In finance , the duration of a financial asset that consists of fixed cash flows , for example a bond , is the weighted average of the times until those fixed cash flows are received. When the price of an asset is considered as a function of yield , duration also measures the price sensitivity to yield, the rate of change of price with respect to yield or the percentage change in price for a parallel shift in yields. 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. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield.

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This chapter discusses one-factor risk metrics, which include DV01, duration, and convexity, as used in the analysis of fixed-income portfolios. We consider these measures to quantify the consequence of a parallel shift in the interest rate term structure: DV01 and duration consider a small parallel shift in the term structure while convexity extends the duration to accommodate larger parallel shifts.

Hedging can, in accordance with these risk metrics, be efficient for a parallel shift of term structure as compared to non-parallel shifts.

The one-factor assumption states that when the rates are driven by one factor, the change of one interest rate can be used to determine the change in all other interest rates over a short period of time. For instance, a one-factor model assumes that all interest rates changes by the same amount.

As such, the shape of the term structure never changes. That is, if a one-year spot rate increases by two basis points, all other spot rates increase by two basis points.

DV01, duration, and convexity are examples of such models. It is worth noting that shifts in the term structure are not always parallel. For instance, a one-factor model might predict that if one-year spot rate increases by five basis points over a short period, then the two year increases by three basis points and the ten-year rate increases by one basis point.

DV01 reflects the effect of a one-basis movement in interest rates on the value of a portfolio. DV01 is defined as:. The spot rates, as shown in the table below:. If the spot rates are each increased by 5 basis points 0. Note that the rise of the spot rates by 5 basis points decreases the bond price by Assume now that the spot rates decrease by 5 basis points so that the six-month spot rate is 5.

Under decreased spot rates, the price of the bond is USD , As such, an increase of spot rates by five basis points causes the price of the bond to increases by It is worth noting that the DV01 for the decrease and increases of the basis points are slightly different because the bond price is not a linear function of interest rates.

We estimate the DV01 by averaging the estimates above:. Assume that now bond yield increases by one basis point. Consider the spot rates, as in the example above. The spot rates, as shown in the table above. Recall that, using the spot rates, we calculated the bond price to be , To find the yield to maturity of the bond bond yield , we solve the equation:. Now, if the bond yield increases by 5 basis point to 7. We can, therefore, calculate the DV Similarly, if the bond yield decreases by 5 basis points to 7.

A decrease of bond yield by 5 basis points increases the bond price by In the cases of the forward rates, the analogy is similar to that of the spot rates. However, this chapter will primarily emphasize on DV01 computed from a one-basis-point parallel shift in the term structure of spot rates.

In any position that depends on the interest, DV01 can be computed efficiently. DV01 can, therefore, be used to hedge a position. By the definition of DV01, the banks will gain from their position if interest rates increases and will undoubtedly lose value if interest rates decrease.

The goal of hedging is to lock in the value of a position even in the face of small changes in yield. The hedge ratio is given by:. Assume that DV01 of the coupon bond is In this case, we need to increase the value of the position by:.

Thus, adding the USD , Effective duration measures the percentage change in the price of a bond or other instruments caused by small changes in all rates. Note that effective duration is different from the DV01 because DV01 measures actual price changes against small changes in all rates. When the change in all rates is measured basis points, the effective duration is equivalent to DV01 divided by the bond price.

Consider the DV01 example on the spot rates where we had calculated the price of the bond USD , Effective duration gives the proportional change in the price of an instrument corresponding to one-basis point change in all interest rates. Typically, the effective duration is stated as a percentage change in the price of an instrument for a basis-change in all rates by multiplying the effect of one-basis-point change by Therefore, our example above would be stated as 1.

However, for the sake of clarity, the duration in this chapter will be reported as a decimal. In the decimal reporting system, one basis point is equivalent to 0. Therefore, the duration calculated above will be:.

A callable bond is one that an issuing party has the right to purchase back the bond at a pre-determined price at a particular time in the future before the maturity period of the bond.

The call feature of the bond should not be ignored as it reduces the duration of the bond. A practical approach to address the callable feature of a bond while calculating duration is outlined as follows:. A puttable bond is one that a holder has a right to ask for early repayment. Calculating the effective duration of a puttable bond is similar to that of a callable bond. That is, the probability of the put options increases with the increase of the interest rates.

