes the relationship between the bond's coupon rate, current yield, and yield to maturity?

Answer A. The bond trades at a discount (price $920 < par $1,000). For discount bonds: coupon rate < current yield < YTM. Current yield = $80/$920 = 8.70%. YTM must be higher than current yield because YTM also captures the capital gain from $920 to $1,000 at maturity. The ordering coupon rate < current yield < YTM always holds for discount bonds.

Your turn, setup given

An investor buys a bond with a 6% annual coupon at a price of $108 per $100 face value. The bond matures in 4 years. The bond's current yield is closest to:

For a nominal-to-effective yield conversion, apply (1 + nominal/m)^m - 1 using the stated compounding frequency m.

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 callable bond has a Z-spread of 180 basis points. If the estimated value of the embedded call option is 25 basis points, the bond's option-adjusted spread (OAS) is closest to:

How sure are you?

Correct: A. For a callable bond: OAS = Z-spread - Option cost = 180 - 25 = 155 bps. The call option benefits the issuer (hurts the investor), so the option cost is subtracted from the Z-spread to isolate the pure credit and liquidity spread. OAS is always less than Z-spread for callable bonds because Z-spread includes compensation for both credit risk AND the option the investor has written.
B. Choosing 180 bps might seem logical if you think the Z-spread and OAS are the same, but this ignores the need to subtract the value of the call option, which you have written, from the Z-spread to get the OAS for a callable bond.
C. Choosing 205 bps might tempt you if you mistakenly add the option cost to the Z-spread, but the OAS calculation for a callable bond actually subtracts the option cost to account for the embedded call option's benefit to the issuer, not add it.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 2Exam level

A bond has a YTM of 7.2% on a semi-annual bond basis (BEY). The bond's effective annual yield (EAY) is closest to:

How sure are you?

Correct: A. EAY = (1 + BEY/2)^2 - 1 = (1 + 0.072/2)^2 - 1 = (1.036)^2 - 1 = 1.07330 - 1 = 7.33%. BEY is the nominal yield (stated as twice the semi-annual rate). EAY accounts for compounding and is always HIGHER than BEY for the same bond. The difference arises from the half-year reinvestment of the first coupon before the second coupon is paid.
B. You might use EAY = BEY/2 x 2 (no compounding), which gives 7.20%. The distractor at answer B. Always apply (1+r/m)^m - 1 to convert from nominal to effective.
C. You might be tempted to choose 7.41% if you incorrectly applied annual compounding instead of semi-annual, leading to an overestimation of the effective annual yield; remember, the effective annual yield (EAY) correctly accounts for semi-annual compounding, resulting in a lower figure than annual compounding would suggest.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 3Exam level

Which of the following yield spread measures is MOST appropriate for comparing bonds with embedded options to option-free bonds?

How sure are you?

Correct: C. OAS removes the value of the embedded option from the spread, leaving only the spread attributable to credit risk and liquidity. G-spread and Z-spread include compensation for the embedded option, making cross-security comparisons misleading. OAS allows apples-to-apples comparison of callable bonds versus option-free bonds of the same issuer.
A. You might choose Z-spread because it sounds more 'sophisticated' than G-spread. Z-spread is better than G-spread for non-flat yield curves, but neither removes option value. Only OAS does.
B. You might choose Z-spread because it sounds more 'sophisticated' than G-spread. Z-spread is better than G-spread for non-flat yield curves, but neither removes option value. Only OAS does.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 4Exam level

A 10-year government bond yields 3.5%. A 10-year corporate bond of equal maturity yields 5.2%. The 10-year swap rate is 3.8%. What is the corporate bond's G-spread and I-spread, respectively, most likely?

How sure are you?

Correct: A. G-spread = Corporate yield - Government benchmark yield = 5.2% - 3.5% = 1.70% = 170 bps. I-spread = Corporate yield - Swap rate = 5.2% - 3.8% = 1.40% = 140 bps. G-spread uses government bond yield as the risk-free benchmark. I-spread uses the swap rate (LIBOR/SOFR-based), which is slightly above government yields, producing a lower spread for the same bond.
B. You might be tempted to choose B if you mistakenly subtracted the swap rate from the government bond yield instead of the corporate bond yield for the G-spread, confusing the roles of the swap rate and government bond yield in calculating these spreads.
C. You might be tempted to think the G-spread and I-spread would be the same if you confuse the government bond yield and the swap rate as equivalent benchmarks, but the I-spread uses the swap rate of 3.8%, resulting in a lower spread of 140 bps compared to the G-spread using the government bond yield of 3.5%.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 5Exam level

YTM assumes which of the following about a bond's cash flows, most likely?

