Most CFA Level 1 candidates lose marks in Fixed Income not on the hard math, but on the small distinctions: YTM versus current yield, Macaulay versus modified duration, and the moment convexity starts to matter. These CFA fixed income notes walk through bond valuation, yield measures, duration and convexity the way the exam tests them, with the formulas you must be able to apply and the traps to avoid. Work through it once, then use the one-page revision box at the end before your mock.
Fixed Income rewards clean calculation habits more than memorization. The whole topic is built from four ideas that repeat everywhere: price is present value, yield is the discount rate, duration is price sensitivity, and convexity is the correction for big moves. Everything below hangs on those four lines. If any of them feel shaky, slow down on that section before moving on. And if you have not read it yet, our 15-minute CFA Ethics notes pair well with this one: Ethics and Fixed Income are the two Level 1 topics where careful reading beats raw speed.
In this note
- How these CFA fixed income notes price a bond
- Bond pricing in action: a worked example
- Yield measures these CFA fixed income notes keep straight
- Spot rates and forward rates in CFA fixed income notes
- Duration: the risk number CFA fixed income notes revolve around
- Effective duration: when cash flows can change
- Convexity: why duration alone misprices big moves
- Exam traps: the distinctions that cost marks
- One-page revision box
How these CFA fixed income notes price a bond
A bond is a bundle of future cash flows: periodic coupons plus the par value at maturity. Its price today is the present value of those cash flows, discounted at one rate. That rate is the yield to maturity, and it is the single most important input in these CFA fixed income notes.
Price = sum of [Coupon / (1 + YTM)t] + Par / (1 + YTM)N
Three pricing rules follow directly and show up as exam one-liners. When the coupon rate equals the YTM, the bond trades at par. If the coupon rate is above the YTM, it trades at a premium. When the coupon rate is below the YTM, it trades at a discount. The price of a premium bond falls toward par as maturity approaches; the price of a discount bond rises toward par. This pull-to-par happens regardless of what market yields do in between, because at maturity the bondholder receives exactly par.
The price-yield relationship is inverse and convex. Yields up, price down, and the curve bends: a given fall in yield raises the price by more than an equal rise in yield lowers it. That asymmetry is the whole reason convexity exists, and we will return to it below. For the official curriculum framing of what a fixed-income instrument is, the CFA Institute refresher reading on fixed-income instrument features is the authoritative starting point.
Bond pricing in action: a worked example
Take a 5-year bond with a 6% annual coupon paid semiannually, par value 1,000, and a YTM of 5% (quoted with semiannual compounding). Pricing it step by step, the way every worked example in these CFA fixed income notes does it: set up the periods, discount, then check the answer against the premium/par/discount rule.
- Semiannual coupon = 1,000 x 6% / 2 = 30
- Number of periods = 5 x 2 = 10; per-period discount rate = 5% / 2 = 2.5%
- Present value of coupons = 30 x [1 – (1.025)-10] / 0.025 = 30 x 8.7521 = 262.56
- Present value of par = 1,000 / (1.025)10 = 781.20
- Price = 262.56 + 781.20 = 1,043.76
The coupon rate (6%) is above the YTM (5%), so the bond trades at a premium, exactly as the pricing rule predicts. Over the next five years, that 43.76 premium will amortize away and the price will pull to par at maturity. The current yield here is 60 / 1,043.76 = 5.75%, which sits between the YTM (5%) and the coupon rate (6%). That ordering, coupon > current yield > YTM for a premium bond, is a favorite exam check, and it reverses for discount bonds.
Yield measures these CFA fixed income notes keep straight
The exam loves asking which yield is being described, so keep these five straight.
Current yield is the simplest: annual coupon divided by the current price. It ignores the capital gain or loss at maturity, so for a premium bond it overstates your true return and for a discount bond it understates it.
Yield to maturity is the internal rate of return of the bond’s cash flows: the single discount rate that equates price to the present value of all future payments. It silently assumes every coupon is reinvested at the YTM itself. That assumption is the source of reinvestment risk: if rates fall after you buy, your coupons compound at less than the YTM and your realized return drops short. The risk is worst for high-coupon, long-maturity bonds, because more of the total return depends on reinvested coupons.
Yield to call matters for callable bonds. Compute the yield assuming the bond is redeemed at the first call date at the call price, and repeat for each call date. Yield to worst is the lowest of the YTM and all the yields to call. For a premium bond trading above its call price, yield to call is typically below YTM, which is exactly why issuers call premium bonds when rates fall.
| Yield measure | What it divides or assumes | Ignores |
|---|---|---|
| Current yield | Annual coupon / current price | Gain or loss at maturity |
| Yield to maturity | IRR of all cash flows | Reinvestment at rates other than YTM |
| Yield to call | IRR assuming redemption at a call date | Cash flows after the call |
| Yield to worst | Minimum of YTM and all yields to call | Nothing extra; it is the conservative pick |
| Spot / forward rates | Rate for each specific maturity | Single-rate averaging |
Spot rates and forward rates in CFA fixed income notes
Spot rates price each cash flow at the rate for its own maturity, while the YTM is roughly the average of the relevant spot rates. Forward rates are the implied future rates locked in by today’s spot curve. The no-arbitrage link the exam tests is:
(1 + S2)2 = (1 + S1) x (1 + f1,2)
where S1 and S2 are the one- and two-year spot rates and f1,2 is the one-year forward rate starting in one year. If you can rearrange that equation, you can answer every forward-rate question on the paper.
