Duration explained
• Duration measures a bond’s sensitivity to interest rate changes and can also be viewed as the present value–weighted average time to receive cash flows, expressed in years.
• Intuition: higher duration means greater price movement for a given yield change; lower duration dampens price swings.
• For a small change in yield Δy, the percentage price change is approximately ΔP/P ≈ -Dmod × Δy, where Dmod is modified duration.
Properties of duration
• Inverse relation to coupon: higher coupons shorten duration because more cash flows arrive earlier, shifting the cash flow ‘center of mass’ forward.
• Direct relation to maturity (usually): longer maturity generally increases duration, but the effect tapers as discounting pushes far dated cash flows’ weights down.
• Inverse relation to yield: higher yields reduce duration by discounting later cash flows more heavily.
• Zero coupon benchmark: a pure discount bond has duration equal to its time to maturity because all value is at maturity.
• Additivity by market value: portfolio duration is a market value weighted average of constituent durations under parallel yield shifts.
• Callability and amortization: embedded options and principal amortization reduce effective duration versus option-free bullets as cash flows can occur sooner or change with rates.
Portfolio duration
• Definition: portfolio duration is the market value–weighted average of the constituent bonds’ durations.
• Computation: Dp = Σ wi Di, with wi = Pi / Σ Pj.
• Use cases: immunization and ALM, tactical duration tilts, and risk budgeting via DV01/PVBP targets.
What duration means
- Duration measures interest rate sensitivity: if the portfolio duration is 6, a 1% increase in yields implies roughly a 6% price drop, assuming a parallel shift of the yield curve.
- For a single bond, Macaulay duration is the present‑value‑weighted average time to receive all cash flows (in years), and modified duration converts that into price sensitivity per 1% yield change.
Symbols and their roles
- Di: Duration of bond i (use Macaulay or modified consistently across all bonds; modified is common for sensitivity).
- Pi: Market value (price × quantity) of bond i; this sets how much bond i contributes to the portfolio’s value.
- Σ Pj: Total market value of the portfolio; the sum of market values across all bonds.
- wi: Weight of bond i in the portfolio by market value, defined as wi = Pi / Σ Pj; the weights sum to 1.
- Dp: Portfolio duration, computed as the weighted average Dp = Σ wi Di.
How to calculate step by step
- Compute each bond’s market value Pi (clean price × units; or dirty price if that’s the chosen convention—be consistent across bonds).
- Compute each bond’s duration Di (all in the same duration type; if using modified duration, ensure the same compounding convention for yield).
- Compute each weight wi = Pi / Σ Pj; verify the weights add to 1 within rounding.
- Multiply and sum: Dp = Σ (wi × Di).
Worked numeric example
- Suppose a portfolio has two bonds:
- Bond A: market value P1 = 300, duration D1 = 2.5.
- Bond B: market value P2 = 700, duration D2 = 6.0.
- Total value Σ Pj = 300 + 700 = 1000; weights: w1 = 0.3, w2 = 0.7.
- Portfolio duration: Dp = 0.3×2.5 + 0.7×6.0 = 0.75 + 4.2 = 4.95.
- Interpretation: for a 1% parallel yield increase, portfolio value drops about 4.95%; for 0.25%, about 1.24% (approximate, linear for small moves).
Practical tips and nuances
- Consistency: Use the same duration type across bonds. If price sensitivity is the goal, use modified or effective duration; if averaging times is the goal, use Macaulay.
- Yield curve moves: The weighted‑average method assumes a parallel shift; for non‑parallel moves, key rate durations or full cash‑flow aggregation are better.
- Floating‑rate notes: They typically have low effective duration (near reset dates); using their market‑value weight naturally reduces Dp.
- Callable/MBS: Use effective duration estimated from models, since cash flows change with rates.
- Rebalancing: As prices and durations change, Dp drifts; managers adjust holdings to target a desired duration.
Common variations
- Value‑weighted Macaulay duration: Dp^Mac = Σ wi Di^Mac.
- Value‑weighted modified duration: Dp^Mod = Σ wi Di^Mod, then price change ≈ −Dp^Mod × Δy.
- If only Macaulay durations are available with yield y (per period), modified ≈ Macaulay / (1 + y) under the chosen compounding basis.