How sure are you?

Correct: B. YTM is calculated assuming all coupon payments are reinvested at a rate equal to the YTM itself. This is the reinvestment assumption. In practice, reinvestment rates change over time (rates may be higher or lower than YTM when coupons are received), so the realized return almost never equals the YTM. The only bond for which YTM = realized return without reinvestment risk is a zero-coupon bond (no coupons to reinvest).
A. You might be tempted by the idea of reinvesting at the risk-free rate because it seems conservative and safe, but YTM specifically assumes coupons are reinvested at the YTM rate itself, not at the risk-free rate, to maintain consistency in the calculation.
C. Choosing C might seem logical if you think about immediate consumption, but YTM specifically assumes coupons are reinvested, not spent, directly contrasting with the assumption that coupons are reinvested at the YTM rate.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 6Exam level

A bond with a 5% semi-annual coupon, $1,000 face value, and 10 years to maturity is priced at $1,050. Using a financial calculator, which of the following inputs correctly sets up the YTM calculation? The value is closest to:

How sure are you?

Correct: B. Semi-annual coupon bond: N = 10 years x 2 = 20 periods; PMT = 5% x $1,000 / 2 = $25 per period; PV = -$1,050 (negative because cash outflow for buyer); FV = $1,000. Solve for I/Y, then multiply by 2 for the BEY. Option A uses annual periods (N=10) and annual coupon ($50). Incorrect for semi-annual. Option C uses correct N=20 but annual coupon $50. The most common mistake.
A. Option A uses annual periods (N=10) and annual coupon ($50). Incorrect for semi-annual.
C. Option C uses correct N=20 but annual coupon $50. The most common mistake.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 7Exam level

A bond is quoted with a bank discount yield of 4.8% on a 90-day Treasury bill with a face value of $1,000. The bill's price is closest to:

How sure are you?

Correct: A. Bank discount yield (BDY) uses face value as the base and a 360-day year. Price = Face x [1 - BDY x (days/360)] = $1,000 x [1 - 0.048 x (90/360)] = $1,000 x [1 - 0.012] = $1,000 x 0.988 = $988.00. True holding period yield or EAY would be higher.
B. You might be tempted by $987.69 if you mistakenly applied the money market yield formula instead of the bank discount yield formula, leading to a calculation that does not account for the face value base and 360-day year requirement of BDY.
C. You might be tempted by choice C if you incorrectly applied the formula for the holding period return instead of the bank discount yield, leading to an overly discounted price that does not reflect the BDY calculation based on a 360-day year.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 8Exam level

For a premium bond, which ordering of yield measures is most likely correct?

How sure are you?

Correct: B. For a bond trading at a premium (price > par): coupon rate > current yield > YTM. Coupon rate is highest because it is based on par (a lower base than market price). Current yield = coupon/price, and since price > par, current yield < coupon rate. YTM is lowest because it incorporates the capital loss from paying premium price to receive only par at maturity. This reduces the total return below the current yield.
A. This reverses the entire ordering for a premium bond. Paying a market price above par means the bond's expected total return, the yield to maturity, falls below both the coupon rate and the current yield, since the investor loses money on the capital side at maturity. The correct order runs coupon rate first, then current yield, then yield to maturity, the opposite direction from what this choice states.
C. This puts current yield ahead of coupon rate, but for a premium bond current yield is the coupon divided by a price that is above par, which makes current yield lower than the coupon rate, not higher. The correct order is coupon rate first, then current yield, then yield to maturity at the bottom, reflecting the capital loss built into paying more than par for a bond that repays only par at maturity.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 9Harder

Which of the following most accurately describes the difference between a Z-spread and a G-spread, most likely?

How sure are you?

