3.3 Bond Yields, Duration & Convexity

Bond Yields, Duration & Convexity

Quick Summary: While a bond's coupon is fixed, its yield fluctuates continuously with market prices. Yield to Maturity (YTM) measures total annualized return assuming coupons are reinvested at the YTM rate. The yield curve plots yields across maturities, acting as an economic barometer. Interest rate sensitivity is quantified by modified duration (percentage price change per 100 bps yield change) and refined by convexity, which accounts for the curvature of the price-yield function.

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1. Fixed Income Yield Measures

Investors utilize multiple yield metrics depending on whether they seek immediate income generation or total annualized holding-period return.

Nominal Yield (Coupon Rate)

The nominal yield is simply the contractual coupon rate expressed as a percentage of nominal par value:

$\text{Nominal Yield} = \frac{\text{Annual Coupon}}{\text{Par Value}} \times 100$

This percentage is fixed at issuance for conventional bonds and does not reflect market fluctuations.

Current Yield (Running / Flat / Income Yield)

The current yield measures the annual cash income generated by the bond relative to its current clean market price:

$\text{Current Yield} = \frac{\text{Annual Coupon}}{\text{Clean Market Price}} \times 100$

• Primary Application: Useful for income-seeking investors (e.g., retirees, endowment funds) needing to project immediate cash distribution from a portfolio.
• Critical Limitation: Current yield completely ignores the capital gain realized when a discount bond matures at par, or the capital loss incurred when a premium bond matures at par. It also ignores the time value of money.

Yield to Maturity (YTM) / Gross Redemption Yield (GRY)

Yield to Maturity (referred to as Gross Redemption Yield in the UK) is the definitive total-return benchmark in bond analysis. Mathematically, YTM is the Internal Rate of Return (IRR) that discounts all remaining future cash flows (coupons and the final principal redemption) to equate precisely to the bond's dirty market price:

$P_{\text{dirty}} = \sum_{t=1}^n \frac{C}{(1 + y)^t} + \frac{M}{(1 + y)^n}$

Where:

• $P_{\text{dirty}}$ = Current settlement price
• $C$ = Periodic coupon payment
• $M$ = Maturity redemption value (par)
• $y$ = Yield to maturity per period
• $n$ = Total number of compounding periods to maturity

Key Theoretical Assumptions of YTM

YTM provides a standardized metric for comparing bonds across different coupons and tenors, but it relies on three stringent assumptions:

1. Held to Maturity: The investor holds the bond until its final redemption date.
2. No Default: The issuer makes all scheduled payments in full on time.
3. Reinvestment at YTM: All intermediate coupon cash flows are reinvested at the exact same YTM rate. This creates reinvestment risk: if market interest rates decline during the bond's life, coupons will be reinvested at lower yields, causing the investor's realized compound return to fall short of the original YTM.

Yield to Call (YTC) & Yield to Worst (YTW)

For callable bonds (where the issuer holds the contractual right to redeem the debt early at a specified call price):

• Yield to Call (YTC): Calculates the IRR assuming the bond is called at the first or next call date.
• Yield to Worst (YTW): For prudent risk management, wealth managers calculate both YTM and YTC across all call dates; the lowest resulting yield is the Yield to Worst. If a callable bond trades at a premium, the issuer has a strong financial incentive to call the debt and refinance at lower rates, making YTC the realistic return metric.

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2. Price-Yield Relationship: Discount, Par, and Premium

The most fundamental axiom of fixed income analysis is the inverse relationship between bond prices and market yields. Because a bond's contractual cash flows are fixed, when prevailing interest rates rise, existing bonds with lower coupons become less attractive, forcing their market price down until their yield matches prevailing rates. Conversely, when rates fall, existing bonds appreciate.

