# Duration And Convexity, With Illustrations And Formulas

Note that i is the change in the term structure of interest rates and not the yield to maturity for the bond, because YTM is not valid for an option-embedded bond when the future cash flows are uncertain. This bond duration tool can calculate the Macaulay duration and modified duration based on either the market price of the bond or the yield to maturity of the bond. The yield-price relationship is inverse, and the modified duration provides a very useful measure of the price sensitivity to yields. For large yield changes, convexity can be added to provide a quadratic or second-order approximation. Alternatively, and often more usefully, convexity can be used to measure how the modified duration changes as yields change. Similar risk measures used in the options markets are the delta and gamma.

### What is effective duration formula?

Y = the estimated change in yield used to calculate P(1) and P(2). The complete formula for effective duration is: Effective duration = (P(1) – P(2)) / (2 x P(0) x Y)

The Macaulay duration is the weighted average time until all the bond’s cash flows are paid. By accounting for the present value of future bond payments, the Macaulay duration helps an investor evaluate and compare bonds independent of their term or time to maturity.

## Usefulness Of Duration

Let us assume that company XYZ Ltd has issued a bond having a face value of \$100,000, carrying an annual coupon rate of 7% and maturing in 5 years. Discounted Future Cash InflowDiscounted cash flow analysis is a method of analyzing the present value of a company, investment, or cash flow by adjusting future cash flows to the time value of money. This analysis assesses the present fair value of assets, projects, or companies by taking into account many factors such as inflation, risk, and cost of capital, as well as analyzing the company’s future performance. Duration measures a bond price’s sensitivity to changes in interest rates—so why is it called duration? A bond with a longer time to maturity will have a price that is more sensitive to interest rates, and thus a larger duration than a short-term bond. The first part is used to find the present value of all future bond cash flows. The second part finds the weighted average time until those cash flows are paid.

Note that modified duration is always slightly less than duration, since the modified duration is the duration divided by 1 plus the yield per payment period. While bond investing is generally less risky than stock investing, factors like duration become important considerations if you decide to sell a bond before its maturity date. “The higher the duration, the more sensitive the bond is to interest rate changes, and thus, the more prone it is to interest rate risk.” From equities, fixed income to derivatives, the CMSA certification bridges the gap from where you are now to where you want to be — a world-class capital markets analyst. A full analysis of the fixed-income asset must be done using all available characteristics.

## What Is Duration?

Bond duration is a measurement that tells us how much a bond’s price might change if interest rates fluctuate. Its full definition is actually a little more technical than that since duration measures how long it will take an investor to be repaid a bond’s price from the cash flows it produces. After all, bond investors are placing loans to a borrower (a corporation, the federal government, etc.), which must be repaid with interest by their maturity date. Key rate durations are a natural extension of the total modified duration to measuring sensitivity to shifts of different parts of the yield curve. Key rate durations might be defined, for example, with respect to zero-coupon rates with maturity ‘1M’, ‘3M’, ‘6M’, ‘1Y’, ‘2Y’, ‘3Y’, ‘5Y’, ‘7Y’, ’10Y’, ’15Y’, ’20Y’, ’25Y’, ’30Y’.

• As a bond’s duration rises, its interest rate risk also rises because the impact of a change in the interest rate environment is larger than it would be for a bond with a smaller duration.
• 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 rates.
• The first person gratefully accepted your loan for a period of 5 years, and the second person took it for 10 years.
• Dollar duration measures the dollar change in a bond’s value to a change in the market interest rate, providing a straightforward dollar-amount computation given a 1% change in rates.

Duration is a linear measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates.

## What Is The Duration Formula?

Otherwise the weighted average of the bond’s durations is just a good approximation, but it can still be used to infer how the value of the portfolio would change in response to changes in interest rates. (\$ per 1 basis point change in yield)The DV01 is analogous to the delta in derivative pricing (one of the “Greeks”) – it is the ratio of a price change in output to unit change in input . Dollar duration or DV01 is the change in price in dollars, not in percentage.

Expression which uses the bond’s yield to maturity to calculate discount factors. The most common are the Macaulay duration, modified duration, and effective duration.

## Example: Compute The Macaulay Duration For A Bond

Bonds with high duration are more prone to the ups and downs of market volatility that happens in an environment of changing interest rates. Bonds with the highest durations are typically long-term bonds with low coupons.

