Macaulay Duration

What is Macaulay Duration?

Macaulay Duration is the length of time taken by the investor to recover his invested money in the bond through coupons and principal repayment. This length of time is the weighted average of the period the investor should remain invested in the security in order to have the present value of the cash flows from the investment match the amount paid for the bond.

Macaulay Duration is a very important factor to consider before buying a debt instrument. It can greatly help investors choose from amongst varied sets of available fixed income securities in the market. As we all know, bond pricesBond PricesThe 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 are inversely related to interest rates. Investors get a good sense of idea in terms of which bond to buy, longer-term or shorter term, if they know the Duration the various coupon bondsCoupon BondsCoupon bonds pay fixed interest at a predetermined frequency from the bond’s issue date to the bond’s maturity or transfer date. The holder of a coupon bond receives a periodic payment of the stipulated fixed interest rate.read more are offering along with the projected interest rate structure.

Macaulay Duration Formula

It can be calculated using the below formula,

Macaulay Duration Formula

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For eg:
Source: Macaulay Duration (wallstreetmojo.com)

Where,

  • t = time period
  • C = coupon payment
  • y = yield
  • n = number of periods
  • M = maturity
  • Current Bond Price = present value of cash flows

Calculation of Macaulay Duration with Example

Let’s see an example of Macaulay’s duration to understand it better.

You can download this Macaulay Duration Excel Template here – Macaulay Duration Excel Template

A $1,000 value bond pays an 8% coupon rate and matures in four years. The coupon rate is 8% p.a. With semi-annual payment. We can expect the following cash flows to occur.

  • 6 months: $40
  • 1 year: $40
  • 1.5 years: $40
  • 2 years: $40
  • years: $40
  • 3 years: $40
  • 3.5 years: $40
  • 4 years: $1,040
Calculate Macaulay DurationCalculate Macaulay 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

Solution:

With the above information, we can calculate the discount factorDiscount FactorDiscount Factor is a weighing factor most often used to find the present value of future cash flows, i.e., to calculate the Net Present Value (NPV). It is determined by, 1 / {1 * (1 + Discount Rate) Period Number}read more. We can use the following semi-annual interest formula to derive the discount factor. 1 / (1 + r)n, where r is the coupon rate, and n is the number of periods compounded.

Discount Factor

Calculation of discount factors for 6 months will be –

Macaulay Duration Example 0.1

Discount factors for 6 months = 1 / (1 + 8%/2)

Discount Factors = 0.9615

Similarly, we can do the calculation of the discount factor for years 1 to 4.

Example 1.0.1

Present Value of Cash Flow

The present value of cash flow for 6 months will be –

Macaulay Duration Example 1.0.2

Now, in order to get the present value of the cash flows, we must multiply each period cash flow with its respective discount factor.

Present value of cash flow for 6 months : 1 x $40 x 0.9615

Present Value of Cash Flow = $38.46

Similarly, we can do the calculation of the present value of cash flow for year 1 to 4.

 Example 1.0.3

Macaulay Duration

Calculation of Macaulay Duration will be –

Macaulay Duration Example 1.0.4
  • Current Bond Price = PV of all the cash flows 6,079.34
  • Macaulay Duration = $ 6,079.34/ $1,000 = 6.07934

You can refer given excel template above for the detailed calculation of Macaulay duration.

Merits of Using Duration

Duration plays an important role in helping investors understand the risk factor for the available fixed-income security. Just as how the risk in equities is measured by deviation from the mean or simply by deriving the beta of the security, the risk in fixed-income instruments are strictly estimated by Macaulay duration of the instrument.

Understanding and comparing Macaulay Duration of the instruments can go a long way in choosing the right fit for your fixed-income portfolio.

Setbacks of Using Duration

Duration is a good approximation of price changes for an option-free bond, but it’s only good for small changes in interest rates. As rate changes become larger, the curvature of the bond price-yield relationship becomes more important. In other words, a linear estimate of price changes, such as duration, will contain errors.

In fact, the relationship between bond price and yield is not linear but convex. This convexity shows that the difference between actual and estimated prices widens as the yields go up. That is, widening error in the estimated price is due to the curvature of the actual price path. This is known as the degree of convexity.

Bottom Line

Macaulay Duration knowledge is paramount in ascertaining the future returns from fixed income instruments. As such, it is highly advisable for investors, especially risk-averse investors, to assess and compare the duration offered by the various bonds in order to reach a minimum variance mix and draw maximum returns with the least risk possible. Also, the interest rate factor should be considered before making a buying decision.

Recommended Articles

This has been a guide to what Macaulay Duration is and its definition. Here we discuss the formula of Macaulay duration along with calculations and examples. You can learn more about fixed income from the following articles –

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