+-----------------------------------------------------------------------------------------+
|                         PRICE AND YIELD RELATIONSHIPS                                   |
+-----------------------------------------------------------------------------------------+
|  DISCOUNT BOND (Price < Par):    Coupon Rate  <  Current Yield  <  Yield to Maturity     |
|  PAR BOND      (Price = Par):    Coupon Rate  =  Current Yield  =  Yield to Maturity     |
|  PREMIUM BOND  (Price > Par):    Coupon Rate  >  Current Yield  >  Yield to Maturity     |
+-----------------------------------------------------------------------------------------+

Mathematical Order of Yields Explained

1. Discount Bond (Quoted at £85, Coupon 5%):

   • $\text{Coupon Rate} = 5.00%$
   • $\text{Current Yield} = \frac{£5}{£85} = 5.88%$
   • $\text{YTM} = \text{Approx } 7.15%$ (Captures the 5.88% cash income PLUS the annualized £15 capital gain accreting to par at maturity).
   • Ordering: $\text{Coupon} < \text{Current Yield} < \text{YTM}$

2. Par Bond (Quoted at £100, Coupon 5%):

   • $\text{Coupon Rate} = 5.00%$
   • $\text{Current Yield} = \frac{£5}{£100} = 5.00%$
   • $\text{YTM} = 5.00%$ (No capital gain or loss at maturity).
   • Ordering: $\text{Coupon} = \text{Current Yield} = \text{YTM}$

3. Premium Bond (Quoted at £115, Coupon 5%):

   • $\text{Coupon Rate} = 5.00%$
   • $\text{Current Yield} = \frac{£5}{£115} = 4.35%$
   • $\text{YTM} = \text{Approx } 3.25%$ (The 4.35% cash income is reduced by the annualized £15 capital loss suffered as the bond depreciates to par at maturity).
   • Ordering: $\text{Coupon} > \text{Current Yield} > \text{YTM}$

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3. The Term Structure of Interest Rates & Yield Curves

The term structure of interest rates represents the relationship between yields to maturity and terms to maturity for default-free sovereign bonds of identical credit quality. The graphical plot of this relationship is the yield curve.

The Four Primary Yield Curve Shapes

   Yield                     Yield                     Yield                     Yield
     ^                         ^                         ^                         ^    
     |      /----              |  \                      |  --------               |    /---\
     |     /                   |   \----                 |                         |   /     \
     |    /                    |        \                |                         |  /       \
     +----------> Maturity     +----------> Maturity     +----------> Maturity     +----------> Maturity
        NORMAL                    INVERTED                  FLAT                      HUMPED

1. Normal (Upward-Sloping): Short-term yields are lower than long-term yields. This is the typical equilibrium state of an expanding economy. Investors demand a higher yield for locking up money in longer maturities to compensate for inflation risk and lower liquidity.
2. Inverted (Downward-Sloping): Short-term yields are higher than long-term yields. An inverted curve is a historically reliable leading indicator of an impending economic recession. It signals that central banks have raised short-term policy rates aggressively to combat inflation, and the market expects economic contraction to force rate cuts in future years.
3. Flat: Short-term and long-term yields are virtually identical. Often represents a transition phase as the economy shifts from expansion to slowdown or vice versa.
4. Humped: Medium-term yields (e.g., 3-7 years) are higher than both short-term and long-term yields, reflecting near-term policy tightness followed by long-term stability.

Theories Explaining the Term Structure

• Pure Expectations Theory: Assumes investors are risk-neutral. Long-term interest rates are simply a geometric average of expected future short-term rates. An upward-sloping curve implies that the market expects short-term interest rates to rise in the future.
• Liquidity Preference Theory: Formulated by John Maynard Keynes, this theory argues that investors inherently prefer liquidity and shorter maturities. To induce them to hold long-dated bonds, borrowers must offer a liquidity premium. Because this premium increases with maturity, the yield curve will naturally slope upward even if future short-term rates are expected to remain constant.
• Market Segmentation Theory: Assumes institutional investors are strictly constrained by regulatory or liability mandates into specific maturity sectors (e.g., commercial banks in short maturities, pension funds and life insurers in 30-year paper). Yields in each maturity bucket are determined solely by supply and demand within that sector, with minimal arbitrage across the curve.
• Preferred Habitat Theory: A realistic compromise: while market participants have preferred maturity habitats to match liabilities, they are willing to deviate into other tenors if offered an adequate yield incentive.

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4. Duration: Quantifying Interest Rate Risk

While maturity tells an investor when the final principal is repaid, it does not measure how fast capital is returned or how sensitive the bond's price is to interest rate changes.