Worked example: the one-year spot rate is 3% and the two-year spot rate is 4.5%. Then (1.045)2 = 1.092025, and dividing by 1.03 gives 1.0602. So the implied one-year rate one year from now is about 6.02%. Notice the forward rate sits above both spot rates because the spot curve is upward sloping; an upward-sloping curve always implies forward rates above the spot rates, and a downward-sloping curve implies the reverse.
Duration: the risk number CFA fixed income notes revolve around
Duration is not really about time. It is a measure of interest rate sensitivity, expressed in years, and it answers the question the exam keeps asking: if yields move, how much does the price move?
Macaulay duration is the weighted average time until each cash flow arrives, with weights equal to each cash flow’s present value as a share of the price. Modified duration converts that into a sensitivity measure:
Modified duration = Macaulay duration / (1 + YTM / m), where m is the number of coupon payments per year. The workhorse approximation for the whole topic:
Approximate percentage change in price = -Modified duration x change in yield
A bond with modified duration of 6 loses about 6% when yields rise 100 basis points, and gains about 6% when yields fall 100 basis points. On our 1,043.76 bond, a 100-basis-point rise would shave roughly 62.6 off the price, leaving it near 981. Duration rises with longer maturity, falls with higher coupons, and falls with higher yields. A zero-coupon bond’s Macaulay duration equals its maturity, which is the maximum duration for that maturity.
Effective duration: when cash flows can change
The duration approximation assumes the cash flows do not change when yields move, which fails for bonds with embedded options. For callable and putable bonds, use effective duration:
Effective duration = (P– – P+) / (2 x P0 x change in yield)
where P– and P+ are the prices after a small fall and rise in yield. Because it is computed from actual repriced values, effective duration captures the option’s effect. A callable bond has lower effective duration than an otherwise identical option-free bond, since the call caps the upside when yields fall.
Two money versions of the same idea show up in practice questions. Money duration (modified duration times price) and the price value of a basis point (PVBP, also called DV01) translate the percentage into currency. PVBP tells you the rupee or dollar change for a one-basis-point move, and it is what traders actually hedge with. For a bond priced at 1,043.76 with modified duration 6.0, the PVBP is about 6.0 x 1,043.76 x 0.0001 = 0.63 per bond. Portfolio duration is the market-value-weighted average of the individual bonds’ durations, and it works only as a first-order approximation, exactly like single-bond duration.
Convexity: why duration alone misprices big moves
Look at the price-yield curve below. Duration draws a straight tangent line at the current yield and uses it to estimate price changes. For small yield moves the tangent is close enough. For large moves it is not, because the true curve bends away from the line, and it always bends upward for an option-free bond. That upward bend is positive convexity, and it always works in the investor’s favor: prices rise more than duration predicts when yields fall, and fall less than duration predicts when yields rise.
The convexity adjustment is a second-order correction added to the duration estimate. At Level 1 you need the intuition and the direction, not the full Taylor expansion: duration underestimates price gains and overestimates price losses for option-free bonds. Callable bonds can show negative effective convexity in the region where the call becomes likely, because the price gets capped near the call price even as yields keep falling.
Exam traps: the distinctions that cost marks
The traps below are the reason these CFA fixed income notes spend as much time on definitions as on formulas: the math is easy once the distinctions are automatic.
- Current yield vs YTM. Current yield ignores the pull to par. For premium bonds it overstates true return; for discount bonds it understates it.
- YTM’s hidden assumption. YTM assumes coupons are reinvested at the YTM. If the question mentions falling rates after purchase, the answer is reinvestment risk, and the realized return falls short of YTM.
- Yield to worst for callable bonds. When a premium bond is callable, compute yield to each call date and take the minimum with YTM. Issuers call when it saves them money, which is exactly when it costs you.
- Spot vs forward. Spot rates discount each cash flow at its own maturity. The forward rate comes from the no-arbitrage equation (1 + S2)2 = (1 + S1) x (1 + f1,2). Rearrange carefully; the exam inverts which rate is asked for.
- Macaulay vs modified vs effective. Macaulay is weighted-average time; modified is Macaulay adjusted for compounding frequency and is the sensitivity measure; effective is computed from repriced values and is the one to use when cash flows can change.
- Duration drivers. Longer maturity, lower coupon, lower yield: longer duration. The one exception to memorize is the zero-coupon bond, whose Macaulay duration equals its maturity.
- Convexity direction. Positive for option-free bonds and always helps the investor. Negative effective convexity appears for callable bonds near the call region, where the price flattens.
- PVBP units. PVBP is currency per basis point, not a percentage. If the question asks for the rupee or dollar impact of a rate move, reach for money duration or PVBP, not modified duration alone.
One-page revision box: CFA fixed income in 60 seconds
Screenshot this box.
- Price is the PV of coupons plus par, discounted at the YTM.
- Coupon above YTM means premium, below means discount, equal means par. Pull-to-par: premiums fall, discounts rise toward par over time.
- Current yield is coupon over price and ignores the maturity gain or loss.
- YTM is the IRR and assumes reinvestment at the YTM. Falling rates create reinvestment risk.
- Yield to worst is the minimum of YTM and all yields to call.
- Forward rate from spots: (1 + S2)2 = (1 + S1) x (1 + f1,2).
- Modified duration gives price sensitivity: % change = -ModDur x change in yield.
- Duration is longer with longer maturity, lower coupons, lower yields. Zero-coupon duration equals maturity.
- Use effective duration for bonds with embedded options; callable bonds have lower effective duration.
- Convexity corrects duration for large moves and is positive for option-free bonds. Callable bonds can show negative effective convexity near the call.
- PVBP (DV01) is the currency change for a one-basis-point move.
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CA Luhsna Awrak is a Chartered Accountant (DISA-qualified) writing on CA exams, tax and finance careers.
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