If a specific portfolio with prices, quantities, coupons, maturities, and yields is available, the calculation can be shown line‑by‑line using these symbols to produce Dp in either Macaulay or modified terms.
Modified duration and price elasticity
• Modified duration: Dmod = DMac / (1 + y/m), where DMac is Macaulay duration, y is yield to maturity, and m is compounding frequency.
• Price elasticity of interest rates: modified duration is the first order price elasticity to yield, linking a 1% (100 bps) change in yield to an approximate Dmod% opposite change in price.
• Practical interpretation: a bond with Dmod = 6 will gain ~6% if yields fall 100 bps and lose ~6% if yields rise 100 bps, before convexity adjustments.
Price volatility characteristics of bonds
• Duration as primary driver: longer duration increases volatility; shorter duration reduces it.
• Coupon and yield effects: low coupon and low yield bonds are more volatile than high coupon and high yield bonds, holding maturity constant.
• Asymmetry via convexity: price gains from yield declines are larger than price losses from equal yield increases due to convexity.
• Credit and liquidity overlays: spread duration adds volatility from credit spread moves; illiquidity can amplify short term price swings and widen bid-ask costs.
Convexity
• Definition: convexity is the second order sensitivity of price to yield, capturing the curvature of the price-yield relationship.
• Approximate price change with convexity: ΔP/P ≈ -Dmod Δy + 1/2 C (Δy)^2, where C is convexity.
• Implications: positive convexity (plain bonds) enhances gains when yields fall and cushions losses when yields rise; negative convexity (callable/MBS) reduces upside in rallies and increases downside in sell-offs.
• Portfolio convexity: market value–weighted average; necessary for accurate scenario estimates beyond small rate moves.
Putting it together: risk measures practitioners use
• DV01/PVBP: currency change for a 1 bp yield move; for position value V and Dmod, PVBP ≈ V × Dmod × 0.0001.
• Key rate duration: sensitivities to shifts at specific maturities on the curve for non-parallel risk management.
• Spread duration: sensitivity to credit spread changes relative to a benchmark curve.
Bond portfolio management
• Policy and benchmarks: define objective (income, total return, liability matching), risk limits (duration bands, tracking error, credit quality), liquidity constraints; select a benchmark aligned to mandate.
• Duration and convexity positioning: strategically align with liabilities for immunization; tactically tilt duration with rate views; maintain positive convexity unless compensated for negative convexity risk.
• Curve and key rate positioning: use steepeners/flatteners and butterflies (barbell vs bullet) to express curve views, optionally duration neutral.
• Credit and spread risks: diversify issuers/sectors; manage spread duration and concentrations; use CDS or futures to separate spread and rate exposures.
• Liquidity and implementation: prefer liquid issues; size positions to exit assumptions; use ETFs, futures, and interest rate swaps to adjust duration quickly; deploy cash bonds for carry and selection alpha.
• Hedging and overlay: hedge duration with futures/swaps/options; restore convexity for negative convexity exposure via options overlays.
• Monitoring and attribution: attribute returns to duration, curve, convexity, spread, selection, and currency; run stress tests and VaR; rebalance to policy ranges.
Illustrative example
• A ₹100 crore portfolio with modified duration 5.2 and convexity 22.
• A +50 bps parallel rise implies first order ≈ −5.2 × 0.50% = −2.60%; convexity adds +0.0275%.
• Net ≈ −2.57%, or −₹2.57 crore; convexity mildly cushions the loss.
• To target PVBP neutrality at ₹5 lakh/bp when current PVBP is ₹5.2 lakh/bp, reduce PVBP by ₹0.2 lakh/bp via futures/swaps while holding key rate exposures steady.
Formulas at a glance
• Macaulay duration (conceptual): PV weighted average time of cash flows.
• Modified duration: Dmod = DMac / (1 + y/m).
• First and second order price estimate: ΔP/P ≈ -Dmod Δy + 1/2 C (Δy)^2.
• Portfolio duration: Dp = Σ wi Di; portfolio convexity: Cp = Σ wi Ci.
• PVBP: PVBP ≈ V × Dmod × 0.0001.
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