Difficulty: Intermediate | Prerequisites: basic time-value-of-money concepts, present value, compounding.
This material sits at the heart of any investments course. Fixed-income securities (bonds) are the primary alternative to equities, and understanding how they are priced, how their prices respond to interest rate changes, and how investors measure and manage that interest rate risk is foundational. You should already be comfortable with present value arithmetic and the idea that a dollar tomorrow is worth less than a dollar today. If that feels shaky, revisit your TVM notes before diving in.
TL;DR
Bond prices are the present value of their future cash flows, discounted at the market yield. When yields rise, prices fall (and vice versa). Duration and convexity measure how sensitive a bond's price is to yield changes, and the term structure of interest rates links today's spot rates to expected future (forward) rates.
Zero-coupon bond ("zero")
A bond that makes no periodic interest payments. It is sold at a discount to its face (par) value and redeems at par on the maturity date. The investor's return comes entirely from the difference between the purchase price and par.
In simple terms: you buy it cheap, get nothing in between, and collect the full face value at the end.
Yield to maturity (YTM)
The single discount rate that equates the present value of all a bond's future cash flows (coupons plus par) to its current market price. It is the bond's internal rate of return if held to maturity and all coupons are reinvested at that same rate.
Think of it as the bond's "all-in" annualised return, assuming you hold it to the end.
Current yield
Annual coupon payment divided by the bond's current market price. It captures income return only, ignoring capital gains or losses.
In simple terms: what percentage of your purchase price you get back each year in coupon payments.
Yield to call (YTC)
The yield calculated assuming the bond is called (redeemed early by the issuer) at the first call date and call price, rather than held to maturity.
Think of it as: the YTM equivalent, but computed to the call date instead of the maturity date.
Spot rate
The yield on a zero-coupon bond for a specific maturity. It represents the market's required return for lending money for that exact period with no interim cash flows.
Forward rate
The interest rate agreed upon today for a loan that begins at a future date. It is implied by the relationship between spot rates of different maturities.
In simple terms: the rate the market "expects" (or prices in) for a single future period, given today's term structure.
Macaulay duration
The weighted average time until a bond's cash flows are received, where each cash flow's weight is its present value as a fraction of the bond's total price. Measured in years (or half-years for semiannual bonds, then converted).
Think of it as the bond's "centre of gravity" in time, the point at which you have, on average, received your money back.
Modified duration
Macaulay duration divided by (1 + yield per period). It estimates the percentage change in a bond's price for a 1 percentage-point change in yield.
In simple terms: the bond's price sensitivity dial. Higher modified duration means the price moves more when rates change.
Convexity
A measure of the curvature in the price-yield relationship. Duration gives a linear (straight-line) estimate of price changes; convexity corrects for the fact that the actual relationship is curved.
Think of it as the "error correction" term you add to the duration estimate, especially important for large yield changes.
Classical immunization
A portfolio strategy that matches the Macaulay duration of a bond (or bond portfolio) to the investor's investment horizon, so that the effects of interest rate changes on price and reinvestment income roughly offset each other.
In simple terms: you set up your bond holdings so that if rates move against you on one front, they help you on the other.
For a zero-coupon bond, the price is simply the present value of the par value, discounted at the yield to maturity. With semiannual compounding, you halve the annual yield and double the number of periods.
Price = Face Value / (1 + y/2)^(2n), where y is the annual YTM and n is years to maturity
A $1,000 par, 15-year zero with 8% YTM (semiannual): Price = 1000 / (1.04)^30 = $308.32
The same bond with only 12 years remaining (i.e. 3 years later, rates unchanged): Price = 1000 / (1.04)^24 = $390.12
If rates rise to 10% with 12 years left: Price = 1000 / (1.05)^24 = $310.07
This is the most fundamental relationship in fixed income. When market yields rise, bond prices fall, because future cash flows are discounted more heavily. When yields fall, prices rise.
For zeros, this effect is especially stark because there are no interim coupons to cushion the price move.
The zero-coupon pricing formula has four variables: price, face value, yield, and maturity. Given any three, you can solve for the fourth.
