Chapter 10

Interest Rate Risk and Return

Comprehensive analysis of duration, convexity, and their applications in measuring and managing interest rate risk

1

Sources of Return in Bond Investing

Bond returns come from three primary sources, each contributing differently to total return depending on the investment horizon and market conditions.
Components of Bond Returns
Return Source Description Risk Factor
Coupon Income Periodic interest payments from the bond Credit risk, default risk
Capital Gain/Loss Change in bond price from purchase to sale Interest rate risk, credit risk
Reinvestment Income Return from reinvesting coupon payments Reinvestment risk
Total Return Formula

Total Return = Coupon Income + Capital Gain/Loss +
Reinvestment Income

Where:
Coupon Income = Periodic interest payments
Capital Gain/Loss = Change in bond price
Reinvestment Income = Interest earned on reinvested coupons
Key Insight
For buy-and-hold investors, reinvestment income becomes increasingly important for longer-maturity bonds and higher-coupon bonds. For a 30-year, 8% coupon bond held to maturity, reinvestment income typically represents about 80% of total return.
2

Interest Rate Risk: Duration

Duration measures a bond's sensitivity to interest rate changes. There are several duration measures, each with specific applications and limitations.
Macaulay Duration
Macaulay duration is the weighted average time to receive the bond's cash flows, where weights are the present values of each cash flow.
Macaulay Duration Formula

MacDur = Σ[t × (CF_t / (1+y)^t)] / Bond Price

Where:
MacDur = Macaulay Duration (in years)
t = Time period
CF_t = Cash flow at time t
y = Yield to maturity
Bond Price = Current market price
Modified Duration
Modified duration measures the price sensitivity of a bond to changes in yield to maturity. It's the most commonly used duration measure for risk management.
Modified Duration Formulas

ModDur = MacDur / (1 + y)

% Price Change ≈ −ModDur × Δy

Where:
ModDur = Modified Duration
MacDur = Macaulay Duration
y = Yield to maturity
Δy = Change in yield
Example: Duration Calculation
A bond has:
  • Macaulay duration = 5.2 years
  • Yield to maturity = 6%
Modified Duration = 5.2 / (1.06) = 4.91

If yields increase by 100 basis points:
Price change ≈ -4.91 × 0.01 = -4.91%
Money Duration and Price Value of a Basis Point
These measures express interest rate sensitivity in dollar terms rather than percentages.
Money Duration and PVBP Formulas

Money Duration = Modified Duration × Bond Price

PVBP = Money Duration × 0.0001

Where:
Money Duration = Dollar duration
PVBP = Price Value of a Basis Point
0.0001 = One basis point (0.01%)
3

Properties of Duration

Key Duration Properties
  • Maturity: Duration increases with maturity (but at a decreasing rate for coupon bonds)
  • Coupon Rate: Duration decreases as coupon rate increases
  • Yield to Maturity: Duration decreases as yield increases
  • Zero-Coupon Bonds: Duration equals time to maturity
  • Perpetual Bonds: Duration = (1 + y) / y
Factor Change Effect on Duration Explanation
Maturity Increase Increase More distant cash flows
Coupon Rate Increase Decrease More cash flows received earlier
Yield Increase Decrease Higher discount rate reduces PV of distant cash flows
Duration Limitations
  • Assumes parallel shifts in yield curve
  • Linear approximation becomes less accurate for large yield changes
  • Does not capture convexity effects
  • Assumes cash flows are fixed (not applicable to bonds with embedded options)
4

Convexity

Convexity measures the curvature of the price-yield relationship and improves duration-based price change estimates, especially for large yield changes.
Understanding Convexity
The price-yield relationship for bonds is not linear but convex. This convexity means that:
  • Bond prices rise more when yields fall than they decline when yields rise by the same amount
  • Duration underestimates price increases and overestimates price decreases
  • Higher convexity is generally beneficial to investors
Convexity Formulas

Convexity = [1 / (P × (1+y)²)] × Σ[t × (t+1) × CF_t / (1+y)^t]

% Price Change ≈ −ModDur × Δy + 0.5 × Convexity × (Δy)²

Where:
P = Bond Price
y = Yield to maturity
t = Time period
CF_t = Cash flow at time t
Δy = Change in yield
Example: Duration vs. Duration + Convexity
A bond has:
  • Modified duration = 7.2
  • Convexity = 65
  • Yield change = -200 basis points (-0.02)
Duration estimate: -7.2 × (-0.02) = +14.4%
Duration + Convexity: 14.4% + 0.5 × 65 × (-0.02)² = 14.4% + 1.3% = 15.7%

The convexity adjustment adds 1.3% to the price change estimate.
Factors Affecting Convexity
  • Maturity: Longer maturity generally increases convexity
  • Coupon Rate: Lower coupon rates increase convexity
  • Yield Level: Lower yields increase convexity
  • Embedded Options: Call options reduce convexity (negative convexity), put options increase it
5

Effective Duration and Convexity

For bonds with embedded options or uncertain cash flows, effective duration and convexity provide better risk measures than traditional duration and convexity.
Effective Duration
Effective duration is calculated using a numerical approach that considers how bond prices change when the entire yield curve shifts up or down.
Effective Duration = (P₋ - P₊) / (2 × P₀ × Δy)
where: P₋ = price when yield decreases, P₊ = price when yield increases, P₀ = initial price
Application to Bonds with Embedded Options
Bond Type Effective vs. Modified Duration Reason
Callable Bonds Effective Duration < Modified Duration Call option limits price appreciation
Putable Bonds Effective Duration < Modified Duration Put option limits price decline
Plain Vanilla Bonds Effective Duration ≈ Modified Duration No embedded options
Key Duration Applications
  • Portfolio Duration: Weighted average of individual bond durations
  • Immunization: Matching portfolio duration to investment horizon
  • Hedge Ratios: Determining futures contracts needed to hedge interest rate risk
  • Active Management: Positioning for expected yield curve changes
6

Interest Rate Risk Management

Immunization Strategies
Immunization protects a portfolio against interest rate risk by structuring it so that the target return will be achieved regardless of interest rate changes.
Classical Immunization Conditions
  • Portfolio duration must equal the investment horizon
  • Present value of assets must equal present value of liabilities
  • Portfolio must be rebalanced when duration drifts from the target
  • Assumes parallel shifts in the yield curve
Duration Matching vs. Cash Flow Matching
Strategy Approach Advantages Disadvantages
Duration Matching Match portfolio duration to horizon Lower cost, flexibility Requires rebalancing, model risk
Cash Flow Matching Match cash flows to liability payments No rebalancing, perfect match Higher cost, less flexibility
Example: Portfolio Duration Calculation
Portfolio with two bonds:
  • Bond A: $2 million, Duration = 4.5 years
  • Bond B: $3 million, Duration = 7.2 years
Portfolio Duration = [(2/5) × 4.5] + [(3/5) × 7.2] = 1.8 + 4.32 = 6.12 years

This portfolio has interest rate sensitivity equivalent to a single bond with 6.12 years duration.