## What is Duration?

Duration is a risk measure used by market participants to measure the interest rate sensitivity of a debt instrument, e.g., a Bond. It tells how sensitive a bond is with respect to the change in interest rates. This measure can be used for comparing the sensitivities of bonds with different maturities. There are three different ways to arrive at duration measures, viz. Macaulay DurationMacaulay DurationMacaulay Duration is the amount of time it takes for an investor to recover his invested money in a bond through coupons and principal repayment. This is the weighted average of the period the investor should stay invested in the security in order for the present value of the cash flows from the investment to be equal to the amount paid for the bond.read more, Modified DurationModified DurationModified Duration tells the investor how much the price of the bond will change given the change in its yield. To calculate it, the investor needs to calculate Macauley duration which is based on the timing of the cash flow.read more, and Effective DurationEffective DurationEffective Duration measures the duration of security with options embedded. It helps evaluate the price sensitivity and risk of hybrid securities (bonds and options) to a change in the benchmark yield curve. The modified duration can be called a yield duration.read more.

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For eg:

Source: Duration (wallstreetmojo.com)

### Top 3 Ways to Calculate Duration

There are three different types to calculate durationCalculate DurationThe duration formula measures a bond’s sensitivity to changes in the interest rate. It is calculated by dividing the sum product of discounted future cash inflow of the bond and a corresponding number of years by a sum of the discounted future cash inflow.read more measures,

#### #1 – Macaulay Duration

**The Mathematical Definition:** “Macaulay Duration of a coupon-bearing bond is the weighted average time period over which the cash flows associated with the bond are received.” * *In simple terms, it tells how long it will take to realize the money spent to buy the bond in the form of periodic coupon payments and the final principal repayment.

where:

- Ct: Cashflow at time t
- r: Interest rates/ Yield to maturity
- N: Residual Tenure in Years
- t: Time/ Period in Years
- D: Macaulay Duration

#### #2 – Modified Duration

**The Mathematical Definition:** “Modified Duration is the percentage change in Price of a BondPrice Of A BondThe bond pricing formula calculates the present value of the probable future cash flows, which include coupon payments and the par value, which is the redemption amount at maturity. The yield to maturity (YTM) refers to the rate of interest used to discount future cash flows.read more for a unit change in yield.” It measures the price sensitivity of a bond to changing interest rates. The interest rates are picked from the market yield curve, adjusted for the riskiness of the bondBondBonds refer to the debt instruments issued by governments or corporations to acquire investors’ funds for a certain period.read more and the appropriate tenure.

**Modified Duration = Macaulay Duration / (1+ YTM/f)**

Where:

- YTM: Yield to MaturityYield To MaturityYield to Maturity refers to the expected returns an investor anticipates after keeping the bond intact till the maturity date. In other words, a bond's expected returns after making all the payments on time throughout the life of a bond.read more
- f: Coupon frequency

#### #3 – Effective Duration

If a bond has some options attached to it, i.e., the bond is puttable or callable before maturity. Effective duration takes into consideration the fact that as interest rate changes, the embedded options may be exercised by the bond issuer or the investor, thereby changing the cash flows and hence the duration.

**D _{effective} = – [P_{up} – P_{down} / 2 * Δi * P]**

Where:

- P
_{up}: Bond price with yield up by Δi - P
_{down}: Bond price with yield down by Δi - P: Bond price at current yield
- Δi: Change in yield (usually taken as 100 bps)

### Example of Duration

**Consider a bond with the face value of 100, paying a semi-annual coupon of 7% PA compounded annually, issued on 1 Jan 19 and with a tenure of 5 years and trading at par, i.e., the price is 100 and yield is 7%.**

