4.4 Fundamental Bond Pricing Properties: Volatility Rules, Duration & Convexity

The Inverse Price-Yield Relationship and Bond Volatility Rules

Because the cash flows of a fixed-rate bond are fixed in nominal terms, bond prices and prevailing market yields exhibit an inverse relationship: when yields rise, prices fall; when yields fall, prices rise. However, the magnitude of price sensitivity differs dramatically across different debt issues.

In 1962, economist Burton Malkiel established foundational principles governing bond price volatility, which remain critical for Canadian fixed-income analysis:

1. The Maturity Effect

Rule: For a given change in market interest rates, bond price volatility is directly related to term to maturity. Longer-term bonds experience greater percentage price swings than shorter-term bonds of identical coupon rate.

• Rationale: A greater proportion of cash flows in a 30-year bond occur in the distant future, making their present values far more sensitive to changes in the discount rate than those of a 2-year bond.

2. The Coupon Effect

Rule: For a given term to maturity, bond price volatility is inversely related to the coupon rate. Lower-coupon bonds experience greater percentage price swings than higher-coupon bonds.

• Rationale: A high-coupon bond returns a substantial portion of its total investment early in the form of hefty semi-annual cash payments, reducing the relative weight of the final principal repayment. A lower-coupon bond (and particularly a zero-coupon strip bond) concentrates total cash return at maturity, maximizing price sensitivity.

3. The Yield Level Effect (Diminishing Marginal Volatility)

Rule: A bond's percentage price sensitivity to a given yield change is greater when prevailing market yields are low than when market yields are high.

• Rationale: When yields drop from 3.00% to 2.00%, the relative change in the discounting factor is far larger ($33.3\%$ drop) than when yields drop from 10.00% to 9.00% ($10.0\%$ drop).

Comparison of Bond Volatility Factors

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The Geometry and Power of Bond Convexity

The inverse relationship between bond prices and yields is not linear; it is curved, or convex to the origin.

Bond Price-Yield Curve (Convexity)
 Price
   ^
   |      * (Steeper slope as yields fall: large price gain)
   |       *
   |         *
   |           *
   |             * (Flatter slope as yields rise: small price loss)
   +-------------------------------------> Yield

Positive Convexity Defined

For standard, non-callable option-free bonds, this curvature is known as positive convexity:

• As market yields decline, the bond price rises at an accelerating rate.
• As market yields rise, the bond price falls at a decelerating rate.

The Asymmetric Advantage of Positive Convexity

Due to positive convexity, for an identical parallel shift in market yields (e.g., $\pm 100$ basis points):

$\text{Percentage Price Gain (Yields Fall by 100 bps)} > \text{Percentage Price Loss (Yields Rise by 100 bps)}$

Practical Example: Consider a 10-year Government of Canada bond yielding 5.00% priced at 100.00:

• If yields drop by 100 bps to 4.00%, the bond's price rises to 108.18 ($+8.18\%$ gain).
• If yields rise by 100 bps to 6.00%, the bond's price falls to 92.56 ($-7.44\%$ loss).
• The price gain exceeds the price loss by 0.74 percentage points due entirely to positive convexity.

Negative Convexity in Callable Bonds

While option-free bonds enjoy positive convexity across all yield levels, callable bonds exhibit negative convexity at low market yields. When yields drop below the bond's coupon rate, the issuer is economically motivated to call the bond at par or a modest call premium. Consequently, the bond's market price is capped near the call price, flattening price appreciation and exposing investors to asymmetrical downside risk if rates rise.

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Duration: Quantifying Interest Rate Sensitivity

While maturity provides the nominal calendar lifespan of a bond, it is an incomplete measure of interest rate risk. For example, a 10-year bond with a 10.00% coupon is far less sensitive to interest rate shifts than a 10-year zero-coupon strip bond, because the coupon-paying bond returns substantial cash flow early in its life.

To quantify a bond's true interest rate risk, fixed-income analysts rely on duration.

Macaulay Duration

Developed by Frederick Macaulay in 1938, Macaulay duration measures the weighted average time (expressed in years) until all cash flows (coupons and principal) are received, where each cash flow is weighted by its present value relative to the total bond price:

$D_{\text{Mac}} = \frac{\sum_{t=1}^{2n} \frac{t \times (C/2)}{(1 + y/2)^t} + \frac{2n \times \text{Par}}{(1 + y/2)^{2n}}}{\text{Bond Price}}$

• Zero-Coupon Strip Bond: Pays only a single cash flow at maturity. Therefore, the Macaulay duration of a zero-coupon bond equals its exact term to maturity ($D_{\text{Mac}} = n$).
• Coupon-Paying Bond: Because coupons are received prior to maturity, the Macaulay duration of a coupon bond is always strictly less than its term to maturity ($D_{\text{Mac}} < n$).

