What Is the Macaulay Duration?
Let me explain the Macaulay duration directly: it's a formula that shows you, as an investor, the time it will take for a bond to reach profitability by measuring the weighted average term to maturity of its cash flows.
You determine the weight of each cash flow by dividing its present value by the bond's price. I find it particularly useful if you're a portfolio manager employing an immunization strategy, where you build a portfolio that's protected from negative shifts in interest rates.
Key Takeaways
Understand this: the Macaulay duration is the weighted average number of years you need to hold a bond until the present value of its cash flows matches what you paid for it. Factors like the bond’s price, maturity, coupon, and yield to maturity all play into the calculation. You can use this formula to gauge how sensitive the bond is to interest rate changes.
Understanding the Macaulay Duration
Think of the Macaulay duration as the economic balance point for a bond's cash flows. It's the weighted average number of years you, the investor, must maintain the bond until the present value of those cash flows equals the amount you paid. This metric comes from Canadian economist Frederick Macaulay, who developed it.
Calculating the Macaulay Duration
You calculate Macaulay duration using this formula: Macaulay Duration = [∑(t=1 to n) (t × C / (1 + y)^t) + (n × M / (1 + y)^n)] / Current Bond Price, where t is the respective time period, C is the periodic coupon payment, y is the periodic yield, n is the total number of periods, and M is the maturity value.
This setup gives you a clear way to compute it step by step.
Factors Affecting Duration
Several elements influence the duration calculation, including the bond’s price, maturity, coupon, and yield to maturity. If everything else stays the same, duration increases with longer time to maturity. As the coupon rises, duration decreases. Higher interest rates also reduce duration, making the bond less sensitive to further rate hikes. Features like a sinking fund, scheduled prepayments before maturity, and call provisions will lower the bond’s duration as well.
Calculation Example
Let's walk through a straightforward example. Suppose you have a $1,000 face-value bond with a 6% coupon that matures in three years. Interest rates are 6% per annum with semiannual compounding. The bond pays coupons twice a year and the principal at the end. Expected cash flows over the next three years are: Period 1: $30, Period 2: $30, Period 3: $30, Period 4: $30, Period 5: $30, Period 6: $1,030.
You need discount factors for each period, calculated as 1 / (1 + r)^n, where r is 3% (6% divided by 2 for semiannual). So, Period 1: 0.9709, Period 2: 0.9426, Period 3: 0.9151, Period 4: 0.8885, Period 5: 0.8626, Period 6: 0.8375.
Now, multiply each period’s cash flow by the period number and its discount factor to get present values: 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. Sum these to $5,579.71, which is the numerator.
The current bond price, or denominator, is the sum of the present values of the cash flows, equaling $1,000 since the bond trades at par (coupon rate matches interest rate).
Finally, Macaulay duration is $5,579.71 / $1,000 = 5.58. For a coupon-paying bond, duration is always less than time to maturity—in this case, 5.58 half-years (or 2.79 years) is less than three years.
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