Duration (finance)

[1][2][3] The dual use of the word "duration", as both the weighted average time until repayment and as the percentage change in price, often causes confusion.

Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received and is measured in years.

[4] Both measures are termed "duration" and have the same (or close to the same) numerical value, but it is important to keep in mind the conceptual distinctions between them.

[5] Macaulay duration is a time measure with units in years and really makes sense only for an instrument with fixed cash flows.

(Price sensitivity with respect to yields can also be measured in absolute (dollar or euro, etc.)

[7] Macaulay duration, named for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows, in which the time of receipt of each payment is weighted by the present value of that payment.

For a set of all-positive fixed cash flows the weighted average will fall between 0 (the minimum time), or more precisely

This represents the bond discussed in the example below - two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%.

[11] Similarities in both values and definitions of Macaulay duration versus Weighted Average Life can lead to confusing the purpose and calculation of the two.

[12] For example, a 5-year fixed-rate interest-only bond would have a Weighted Average Life of 5, and a Macaulay duration that should be very close.

[17] To see this, if we take the derivative of price or present value, expression (2), with respect to the continuously compounded yield

Modified duration is defined above as a derivative (as the term relates to calculus) and so is based on infinitesimal changes.

Modified duration is also useful as a measure of the sensitivity of a bond's market price to finite interest rate (i.e., yield) movements.

Fisher–Weil duration calculates the present values of the relevant cashflows (more strictly) by using the zero coupon yield for each respective maturity.

Reitano covered multifactor yield curve models as early as 1991 [23] and has revisited the topic in a recent review.

Consider a bond with a $1000 face value, 5% coupon rate and 6.5% annual yield, with maturity in 5 years.

Compare the total from step 2 with the bond value (step 1) Macaulay duration: 4246.63 / 937.66 = 4.53 The money duration, or basis point value or Bloomberg Risk[citation needed], also called dollar duration or DV01 in the United States, is defined as negative of the derivative of the value with respect to yield: so that it is the product of the modified duration and the price (value): or The DV01 is analogous to the delta in derivative pricing (one of the "Greeks") – it is the ratio of a price change in output (dollars) to unit change in input (a basis point of yield).

This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms.

Thinking of risk in terms of interest rates or yields is very useful because it helps to normalize across otherwise disparate instruments.

The annuity has the lowest sensitivity, roughly half that of the zero-coupon bond, with a modified duration of 4.72%.

The BPV will make sense for the interest rate swap (for which modified duration is not defined) as well as the three bonds.

If we want to measure sensitivity to parts of the yield curve, we need to consider key rate durations.

For large yield changes, convexity can be added to provide a quadratic or second-order approximation.

Similar risk measures (first and second order) used in the options markets are the delta and gamma.

Modified duration and DV01 as measures of interest rate sensitivity are also useful because they can be applied to instruments and securities with varying or contingent cash flows, such as options.

No matter how high interest rates become, the price of the bond will never go below $1,000 (ignoring counterparty risk).

The effective duration is a discrete approximation to this latter, and will require an option pricing model.

These values are typically calculated using a tree-based model, built for the entire yield curve (as opposed to a single yield to maturity), and therefore capturing exercise behavior at each point in the option's life as a function of both time and interest rates; see Lattice model (finance) § Interest rate derivatives.

On the other hand, a bond with call features - i.e. where the issuer can redeem the bond early - is deemed to have negative convexity as rates approach the option strike, which is to say its duration will fall as rates fall, and hence its price will rise less quickly.

[30] The ratio is simply the yield offered (as a percentage), divided by the bond duration (in years).