Use the yield curve and duration to apply maturity strategies and the tools for managing maturity in an investment portfolio.
The idea
The yield curve guides maturity choice and duration measures the price risk of that choice. Duration is the weighted average time to cash flows. Modified duration gives the percent price change per unit yield change, so the price change is about minus modified duration × yield change. Portfolio duration is the market-value-weighted average of holdings' durations. Maturity strategies are a ladder (equal amounts across maturities), a barbell (short and long, nothing in the middle), a bullet (one maturity) and the rate-expectations approach. That approach shortens maturities when rates are expected to rise and lengthens them when rates are expected to fall. On an upward-sloping curve a bank can ride the yield curve, buying longer paper and selling it after it has rolled down to a lower yield. Tools include swaps and futures. Let Dt be the target duration, Dp the portfolio duration, Vp the portfolio value, Df the futures duration and Vf the contract value. N = (Dt − Dp) × Vp ÷ (Df × Vf). A negative result means sell contracts. Match duration, not maturity.
Formulas
Portfolio duration
portfolio duration = Σ wᵢ × Dᵢ
- wᵢ
- the share of the portfolio's market value held in bond i
- Dᵢ
- the duration of bond i
Market-value weights, not a simple average. It assumes all yields move by the same amount.
Futures contracts to reach a target duration
N = (Dt − Dp) × P ÷ (Df × F)
- Dt
- the target duration
- Dp
- the portfolio's current duration
- P
- the portfolio value
- Df
- the duration of the futures, usually the cheapest-to-deliver
- F
- the value of one futures contract
Assumes equal yield moves for portfolio and futures. A negative N means sell contracts to shorten duration.
A practice question
Which description matches a laddered maturity strategy?
- Holding securities only at the short and long ends of the maturity range, with none in the middle.Why not: Describes a barbell strategy, not a ladder.
- Placing the whole portfolio in securities of a single maturity.Why not: Describes a bullet strategy, concentrated at one maturity point.
- Moving maturities longer or shorter as the bank's interest rate forecast changes.Why not: Describes the rate-expectations (active) approach, not a ladder.
- Spreading holdings about equally across a range of maturities, so a similar amount matures each period and is reinvested. - correct
A ladder divides funds roughly evenly across maturities, so each period a rung matures and is rolled to the longest rung. It needs no rate forecast and smooths reinvestment.
A harder one
The yield curve is upward sloping and expected to stay unchanged. Six-month and one-year bills (simple bond-equivalent yields, price = 1 / (1 + y x t)) yield 4.00% and 4.60%. A bank buys the one-year bill, sells it after six months (when it has six months left and yields 4.00%) and annualizes the six-month return by doubling it. The annualized holding return is closest to:
- 4.60%Why not: Reports the one-year bill's yield and ignores the price gain from rolling down the curve.
- 4.00%Why not: Reports the six-month yield, which is the return on buying the short bill, not on riding the curve.
- 5.10% - correct
- 2.55%Why not: Reports the six-month return without annualizing it.
Buy price = 1/1.046 = 0.95602. Sell price = 1/(1 + 0.04 x 0.5) = 0.98039. Six-month return = 2.549%; doubled = 5.10%. Riding the yield curve earns more than 4.00% because the bill rolls down to a lower yield.
The hardest tier
A bank wants to hold USD 100 million in a barbell of 2-year notes (modified duration 1.9) and 10-year notes (modified duration 8.2) to reach a portfolio duration of 5.5. The amount that must be invested in the 10-year notes is closest to:
- USD 42.86 millionWhy not: Computes the weight of the 2-year notes (0.4286) rather than the 10-year weight.
- USD 50.00 millionWhy not: Splits equally between the two notes, as if the target were the midpoint of maturities.
- USD 57.14 million - correct
- USD 67.07 millionWhy not: Divides the target duration by the long duration (5.5/8.2), ignoring the short note's duration contribution.
Let w be the 10-year weight: 1.9 x (1 - w) + 8.2 x w = 5.5, so w = (5.5 - 1.9)/(8.2 - 1.9) = 0.5714. Amount = USD 57.14 million. Duration is linear in weights, so the target can be reached by many barbell mixes.
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