Modified Duration

While Macaulay duration measures time in years, portfolio managers require a direct measure of price volatility. Modified duration converts Macaulay duration into a percentage price sensitivity metric for a 1% (100 basis point) parallel change in yield to maturity:

$\text{Modified Duration} = \frac{D_{\text{Mac}}}{1 + \frac{y}{m}}$

Where $y$ is the nominal annual yield to maturity and $m$ is the annual compounding frequency ($m = 2$ for semi-annual Canadian bonds).

The Duration Price Sensitivity Approximation Formula

Modified duration allows analysts to estimate percentage and dollar price changes instantly:

$\frac{\Delta P}{P} \approx - \text{Modified Duration} \times \Delta y$ $\Delta P \approx - \text{Modified Duration} \times \Delta y \times P$

Where:

• $\Delta P / P$ = Approximate percentage change in bond price
• $\Delta P$ = Approximate dollar change in bond price
• $\Delta y$ = Parallel change in market yield expressed as a decimal (e.g., $+50\text{ bps} = +0.0050$; $-100\text{ bps} = -0.0100$)
• The negative sign reflects the fundamental inverse price-yield relationship.

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Worked Numerical Example: Estimating Price Impact via Duration

Problem: A Canadian institutional fixed-income portfolio has a market value of $10,000,000 and an average portfolio modified duration of 6.50 years. The Bank of Canada unexpectedly reduces its policy interest rate, causing benchmark market yields across all maturities to drop by 40 basis points (-0.40%).

Step 1: Identify Input Variables

• Portfolio Market Value ($P$) = $10,000,000
• Modified Duration = 6.50 years
• Change in Yield ($\Delta y$) = $-0.0040$ ($-40\text{ bps}$)

Step 2: Calculate Percentage Price Change

$\frac{\Delta P}{P} \approx - 6.50 \times (-0.0040) = +0.0260 = +2.60\%$

Step 3: Calculate Dollar Value Change

$\Delta P \approx \$10,000,000 \times (+0.0260) = +\$260,000$

Step 4: Determine New Portfolio Value

$\text{New Market Value} = \$10,000,000 + \$260,000 = \$10,260,000$

The portfolio increases in value by approximately $260,000 (2.60%) as a result of the 40 basis point yield decline.

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Determinants of Duration and Portfolio Strategies

Fixed-income managers manage interest rate risk by monitoring the three core factors that determine duration:

The Three Rules of Duration

1. The Maturity Effect: Holding coupon and yield constant, duration increases as maturity increases (though at a decreasing rate for coupon-paying debt).
2. The Coupon Effect: Holding maturity and yield constant, duration is inversely related to the coupon rate. Higher coupons front-load cash flows, shortening duration and dampening price volatility. Zero-coupon strip bonds have the maximum possible duration for any given maturity.
3. The Yield Level Effect: Holding maturity and coupon constant, duration is inversely related to prevailing yields to maturity. Higher yields discount distant cash flows more severely, increasing the relative present value weighting of earlier coupons and shortening duration.

Summary of Duration Determinants

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Portfolio Management Applications of Duration

Professional fixed-income managers utilize duration for two primary strategic applications: tactical rate positioning and liability immunization.

1. Tactical Duration Positioning (Interest Rate Anticipation)

Portfolio managers actively adjust their portfolio duration relative to benchmark indices based on macroeconomic interest rate forecasts:

• Bullish Outlook (Expecting Interest Rates to Fall): The manager actively lengthens portfolio duration by selling short-term paper and buying long-term, low-coupon bonds or zero-coupon strips. When yields fall, the high duration amplifies capital gains.
• Bearish Outlook (Expecting Interest Rates to Rise): The manager actively shortens portfolio duration by shifting into short-term T-bills, floating-rate notes, or cash equivalents. When yields rise, the reduced duration shields the portfolio from severe capital losses.

2. Fixed-Income Immunization

Institutional managers (such as pension plan administrators) use duration to immunize a bond portfolio against interest rate fluctuations. Immunization involves structuring the portfolio such that the Macaulay duration of the asset portfolio exactly matches the duration of the future liability obligation:

$D_{\text{Assets}} = D_{\text{Liabilities}}$

• Balancing Opposing Risks: When interest rates change, bond prices and coupon reinvestment rates move in opposite directions:
  • If rates rise: The portfolio suffers a capital loss on market price, but intermediate coupons are reinvested at higher yields.
  • If rates fall: Reinvestment returns decline, but the portfolio captures an offsetting capital gain on market price.
• By matching duration, the capital price effect and reinvestment effect precisely cancel each other out, ensuring that the target accumulated sum is fully realized regardless of where interest rates move.