Portfolio Duration

Portfolio duration is a measure of how sensitive the value of a portfolio of bonds, loans or other fixed-income assets is to changes in interest rates. It is usually calculated as the average of the durations of the individual holdings, weighted by their market values, and is expressed in years. A portfolio duration of 5 means that, for a small parallel rise of one percentage point in yields, the portfolio's value would fall by roughly 5 per cent.

Fund managers, insurers, pension schemes, bank treasuries and corporate cash managers use portfolio duration to set and report interest rate risk, often against a benchmark index. It is a percentage measure; its money equivalent is basis point value. Like that measure, it assumes all yields move together, so it is complemented by analysis of yield curve risk for moves that change the curve's shape.

Think of a seesaw. The further a weight sits from the middle, the more it moves when the seesaw tips. Bonds that pay back their money far in the future sit far out on the seesaw: when interest rates change, their prices move a lot. Bonds that repay soon sit near the middle and barely move.

Portfolio duration describes the whole seesaw. It takes all the bonds in a fund, notes how far out each one sits and how big it is, and gives one figure for how much the fund as a whole will swing if rates change. A higher number means bigger swings, up or down.

Portfolio duration is the market-value-weighted average of holding durations:

• $D_p$ is the portfolio duration.
• $w_i$ is the market value of holding $i$, including accrued interest, divided by the total portfolio value.
• $D_i$ is the duration of holding $i$, usually modified or effective duration.

Suppose a $100 million portfolio holds $50 million of a two-year note with a duration of 1.9, $30 million of a seven-year bond with a duration of 6.2 and $20 million of a 20-year bond with a duration of 13.5. Then $D_p$ = 0.5 × 1.9 + 0.3 × 6.2 + 0.2 × 13.5 = 0.95 + 1.86 + 2.70 = 5.51. The estimated price change for a change in yield is:

so a 0.50 percentage point rise in yields implies a fall of about 5.51 × 0.005 = 2.76 per cent, or $2.76 million. The same portfolio's basis point value is about $100,000,000 × 5.51 × 0.0001 = $55,100.

• Macaulay duration. The weighted average time to receive the cash flows, in years; for a zero-coupon bond it equals the time to maturity.
• Modified duration. Macaulay duration adjusted for the yield, giving the percentage price change for a small change in yield to maturity.
• Effective duration. Calculated by revaluing holdings under shifted rates, which captures instruments whose cash flows change with rates, such as a callable bond or mortgage-backed securities.

Portfolio duration differs from weighted average maturity, which considers only final maturity dates and ignores the timing of coupons and the effect of yields.

Active managers lengthen duration when they expect rates to fall and shorten it when they expect rates to rise, within limits measured against a benchmark index. Insurers and pension funds match the duration of their assets to that of their liabilities, a strategy known as immunisation, so that changes in rates affect both sides similarly. Managers can adjust duration quickly without trading bonds by buying or selling interest rate futures or interest rate swaps, whose sensitivities are often measured on the interest rate swap curve. Money market funds and corporate treasuries cap duration to keep cash investments stable. Banks compare the duration of assets and liabilities in the banking book as part of managing interest rate risk.

Duration is a linear approximation that works best for small, parallel rate changes. For large moves the curvature of bond prices, called convexity, makes actual changes differ from the estimate. Averaging durations hides how exposure is spread along the curve: a barbell of very short and very long bonds can have the same duration as a portfolio of medium bonds but behave quite differently when the curve twists. Duration also ignores changes in credit spreads, liquidity and currency, and for instruments with embedded options it depends on model assumptions.