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# What is duration?

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Often confused with maturity, duration is the weighted average maturity period of the cash flow or of any series of linked cash flows on a rate-sensitive instrument.  It reflects the timing and size of cash flows that occur before the instrument’s contractual maturity and is expressed either as Macaulay duration or modified duration.

Macaulay duration (DMAC) is the sum of the present value of all future cash flows of a rate-sensitive instrument after multiplying each of its cash flows by the time period until it occurs and then dividing that product by the sum of the present value of the total future cash flows, where PvCf is the present value of the periodic cash flow, and n is the total number of discounting periods:

DMAC = (PvCf1 x 1) + (PvCf2 x 2) + … + (PvCfn x n)/PvCfTotal

Modified duration (DMOD) is a metric that estimates the price change of an option-free rate-sensitive instrument or portfolio for interest rate changes of up to 100 bps (1%).  It is a derivative of Macaulay duration (DMAC) and is computed by dividing Macaulay duration by the yield to maturity (y) and the number of discounting periods (n):

DMOD = DMAC/(1 + y/n)

DMOD expresses the percentage change in the value of a fixed-income instrument that has no option features, for each one-percentage point change in interest rates.  An instrument’s DMOD multiplied by the percentage change in interest ratesr%) approximates the percentage change in the instrument’s price (ΔP%):

ΔP% ≅ DMOD x Δr%

If, for example, an asset has a modified duration of 5 years and interest rates rise by 100 basis points (bps), from 5% to 6%, the asset’s drop in price will be roughly 5%.  Because modified duration is additive, it can be used to measure and limit the risk of a bond portfolio.

 Duration of a Portfolio (Example) Assuming the weighted duration of a portfolio is 4.04 and interest rates were to increase by 1%, the market value of the portfolio would decline by approximately 4.04%. Instrument YTM Price/ Amount DMOD Weighted Duration 7.5% 2-year note 10% 95,567 1.8 0.6 8% 5-year note 10% 92,278 3.98 1.28 10% 10-year note 10% 100,000 6.23 2.16 Duration of Portfolio 4.04

Effective duration allows for the calculation of a bond’s price movement given the existence of an embedded call or put option, it taking into account the effect of option exercise on the expected life of the bond.  It weighs the probability that the option will be exercised based on the spread between its coupon rate and its yield as well as the volatility of interest rates.  It is commonly used to report the duration of portfolios containing mortgage-backed securities.

All duration measures assume a linear price/yield relationship, although the relationship actually is curvilinear.  Therefore, duration may only accurately estimate price sensitivity for rather small interest-rate changes of up to 100 basis points.  Convexity-adjusted duration should be used to more accurately estimate price sensitivity for larger interest-rate changes of over 100 bps.