For example, a standard ten-year coupon bond will have a Macaulay duration of somewhat but not dramatically less than 10 years and from this, we can infer that the modified duration will also be somewhat but not dramatically less than 10%. Similarly, a two-year coupon bond will have a Macaulay duration of somewhat below 2 years and a modified duration of somewhat below 2%. Now, the total number of periods till maturity is computed by multiplying the number of years till maturity and the frequency of the coupon payments in a year. Also, the time of the periodic payment is noted, which is denoted by i. However, a long-duration strategy describes an investing approach where a bond investor focuses on bonds with a high duration value. In this situation, an investor is likely buying bonds with a long time before maturity and greater exposure to interest rate risks.

## Average Duration

Convexity is the rate that the duration changes along the price-yield curve, and, thus, is the 1st derivative to the equation for the duration and the 2nd derivative to the equation for the price-yield function. Consequently, bonds with higher convexity will have greater capital gains for a given decrease in yields than the corresponding capital losses that would occur when yields increase by the same amount.

In terms of standard bonds , this means the Macaulay duration will equal the bond maturity only for a zero-coupon bond. The modified duration figure indicates the percentage change in the bond’s value given an X% interest rate change. Unlike the Macaulay duration, modified duration is measured in percentages. In case investors are seeking benefits from a fall in interest rate, the investors will intend to buy bonds with a longer duration, which is possible in the case of bonds with lower coupon payment and long maturity. On the other hand, investors who want to avoid the volatility in interest rate, the investors will be required to invest in bonds that have a lower duration or short maturity and higher coupon payment. Convexity is a measure of the relationship between bond prices and bond yields that shows how a bond’s duration changes with interest rates.

When yields are continuously compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically compounded Macaulay and modified duration will differ slightly, and there is a simple relation between the two. Fisher–Weil duration is a refinement of Macaulay’s duration which takes into account the term structure of interest rates.

• Macaulay duration has the diagrammatic interpretation shown in figure 1.
• Market PriceMarket price refers to the current price prevailing in the market at which goods, services, or assets are purchased or sold.
• A bond with a longer time to maturity will have a price that is more sensitive to interest rates, and thus a larger duration than a short-term bond.
• If the YTM rises, the value of a bond with 20 years to maturity will fall further than the value of a bond with five years to maturity.
• Bonds are fixed-income securities that are issued by corporations and governments to raise capital.
• Speaking of maturity dates, it’s important to note that bond duration and maturity are not the same thing.

In order to understand modified duration, keep in mind that bond prices are said to have an inverse relationship with interest rates. Therefore, rising interest rates indicate that bond prices are likely to fall, while declining interest rates indicate that bond prices are likely to rise. Unlike Macaulay’s duration, modified duration is not measured in years. Modified duration measures the expected change in a bond’s price to a 1% change in interest rates. Duration is an effective analytic tool for the portfolio management of fixed-income securities because it provides an average maturity for the portfolio, which, in turn, provides a measure of interest rate risk to the portfolio.

It is a measure of the time required for an investor to be repaid the bond’s price by the bond’s total cash flows. Price Of The 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 refers to the rate of interest used to discount future cash flows. This gives the well-known relation between Macaulay duration and modified duration quoted above. It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay duration is a weighted average time until repayment while modified duration is a price sensitivity measure when the price is treated as a function of yield, the percentage change in price with respect to yield. The duration of a bond in practice can refer to two different things.

Alternatively, for every 1% decrease in interest rates, the bond will increase 1% for every year of duration. When a coupon is added to a bond, the duration will always be less than its maturity. It is easier to immunize a bond portfolio when the duration of the portfolio is matched to the need for funds. For example, if funds are needed at year five, then the duration of the portfolio should equal five. Is the present value of all cash payments from the asset until maturity. For this bond, the Macaulay duration is 2.856 years, heavily weighted towards maturity . Yield to Maturity (%) – The yield of the bond if held until maturity , as implied by the market.

For small changes in yield, it is very accurate, but for larger changes in yield, it always underestimates the resulting bond prices for non-callable, option-free bonds. Interest rates vary continually from high to low to high in an endless cycle, so buying long-duration bonds when yields are low increases the likelihood that bond prices will be lower if the bonds are sold before maturity. This is sometimes called duration risk, although it is more commonly known as interest rate risk.

It is equal to the maturity if and only if the bond is a zero-coupon bond. Bond managers will often want to know how much the market value of a bond portfolio will change when interest rates change by 1 basis point. At bond trading desks, trading exposure is often set in terms of the BPV. On the surface, it can seem pretty confusing, but for most investors, the main takeaway is that bond duration predicts how sharply the market price of a bond will change as a result of changes in interest rates.