Known price and yield, solve for maturity: rearrange to n = ln(FV/Price) / (2 × ln(1 + y/2))
Known price and maturity, solve for yield: y = 2 × [(FV/Price)^(1/(2n)) - 1]
For example, a $1,000 zero priced at $601 with an 8% yield: n = ln(1000/601) / (2 × ln(1.04)) = approximately 6.5 years. A $1,000 zero priced at $350 with 9 years to maturity: y = 2 × [(1000/350)^(1/18) - 1] = approximately 6.0%.
These three yield measures each capture something different, and exams love testing whether you know which is which.
Current yield = Annual coupon / Market price. It tells you income return only. A bond trading at a discount has a current yield above its coupon rate; a premium bond has current yield below its coupon rate.
YTM factors in both coupon income and capital gains or losses (the pull to par). For a discount bond, YTM > current yield > coupon rate. For a premium bond, the ordering reverses.
YTC replaces the maturity date with the call date and the par value with the call price. For callable bonds trading at a premium (likely to be called), YTC is often the more relevant measure.
Worked comparison: Bond A, 20-year 9% semiannual, priced to yield 10.5%. Its price is below par (discount bond). Bond B, 20-year 8% annual, priced to yield 7.5%. Its price is above par (premium bond). Bond B has the higher current yield only if its coupon/price ratio exceeds Bond A's. You need to compute both prices, then the ratio.
To find current yield: first compute the bond's price via the PV formula, then divide the annual coupon by that price.
The term structure of interest rates describes the relationship between yields and maturities. Forward rates are the building blocks.
The spot rate for year n is the geometric average of the forward rates from year 0 through year n-1.
(1 + spot_n)^n = (1 + f_0)(1 + f_1)...(1 + f_{n-1})
Equivalently, the forward rate between year n-1 and year n is: f_n = [(1 + spot_n)^n / (1 + spot_{n-1})^{n-1}] - 1
When you have the full set of forward (or spot) rates, you discount each cash flow at the appropriate rate, not at a single flat YTM.
Price of a zero-coupon bond = Par / [(1 + f_0)(1 + f_1)...(1 + f_{n-1})]
Price of a coupon bond = sum of each coupon discounted at the product of forward rates to that date, plus the par discounted to maturity
Example: given forward rates of 5%, 7%, 9%, 10% for years 0 through 3, the price of a 3-year zero ($1,000 par) = 1000 / (1.05 × 1.07 × 1.09) = $816.30.
A 2-year 10% coupon bond: price = 100/(1.05) + 1100/(1.05 × 1.07) = $95.24 + $979.00 = $1,074.24.
Under the expectations hypothesis, the expected return on any bond over one period equals the current one-period spot rate, regardless of the bond's maturity. For a 4-year zero held for one year, the expected return is the year-0 forward rate (5% in the example above).
Macaulay duration is a weighted-average time to receipt of cash flows. For a semiannual coupon bond:
List each semiannual period (t = 1, 2, ..., 2n)
Compute the cash flow at each period (coupon = C/2 each period, plus par at the final period)
Discount each cash flow to present value using the semiannual yield: PV_t = CF_t / (1 + y/2)^t
Weight each period by PV_t / Bond Price
Macaulay duration (in half-years) = sum of t × (PV_t / Price)
Convert to years by dividing by 2
Example: an 8% coupon, 8% YTM, 5-year semiannual bond sells at par ($1,000). Each semiannual coupon is $40. You compute PV-weighted time for each of the 10 periods, sum them, then divide by 2 to get years. The Macaulay duration comes to approximately 4.218 years.
Modified duration = Macaulay duration / (1 + y/2) for semiannual bonds.
This gives the approximate percentage price change for a 1 percentage-point change in yield:
%ΔP ≈ -Modified duration × Δy
For the bond above: Modified duration = 4.218 / 1.04 ≈ 4.056.
Maturity tells you only when the last cash flow arrives. It ignores the timing and size of all interim cash flows. Two bonds with the same maturity can have very different price sensitivities if their coupon rates differ. Modified duration captures the full cash-flow profile and gives you a direct, usable estimate of price sensitivity.