Calculation of three types of duration is as follows –

**Please download the above Excel template for detailed calculation.**

### Important Points

- As bond price is inversely proportional to yield, it is highly sensitive to how yield changes. The duration measures defined above quantify the impact of this sensitivity on bond price.
- A bond with a longer maturity will have a longer duration; hence, it is more sensitive to changes in interest rates.
- A bond with a lower coupon rate will be more sensitive than a bond with a bigger coupon. However, the reinvestment riskThe Reinvestment RiskReinvestment risk refers to the possibility of failing to induce the profits earned or cash flows into the same scheme, financial product or investment. It even states the uncertainty of not getting the similar returns when such funds are invested in a new investment opportunity.read more will be higher in the case of a small coupon bond.
- Effective duration is an approximate measure of duration, and for an option-free bond, the modified and effective duration will be almost the same.
- Modified duration quantifies the sensitivity by specifying the percentage change in bond price for every 100-bps change in the interest rates.

### Limitations

Although highly used and one of the prominent risk measures for fixed income securities, The duration is restricted for wider use because of underlying assumptions of interest rates movement. It assumes:

- Market yield will be the same for the entire tenure of the bond
- There will be a parallel shift in market yield, i.e., Interest rates changes by the same amount for all the maturities.

Both limitations are handled by considering regime-switching models, which provide for the fact that there can be different yields and volatility for a different period, thereby ruling out the first assumption. And by dividing the tenure of bonds into certain key periods basis, the availability of rates or basis the majority of cash flows lying around certain periods. This helps in accommodating nonparallel yield changes, hence taking care of the second assumption.

### Advantages of Duration Measures

As discussed earlier, a bond with longer maturity is more sensitive to changes in interest rates. This understanding can be utilized by a bond investor to decide whether to stay invested in or sell off the holding. e.g., If Interest rates are expected to go low, an investor should plan to stay long in long term bonds. And if interest rates are expected to go high, short term bonds should be preferred.

These decisions become easier with the use of Macaulay duration as it helps in comparing the sensitivity of bonds with different maturities and coupon rates. Modified duration gives one level deeper analysis of a particular bond by giving the exact percentage by which the prices can change for a unit change in yield.

It measures are one of the key risk measures along with DV01DV01DV01, or dollar value of 1 basis point, measures the interest rate risk of a bond or a portfolio of bonds by estimating the price change in dollar terms in response to a single basis point change in yield (1% comprising 100 basis points).read more PV01s. Thereby, monitoring of portfolio duration becomes all the more important in deciding what kind of portfolio will better suit the investment needs of any financial institution.

### Disadvantages of Duration Measures

As discussed under limitations, duration being one-factor risk metric can go awry in highly volatile markets, in troubled economies. It measures also assume a linear relationship between the price of the bond and interest rates. However, the price – interest rate relation is convex. Hence, this measure alone is not sufficient to estimate sensitivity.

Even after certain underlying assumptions, the duration can be used as an appropriate risk measure in normal market conditions. To make it more accurate, convexity measures can also be incorporated, and an enhanced version of the price sensitivity formula can be used to measure the sensitivity.

**ΔB/B = -D Δy + 1/2 C(Δy) ^{2}**

Where

- ΔB: Change in bond price
- B: Bond Price
- D: Duration of bond
- C: Convexity of the bondConvexity Of The BondConvexity of a bond is a measure that shows the relationship between bond price and yield, and it helps risk management tools to measure and manage a portfolio's exposure to interest rate risk and loss of expectation.read more
- Δy: Change in yield (usually taken as 100 bps)

The Convexity in the above formula can be calculated using the below formula:

**C _{E} = P_{–} + P_{+} – 2P_{0} / 2(Δy)^{2} P_{0}**

Where

- C
_{E}: Convexity of the bond - P_: Bond Price with yield down by Δy
- P
_{+}: Bond Price with yield up by Δy - P
_{o}: Original bond price - Δy: Change in yield (usually taken as 100 bps)

### Recommended Articles

This has been a guide to what is Duration and its definition. Here we discuss the 3 different ways to arrive at duration measures along with an example, advantages & disadvantages. You can learn more about fixed income from the following articles –

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