The Macaulay duration represents the average time until a bond’s cash flows are received and is calculated as the weighted sum of their maturities. Each cash flow’s weight is determined by dividing its present value by the bond’s price. Portfolio managers employing an immunization strategy often rely on Macaulay duration as a crucial metric.
The metric derives its name from its originator, Frederick Macaulay. It serves as the economic equilibrium point for a set of cash flows. Alternatively, the statistic represents the average duration an investor needs to hold a bond until the total present value of its cash flows equals the bond’s purchase price.
To calculate the Macaulay duration, you multiply each time period by its corresponding periodic coupon payment and then divide the result by 1 plus the periodic yield raised to the time to maturity.
D = 1 + 1 / YTM
This calculation is performed for each period, and the values are then added together to obtain the Macaulay duration.
The duration calculator employs an annual compounding period to standardize duration values across various bonds. Duration is typically expressed in years in international markets (e.g., Bloomberg), while in the Russian and Ukrainian markets, it is mostly measured in days.
Beyond merely reflecting the average payment flow timeline of bonds, duration serves as a reliable indicator of price sensitivity to changes in interest rates.
The calculation of Macaulay duration is a straightforward process. Let’s consider a $1,000 face-value bond with a 6% coupon, maturing in three years, and subject to a 6% per annum interest rate with semiannual compounding. The bond pays its coupon twice a year and the principal at the end. Here are the expected cash flows over the next three years:
Period 1: $30
Period 2: $30
Period 3: $30
Period 4: $30
Period 5: $30
Period 6: $1,030
To compute the discount factor for each period, use the formula 1 / (1 + r)^n, where r is the interest rate, and n is the period number. With a 6% interest rate compounded semiannually (3% per period), the discount factors are as follows:
Period 1 Discount Factor: 1 / (1 + 0.03)^1 = 0.9709
Period 2 Discount Factor: 1 / (1 + 0.03)^2 = 0.9426
Period 3 Discount Factor: 1 / (1 + 0.03)^3 = 0.9151
Period 4 Discount Factor: 1 / (1 + 0.03)^4 = 0.8885
Period 5 Discount Factor: 1 / (1 + 0.03)^5 = 0.8626
Period 6 Discount Factor: 1 / (1 + 0.03)^6 = 0.8375
Next, calculate the present value of each cash flow by multiplying the period’s cash flow by the period number and its corresponding discount factor:
Period 1: 1 × $30 × 0.9709 = $29.13
Period 2: 2 × $30 × 0.9426 = $56.56
Period 3: 3 × $30 × 0.9151 = $82.36
Period 4: 4 × $30 × 0.8885 = $106.62
Period 5: 5 × $30 × 0.8626 = $129.39
Period 6: 6 × $1,030 × 0.8375 = $5,175.65
Summing up the present values of all cash flows: $29.13 + $56.56 + $82.36 + $106.62 + $129.39 + $5,175.65 = $5,579.71
Finally, calculate the Macaulay Duration by dividing the sum of the present values by the bond’s price: Macaulay Duration = $5,579.71 ÷ $1,000 = 5.58
The result, 5.58 half-years, is less than the time to maturity of six half-years, which equates to 2.79 years. Thus, the bond’s duration is indeed less than its time to maturity, as expected for a coupon-paying bond.
More information about duration calculation and examples of calculations can be found at Bond Calculator Guide.
Macaulay duration represents the weighted average time to maturity of a bond’s cash flows.
Modified duration, on the other hand, quantifies a bond’s price sensitivity to fluctuations in interest rates. It is derived from Macaulay duration but considers the bond’s yield to maturity (YTM) in its calculation.
In asset-liability portfolio management, duration matching serves as a method for interest rate immunization. Variations in interest rates influence the present value of cash flows and, consequently, impact the value of a fixed-income portfolio. By aligning the durations of assets and liabilities in a company’s portfolio, a change in interest rates will cause the value of assets and liabilities to move by the exact same amount but in opposite directions.
As a result, the total value of the portfolio remains constant. However, it’s important to note that duration matching has its limitations. While it immunizes the portfolio against minor interest rate changes, it becomes less effective when dealing with significant fluctuations.
Duration is a measure of the weighted average time it takes to receive the present value of a bond’s cash flows, including both coupon payments and the bond’s face value (principal) at maturity. It provides an estimate of how long it takes, in years, for an investor to recover the price of the bond through its cash flows. Duration is used to assess a bond’s price sensitivity to changes in interest rates. It takes into account both the timing and size of cash flows.
Macaulay duration, named after Frederick Macaulay, is a specific type of duration. It is the weighted average time until a bond’s cash flows are received, measured in years. The calculation involves dividing the present value of each cash flow by the bond’s current price and then summing the results. Macaulay duration is useful for comparing bonds with different coupon rates, maturities, and prices. It represents the economic balance point of a bond’s cash flows.
For a zero-coupon bond, the Macaulay duration is equivalent to its time to maturity. This fixed-income security does not offer any interest payments on the principal amount. Thus, its Macaulay duration is equal to the time remaining until the bond matures.
Set up your data. Prepare your cash flow data, including the bond’s periods and corresponding cash flows.
Determine the bond’s price. Note the current price of the bond.
Calculate the present value for each cash flow. Use Excel’s present value (PV) function to find the present value of each cash flow using the bond’s yield to maturity (YTM) as the discount rate.
Calculate the weighted sum of present values. Multiply the present value of each cash flow by its corresponding period number and sum them up.
Divide the weighted sum by the bond’s price. Finally, divide the weighted sum of present values by the bond’s price to get the Macaulay duration.
Suppose you have the following data for a bond.
Cash flows. $30 per period for 5 periods and $1,030 at the end (period 6).
Yield to Maturity (YTM). 6% per annum, compounded semiannually.
Bond price. $1,000.
In Excel, enter your cash flows in a column (let’s say column A) starting from cell A2 (ignore cell A1).
Enter the corresponding periods in another column (e.g., column B) starting from cell B2.
In cell C2, use the PV function to calculate the present value of each cash flow: =PV(YTM/2, B2, -A2)
Drag this formula down to calculate the present values for all cash flows.
In cell D2, calculate the weighted sum of present values: =SUMPRODUCT(A2:A6, B2:B6, C2:C6)
Finally, calculate the Macaulay duration in cell E2 by dividing the weighted sum by the bond’s price: =D2 / $1,000.
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