DV01 is useful in measuring the effect of all rate changes on the value of a position. DV01 is also appropriate in measuring the changes in swaps and interest rate futures. Effective duration is appropriate in measuring the effect of rate changes on the value of a position as a percentage. Effective duration is an appropriate measure in bond valuation.

DV01 increases with an increase in the position size, but effective duration does not. In other words, if the value of a position is doubled, DV01 doubles, but effective duration does not.

Convexity measures the sensitivity of duration measure to movement in the interest rates. Denoted convexity by C and value of a position by P, convexity is defined as by:. As computed earlier, the price of a bond with no interest spread is USD , The duration of a bond is a linear relationship between the bond price and interest rates where, as interest rates increase, bond price decreases.

However, for much larger changes in yield, the duration is not so desirable since the relationship between price and interest rates tends to be non-linear. As seen on the graph, the linear approximation by effective duration becomes less efficient as the rate changes become large. The combination of the duration and convexity is a quadratic function which provides a better estimation.

As such the price change is given by. By combining the duration and the convexity, we can reach a convenient estimate, even for large changes.

Consider the example in the DV01 section. Note that we had calculated the DV01 for the 5 basis point movement of all the spot rates to be Because all the spot rates increase by 30 basis points 0. So the price due to new rates is given by the formula:.

The estimated price change provided by the duration is a good approximation of the price change, but we can improve it by combining duration and the convexity. Hedging using effective duration analogous to that of DV By the definition of effective duration,. These small parallel shifts are hedged against if:. A bank has a position of USD 12 million with an effective duration of 5 and a bond with an effective duration of 6. How will the bank hedge against its position?

Therefore, the bank should short bonds worth 10 million to hedge against its position. This is true because:. Hedging based on both duration and convexity is a complex undertaking where we are trying to make both effective duration and convexity. We need two bonds.

Now by the definition of approximating the price change using a combination of the duration and convexity we have:. A bank has a position of USD 12 million with an effective duration of 5 and a convexity of 4.

The bank wishes to hedge its position with two bonds where the first bond has an effective duration of 6 and a convexity of 9. The second bond has a duration of 4 and a convexity of 7. Therefore, for the bank to hedge its position, it must take a position of USD 8. In other words, by combining these positions in bonds, there is no duration or convexity exposure, and thus the bank is hedged against large parallel shifts in the term structure though it will be exposed to non-parallel shifts.

Consider a bond whose price is P and yield is y. Now, the yield-based duration is defined as the proportional change in the bond price due to a small change in the yield. More precisely,. As such, the duration will measure the time an investor must wait to receive cash flows.

Typically, the yield on the bond is given as a semi-annual compounding rate rather than continuous compounding. Modified duration is slightly different from an effective duration.

Recall convexity measures the sensitivity of duration measure to movement in the interest rates. In the calculus context, yield based convexity is the derivative if the duration. The last expression is termed as Modified convexity, and it is slightly different from the effective convexity.

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## Quick Guides > Basis Point Value (BPV, DV01)

DV01 or Dollar Value of 1 basis point measures the interest rate risk of bond or portfolio of bonds by estimating the price change in dollar terms in response to change in yield by a single basis point One percent comprise of basis points. The calculation of Dollar Value of one basis point aka DV01 is very simple and there are multiple ways to calculate it. One of the most common formulas used to calculate DV01 is as follows:. It is important to note here that we are dividing by because DV01 is based on linear approximation but is one basis point which is 0. Ryan is holding a US Bond with a yield of 5. The yield on the Bond declines to 5. Based on the information lets calculate DV01 using the formula stated above:.

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## Dollar Duration

The DV01 or the dollar value of 1 basis point, also referred to as bpv or basis point value. This is a duration related metric in determining the interest rate sensitivity of a bond. The metric shows by how much the price in the bond changes by 1 basis point change in the interest rate. Its use is widespread among dealers and traders since they require to know the dollar amount of their portfolio on a daily basis. The DV01 of a portfolio can simply be calculated as the weighted average of all DV01 of every positions in which the weights represent the value of the position compared to the whole portfolio. The DV01 value is calculated by taking the negative of the change in the bond price divided by