Correct: A. Z-spread (zero-volatility spread) adds a constant spread Z to each relevant spot rate on the benchmark zero-coupon yield curve so that the present value of cash flows equals the bond's price. G-spread simply subtracts the government benchmark bond's yield from the bond's YTM. A single-number subtraction using par yields. Z-spread is more accurate for non-flat yield curves because it respects the timing of each cash flow. G-spread uses a flat-curve approximation. Neither adjusts for option value. That is OAS.
B. You might be thinking that Z-spread accounts for option-adjusted values, but both Z-spread and G-spread do not adjust for embedded option values; that adjustment is actually done by the option-adjusted spread (OAS), not the Z-spread or G-spread.
C. You might think the G-spread uses spot rates for accuracy, but the G-spread actually relies on a single benchmark yield, making it less precise for non-flat yield curves compared to the Z-spread, which adds a constant spread to each spot rate.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 10Exam level

A floating-rate note (FRN) resets its coupon every 6 months based on 6-month SOFR + 120 bps. If current 6-month SOFR is 4.5%, the current coupon rate on the FRN is closest to:

How sure are you?

Correct: A. FRN coupon = Reference rate + Quoted margin = 4.5% + 1.20% = 5.70%. For FRNs, the key yield measure is the quoted margin (QM) set at issuance and the discount margin (DM), which is the margin required by the market currently. If the bond trades at par, QM = DM. If credit quality deteriorates, DM > QM and the bond trades below par.
B. You might be tempted to choose 4.50% because it reflects the current 6-month SOFR rate, but this overlooks the additional quoted margin of 120 bps that must be added to the reference rate to determine the FRN coupon rate.
C. Choosing 1.20% might seem right if you mistakenly thought the quoted margin alone was the coupon rate, but the coupon rate for an FRN is the sum of the reference rate and the quoted margin, not just the margin itself.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 11Above the exam

Bond A has a G-spread of 150 bps over the interpolated government yield curve. Bond B has a Z-spread of 140 bps and an OAS of 90 bps, reflecting a meaningful embedded call option. Combining the definitions of G-spread, Z-spread, and OAS, an analyst comparing the credit and liquidity risk compensation embedded in the two bonds should most likely:

How sure are you?

Correct: B. The Z-spread includes compensation for ALL of a bond's risks, including any embedded option; the OAS specifically REMOVES the portion of the spread attributable to the option, leaving a cleaner measure of credit and liquidity compensation. When comparing an option-free bond's spread (like Bond A's G-spread, a reasonable option-free benchmark-relative spread) to a callable bond's spread, the OAS is the appropriate, apples-to-apples comparison, not the Z-spread, which still has the option's effect baked in.
A. G-spread and Z-spread are conceptually similar but not always numerically identical (G-spread uses a linearly interpolated government yield, Z-spread uses the full spot curve), and neither is the right comparison for a bond with an embedded option; comparing a Z-spread (with the option's effect still included) to an option-free spread mixes in the option's distorting effect on Bond B's numbers.
C. The comparison in this choice garbles the actual figures and the underlying logic; the correct comparison uses OAS (90 bps) for the option-embedded bond specifically because the raw Z-spread or a simple 'bigger number wins' comparison would be distorted by the value of Bond B's embedded call option, not a valid apples-to-apples risk comparison.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds

Question 12Above the exam

A floating-rate note has a quoted margin of 80 bps over the reference rate and a discount margin (DM), computed at the current price, of 120 bps. Combining the meaning of quoted margin with the meaning of discount margin, the FRN is most likely:

How sure are you?

Correct: B. The quoted margin is the FIXED spread over the reference rate set at issuance; the discount margin is the spread that equates the FRN's price (using its current market price) to its cash flows, reflecting the market's CURRENT required margin. When the market's required margin (DM, 120 bps) is higher than the note's fixed quoted margin (80 bps), it means the market now demands more compensation than the note pays, which can only be achieved by the note trading BELOW par (a discount), so that the extra price appreciation-to-par over time makes up the difference.
A. A discount margin higher than the quoted margin indicates the market requires MORE compensation than the note's fixed spread provides, which drives the price BELOW par, a discount, not a premium; a premium would instead occur if the discount margin were BELOW the quoted margin.
C. Quoted margin and discount margin are equal only when the FRN happens to be trading exactly at par; they are not equal by definition in general. The whole point of computing a separate discount margin is to reflect how the market's required spread may have diverged from the note's fixed quoted margin since issuance.

Unit: yield-and-yield-spread-measures-for-fixed-rate-bonds