Macaulay Duration ($D_{\text{mac}}$)

Developed by Frederick Macaulay in 1938, Macaulay duration calculates the weighted average time (in years) until a bond's cash flows are received, where the weights correspond to the present value of each cash flow divided by the total bond price:

$D_{\text{mac}} = \frac{\sum_{t=1}^n \frac{t \cdot C_t}{(1+y)^t} + \frac{n \cdot M}{(1+y)^n}}{P_{\text{dirty}} }$

Fundamental Properties of Macaulay Duration

• For a zero-coupon bond, Macaulay duration is exactly equal to its time to maturity ($D_{\text{mac}} = \text{Maturity}$), because 100% of cash flows arrive at the redemption date.
• For a coupon-paying bond, Macaulay duration is always strictly less than its maturity, because interim coupons return capital early.
• Higher coupon payments result in a shorter duration (more cash weight received in early years).
• Higher market yields (YTM) result in a shorter duration (distant cash flows are discounted more heavily, shifting weight toward earlier payments).

Modified Duration ($D_{\text{mod}}$)

Modified duration adapts Macaulay duration into a direct measure of percentage price volatility for a given change in yield:

$D_{\text{mod}} = \frac{D_{\text{mac}}}{1 + \frac{y}{m}}$

Where $y$ is the annualized YTM and $m$ is the number of coupon compounding periods per year.

Estimating Bond Price Changes

Modified duration provides a linear estimate of percentage price change for a specified shift in yield ($\Delta y$):

$\% \Delta P \approx -D_{\text{mod}} \times \Delta y$

Worked Example: A bond portfolio has a modified duration of 6.2 years.

• If market yields rise across the curve by 50 basis points (+0.50%): $\% \Delta P \approx -6.2 \times (+0.0050) = -0.031 = -3.10\%$ The portfolio value will fall by approximately 3.10%.
• If market yields fall by 100 basis points (-1.00%): $\% \Delta P \approx -6.2 \times (-0.0100) = +0.062 = +6.20\%$ The portfolio value will rise by approximately 6.20%.

Dollar Value of an 01 (DV01) / PV01

Institutional desks frequently express interest rate risk in monetary terms rather than percentages. DV01 (Dollar Value of an 01) measures the absolute cash change in a bond's value resulting from a 1 basis point (0.01%) parallel shift in yield:

$\text{DV01} = D_{\text{mod}} \times P_{\text{dirty}} \times 0.0001$

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5. Convexity: The Non-Linear Reality

Modified duration assumes that the relationship between bond price and yield is a straight tangent line. For very small yield changes (e.g., 5-10 bps), this linear approximation is exceptionally accurate. However, for larger yield shifts (e.g., 100-200 bps), the linear model introduces substantial estimation error because the true price-yield curve is convex to the origin.

    Price
      ^                    Actual Price-Yield Curve (Convex)
      |                       / 
      |                      /     Duration Linear Tangent
      |                     /    /
      |                    /   /
      |                   /  /
      |                  / /
      |                 //
      |                /o----------------- Tangent Point (Current Price & Yield)
      |              // 
      |             /  \
      |            /     \
      |           /        \
      +------------------------------> Yield

The Asymmetric Benefit of Positive Convexity

Most conventional fixed-rate bonds possess positive convexity. Because the curve bends upward:

• When yields fall, the actual bond price rises by MORE than predicted by duration alone.
• When yields rise, the actual bond price falls by LESS than predicted by duration alone.

This asymmetric property is highly attractive to bondholders: it magnifies capital gains in a bull market (falling rates) while cushioning capital losses in a bear market (rising rates).

Combined Price Change Approximation

To capture both the linear duration effect and the curvature convexity effect, analysts employ a second-order Taylor series expansion:

$\% \Delta P \approx \left( -D_{\text{mod}} \times \Delta y \right) + \left( \frac{1}{2} \times \text{Convexity} \times (\Delta y)^2 \right)$

Because $(\Delta y)^2$ is always positive regardless of whether yields rise or fall, the convexity term always adds to the estimated price, correcting for duration's underestimation.

Negative Convexity

Certain fixed income instruments exhibit negative convexity over specific yield ranges, notably callable bonds and mortgage-backed securities (MBS). When interest rates fall significantly, borrowers exercise their prepayment or call options. This places a ceiling on price appreciation, causing the price-yield curve to flatten or bend downward, exposing investors to limited upside with full downside risk.