Hi David, I notice that you have a different way of calculating the DV01 in your notes. Indeed, Tuckman defines it as follows: (1/)*(dP/dy). There are two items that must be clarified with respect to your question: Are you assuming an interest rate swap (IRS) at mid-market, i.e. at-the-money (ATM) or. In finance, the duration of a financial asset that consists of fixed cash flows, for example a bond, The formula can also be used to calculate the DV01 of the portfolio (cf. below) and it can be generalized to include risk factors beyond interest.

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Many technical issues arise for key rate durations partial DV01s that do not arise for the standard total modified duration because of the dependence of the key rate durations on the specific type of the yield curve used to value the instruments see Coleman, [3]. Resources Latest reviews Search resources.

For large yield changes, calculaton can be added to provide a quadratic or second-order approximation. It should be remembered that, even though Macaulay duration and modified duration are closely related, calculayion are conceptually distinct. It is possible to derive the calcupation you seek in analytic terms but is rather complicated. What’s new New posts New resources New profile posts Latest activity.

Modified duration can be extended to instruments with non-fixed cash flows, while Macaulay duration applies only to fixed cash flow instruments. The circles represent the present value of the payments, with the coupon payments getting smaller the further in the future they are, and the final large payment including both the coupon payment and the final principal repayment. Specifically, duration can db01 formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative.

The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash calculationn, and can thus be applied to a wider range of instruments than can Macaulay duration. Duration is a linear measure of how the price of a bond changes in response to interest rate changes.

By using this site, you agree to the Terms of Use and Privacy Policy. Modified duration is a useful measure to compare interest rate sensitivity across the three. Many thanks for your prompt answer David. Modified duration applies when a bond or other asset is considered as a function of yield.

Consider some set of fixed cash flows. This formula can be used to calculate the VaR of the portfolio by ignoring higher order terms. 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. These values are typically calculated using a tree-based model, built for the entire yield curve as opposed to a single yield to maturityand 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.

Why Take the Exam? Macaulay durationnamed for Frederick Macaulay who introduced the concept, is the weighted average maturity of cash flows. In terms of standard bonds for which cash flows are fixed and positivethis means the Macaulay duration will equal the bond maturity only for a zero-coupon bond. Just as importantly, IMO it is worth the trouble to understand what is going on here.

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Check date values in: The BPV will make sense for the interest rate swap for which modified duration is not defined as well as the three bonds. This page was last edited on 21 Octoberat Convexity also gives an idea of the spread of future cashflows.

There are two items calculztion must be clarified with respect to your question: To price such bonds, one must use option pricing to determine the value of the bond, and then one can compute its delta and hence its lambdawhich is the duration.

Note that convexity can be positive or negative. In this case one can measure the logarithmic derivative with respect to yield:. For example, the annuity above has Macaulay duration of 4.

### formula for physical DV01 of interest rate swap – Quantitative Finance Stack Exchange

Then expression 2 becomes:. Thus the index, or underlying yield curve, remains unchanged. For every-day use, the equality or near-equality of the values calculahion Macaulay and modified duration can be a useful aid to intuition.

The yield-price relationship is inverse, and the modified duration provides a very useful measure of the price sensitivity to yields.

## Bond duration

This bond’s price sensitivity to interest rate changes is different from a non-puttable bond with otherwise identical cashflows. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates.

In symbols, if cash flows are, in order, t 1. Quadratic terms, when included, can be expressed in terms of multi-variate bond convexity.

The DV01 is analogous to the delta in derivative pricing The Greeks — it is the ratio of a price change in output dollars to unit change in input a basis point of yield.

## DV01 Computation

In financethe duration of a financial asset that consists of fixed cash flowsfor example a bondis the weighted average of the times until those fixed cash flows are received. Strictly speaking, Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Sign up or log in Sign up using Google. Weil; Journal of Business,44 4pp. Are you interested in a risk approximation or a more accurate formula that reflects the truest risk sense of a market curve shifted up or down?

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Modified duration is defined as the logarithmic derivative of price with respect to yield, and such a definition will apply to instruments that depend on yields, whether or not the cash flows are fixed.

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

Modified duration is a rate of change, the percent change in price per change in yield. The duration of a portfolio equals the weighted average maturity of all of the cash flows in the portfolio.

One can make assumptions about the joint distribution of the interest rates and then calculate VaR by Monte Carlo simulation or, in some special cases e.

Post as a guest Name. Foreign exchange Currency Exchange rate. Modified duration is also useful as a measure of the sensitivity of a bond’s market price to finite interest rate i.