Lower coupon → higher duration. Fewer interim cash flows means more weight on the distant par payment. A 4% coupon bond is more rate-sensitive than an 8% coupon bond of the same maturity.
Longer maturity → higher duration. Cash flows stretch further into the future, raising the weighted-average time.
Shorter maturity → lower duration. The reverse: a 3-year bond has lower duration than a 5-year bond, other things equal.
Higher yield → lower duration. Higher discount rates reduce the present value of distant cash flows relative to near ones, pulling the centre of gravity earlier.
Convexity measures the curvature of the price-yield curve. Duration alone gives a straight-line approximation, which underestimates price increases (when yields fall) and overestimates price decreases (when yields rise). Convexity corrects for this.
The full price-change approximation for a yield change Δy:
%ΔP ≈ -Modified duration × Δy + (1/2) × Convexity × (Δy)²
Convexity is always positive for standard (option-free) bonds, meaning the correction always adds to the price estimate. This is why investors like convexity: for the same duration, a more convex bond gains more when rates fall and loses less when rates rise.
To compute convexity, you sum t(t+1) × PV(CF_t) / [(1 + y/2)² × Price] across all semiannual periods, then divide by 4 to annualise if working in half-years.
Immunization matches your portfolio's Macaulay duration to your investment horizon. If rates shift, the capital gain (or loss) on the bond is offset by the reinvestment loss (or gain) on coupons.
The key insight: when yields rise, the bond's price falls, but coupon reinvestment earns more; when yields fall, the opposite happens. At the duration-matched horizon, these two effects cancel.
Immunization must be rebalanced because duration changes as time passes and as yields shift. A shift from 8% to 12% yield changes the Macaulay duration of a 4-year 10% coupon bond from roughly 3.56 years (at 8%) to roughly 3.52 years (at 12%). The portfolio manager must adjust holdings to keep duration aligned with the (now shorter) remaining horizon.
P_{\text{zero}} = \frac{FV}{(1 + y/2)^{2n}}P_{\text{coupon}} = \sum_{t=1}^{2n} \frac{C/2}{(1 + y/2)^t} + \frac{FV}{(1 + y/2)^{2n}}\text{Current Yield} = \frac{\text{Annual Coupon}}{\text{Market Price}}(1 + s_n)^n = (1 + f_0)(1 + f_1) \cdots (1 + f_{n-1})D_{\text{Mac}} = \frac{1}{P} \sum_{t=1}^{2n} t \times \frac{CF_t}{(1 + y/2)^t} \quad \text{(in half-years; divide by 2 for years)}D_{\text{Mod}} = \frac{D_{\text{Mac}}}{1 + y/2}\%\Delta P \approx -D_{\text{Mod}} \times \Delta y + \tfrac{1}{2} \times \text{Convexity} \times (\Delta y)^2\text{Convexity} = \frac{1}{P(1 + y/2)^2} \sum_{t=1}^{2n} \frac{t(t+1) \times CF_t}{(1 + y/2)^t} \quad \text{(then } \div 4 \text{ to annualise)}"A bond's maturity tells you its interest rate risk." It does not, on its own. Two 10-year bonds with different coupon rates have different durations and therefore different sensitivities. Always look at duration, not maturity alone.
"YTM is the return you will earn." Only if you hold to maturity and reinvest every coupon at exactly the YTM rate, which rarely happens. YTM is a standardised comparison measure, not a prediction.
"Duration works for any size yield change." Duration gives a linear approximation. For large yield moves (say, 2%+), the linear estimate overshoots or undershoots. That is exactly where convexity matters.
"Forward rates are predictions of future spot rates." Under the pure expectations hypothesis they are, but in practice forward rates also embed a liquidity (or term) premium. The expectations hypothesis is a useful framework, not established fact.
⚠️ Zero-coupon bond pricing with semiannual compounding is a near-certainty on exams. Be fast with it.
⚠️ Know how to solve for any missing variable (price, yield, maturity) in the zero-coupon formula.
