Alternatively, and often more usefully, convexity can be used to measure how the modified duration changes as yields change. Similar risk measures (first and second order) used in the options markets are the delta and gamma. The Macaulay duration is the weighted average term to maturity of the cash flows from a bond. The weight of each cash flow is determined by dividing the present value of the cash flow by the price.

The modified duration for each series of cash flows can also be calculated by dividing the dollar value of a basis point change of the series of cash flows by the notional value plus the market value. The modified duration is an adjusted version of the Macaulay duration, which accounts for changing yield to maturities. The formula for the modified duration is the value of the Macaulay duration divided by 1, plus the yield to maturity, divided by the number of coupon periods per year. The modified duration determines the changes in a bond’s duration and price for each percentage change in the yield to maturity.

This includes the periodic coupon payments as well as the bond’s maturity value, all adjusted for the time at which they are received. The purpose of Macaulay Duration is to provide investors with a precise macaulay duration and modified duration measure of risk and return trade-off, essential for effective portfolio management. Understanding this concept impacts investors as it helps gauge the potential volatility of a bond or bond portfolio.

Lastly, we want to think about the relationship between Macaulay duration and yield to maturity. If we look at coupon payments of a fixed-rate bond, we can also see how two similar bonds with different coupon rates can have different duration measures. However, in the markets, duration can also be understood to be a measure of how much a bond price will move given changes in the yield-to-maturity. This interpretation is more correctly called “dollar duration” but market participants stubbornly tend to use this duration definition the most. Since the interest rate is one of the most significant drivers of a bond’s value, duration measures how much changes in the yield-to-maturity (YTM) of the instrument will ultimately impact the bond’s price. Bond duration is a linear estimate of a bond’s price sensitivity to changes in market yield.

- For those who work on Wall Street trading desks though, the concept of modified duration was still not direct enough.
- The purpose of Macaulay Duration is to provide investors with a precise measure of risk and return trade-off, essential for effective portfolio management.
- This is due to their higher Interest Rate Sensitivity and the inverse relationship between Bond Prices and Interest Rates, i.e., an increase in Interest Rates leading to lower Bond Prices.

In contrast, the modified duration identifies how much the duration changes for each percentage change in the yield while measuring how much a change in the interest rates impacts the price of a bond. Modified duration determines the change in the value of a fixed income security in relation to a change in the yield to maturity. The formula used to calculate a bond’s modified duration is the Macaulay duration of the bond divided by 1 plus the bond’s yield to maturity divided by the number of coupon periods per year. In this case the BPV or DV01 (dollar value of an 01 or dollar duration) is the more natural measure. The BPV in the table is the dollar change in price for $100 notional for 100bp change in yields.

## Impact of Changes in Interest Rates on Bonds With Different Macaulay Durations

For this bond, the Macaulay duration is 2.856 years, heavily weighted towards maturity (3 years). If you have all of the details of the bond and know the market price, click the blue “You Know Market Price” button. Most investors include Debt Funds in their investment portfolio in order to improve the overall stability of their portfolio. This is because Debt investments can help cushion the potential volatility of an investment portfolio due to Equity exposure. That is why investors get very confused when they suddenly see their Debt investments giving negative returns.

## Example: Compute the Macaulay Duration for a Bond

In summary, Macauley duration is a weighted average maturity of cash flows (measured in units of time) and is useful in portfolio immunization where a portfolio of bonds is used to fund a known liability. Modified duration is a price sensitivity measure and is the percentage change in price for a unit change in yield. Modified duration is more commonly used than Macauley duration and is a tool that provides an approximate measure of how a bond price will change given a modest change in yield. For larger changes in yield, both the modified duration and convexity are used to better approximate how a bond price will change for a given change in yield.

It should be remembered that, even though Macaulay duration and modified duration are closely related, they are conceptually distinct. Macaulay Duration serves as a link between bond prices and interest rates, as it measures how sensitive a bond’s price is to changes in interest rates. It is a measure of the weighted average time until a bond’s cash flows are received. The duration of a bond measures sensitivity of a bond’s full price (inclusive of accrued interest) to changes in interest rates if all other factors are held constant.

## What is the approximate value of your cash savings and other investments?

Formally, modified duration is a semi-elasticity, the percent change in price for a unit change in yield, rather than an elasticity, which is a percentage change in output for a percentage change in input. Modified duration is a rate of change, the percent change in price per change in yield. This represents the bond discussed in the example below – two year maturity with a coupon of 20% and continuously compounded yield of 3.9605%. 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. If these circles were put on a balance beam, the fulcrum (balanced center) of the beam would represent the weighted average distance (time to payment), which is 1.78 years in this case.

Third, as interest rates increase, duration decreases, and the bond’s sensitivity to further interest rate increases goes down. The modified duration provides a good measurement of a bond’s sensitivity to changes in interest rates. The higher the Macaulay duration of a bond, the higher the resulting modified duration and volatility to interest https://1investing.in/ rate changes. Modified duration illustrates the concept that bond prices and interest rates move in opposite directions – higher interest rates lower bond prices, and lower interest rates raise bond prices. This occurs when the duration and the yield of a bond decrease or increase together, thus, they are positively correlated.

The yield curve for bonds with positive convexities usually follows an upward movement. Duration is commonly used in the portfolio and risk management of fixed income instruments. Using interest rate forecasts, a portfolio manager can change a portfolio’s composition to align its duration with the expected movement of interest rates. This is because the issuer can redeem the old bond at a high coupon and re-issue a new bond at a lower rate, thus providing the issuer with valuable optionality. Similar to the above, in these cases, it may be more correct to calculate an effective convexity.

As you can see in the figure above, the duration of a bond is not the same as its maturity. As a matter of fact, for coupon-paying bonds, the duration of that bond will always be shorter than the term to maturity of that bond. Next, we give a time weight to each of the present values by multiplying the PVCFt by the time period (i.e., 1 for the first year’s cash flows, 2 for the second year’s cash flows, and 3 for the last year’s cash flows). If you imagine this entire cash flow diagram being put on a see-saw, duration is the point where the cash flow balances, also known as a fulcrum.

Hence, if interest rates are expected to rise, an investor may prefer bonds with shorter Macaulay Durations to limit price volatility. Effective Duration takes into consideration the potential changes in cash flows that result from changes in interest rates. Macaulay Duration quantifies this inverse relationship and helps investors understand how much a bond’s price will change with a change in interest rates.

## Duration

For example, if investors expect interest rates to decrease, they may want to invest in bonds with longer durations to maximize price appreciation. I believe the Macaulay duration is the effective time a bond is due to be repaid in years using a weighted average of future coupons/cashflows. Modified duration could be extended to calculate the amount of years it would take an interest rate swap to repay the price paid for the swap. An interest rate swap is the exchange of one set of cash flows for another and is based on interest rate specifications between the parties.

Macaulay duration is the weighted average of the time to receive the cash flows from a bond. Macaulay duration tells the weighted average time that a bond needs to be held so that the total present value of the cash flows received is equal to the current market price paid for the bond. This equation approximates the slope of a line tangent to the price-yield curve by calculating price when yield decreases PV− and price when yield increases PV+.