⚠️ The ordering of current yield, coupon rate, and YTM for discount vs premium bonds is a classic multiple-choice target.
⚠️ YTC calculations: remember to use the call price (not par) and the call date (not maturity).
⚠️ Forward rate arithmetic: build up the spot rate from the forward rates, or extract a forward rate from two spot rates. Both directions get tested.
⚠️ Duration computation is labour-intensive. On timed exams, set up a table (period, cash flow, PV of CF, PV × t) and sum the last column.
⚠️ The convexity correction is tested as a concept ("explain why duration alone is insufficient") and as a calculation (applying the formula with a given Δy).
True or False: A zero-coupon bond's Macaulay duration equals its maturity. Answer: True. With no interim cash flows, the entire weight falls on the single payment at maturity.
Fill in the blank: When market yields rise, bond prices ______. Answer: Fall.
True or False: Modified duration is always less than Macaulay duration. Answer: True (for positive yields). You divide Macaulay duration by (1 + y/period), which is greater than 1.
Fill in the blank: The convexity adjustment is always ______ (positive/negative) for option-free bonds, meaning it ______ the estimated price. Answer: Positive; increases.
True or False: A bond with a higher coupon rate has a higher duration than an otherwise identical bond with a lower coupon rate. Answer: False. Higher coupons mean more cash flow arrives earlier, reducing duration.
Q: A $1,000 par zero-coupon bond has 20 years to maturity and a 12% yield (semiannual compounding). What is its price?
A: Price = 1000 / (1.06)^40 = $97.22.
Q: Given forward rates of 5%, 7%, 9%, and 10% for years 0 through 3, what is the yield to maturity of a 3-year zero-coupon bond?
A: (1 + YTM)^3 = 1.05 × 1.07 × 1.09 = 1.22504. So YTM = (1.22504)^(1/3) - 1 = 7.00% (approximately).
Q: An 8% semiannual coupon bond with 5 years to maturity has an 8% YTM (priced at par). If the yield rises by 0.25%, what is the approximate percentage price change using both duration and convexity?
A: With modified duration ≈ 4.056 and convexity ≈ 19.8 (annualised), %ΔP ≈ -4.056 × 0.0025 + 0.5 × 19.8 × (0.0025)² = -1.014% + 0.006% = -1.008%. The bond's price falls by approximately 1.01%.
Q: A bond with a 10% annual coupon, 4-year maturity, and 8% yield has a Macaulay duration of about 3.56 years. If yields jump to 12%, duration changes to about 3.52 years. What does this imply for immunization?
A: A portfolio immunized at the 3.56-year horizon must be rebalanced after the yield shift, because the bond's duration has shortened. The manager needs to adjust holdings to realign duration with the remaining horizon.
Q: Explain why an investor might prefer a bond with greater convexity, all else equal.
A: Greater convexity means the bond gains more when yields fall and loses less when yields rise, compared to a less convex bond with the same duration. It is a desirable property, and investors may accept a lower yield for it.
This material connects directly to equity valuation (Ch. 18): the same present-value machinery used to price bonds is used to discount expected dividends and cash flows for stocks. The concept of required return (driven by risk, via CAPM) parallels the role of YTM in bond pricing.
Duration and convexity reappear in portfolio management and risk management. Immunization strategies are the fixed-income analogue of hedging in derivatives (Ch. 20-23), where delta hedging plays a similar "match the sensitivity" role.
The term structure and forward rates are also foundational for derivatives pricing, since forward and futures prices are built on the same no-arbitrage logic that links spot rates to forward rates.
Related Terms / Search Tags
Zero-coupon bond, discount bond, par bond, premium bond, yield to maturity, YTM, current yield, yield to call, YTC, spot rate, forward rate, term structure of interest rates, expectations hypothesis, Macaulay duration, modified duration, convexity, bond price sensitivity, interest rate risk, immunization, classical immunization, semiannual compounding, present value of bond, bond pricing formula, callable bond, call deferment, pull to par, reinvestment risk, duration matching