What is Macaulay Duration?
The Macaulay duration represents the weighted average term to maturity of the cash flows from a bond. Here, each cash flow is weighted by dividing its present value by the bond’s price. Portfolio managers frequently use Macaulay duration as a part of an immunization strategy to protect portfolios from interest rate changes.
Macaulay duration can be calculated with the following formula:
( \text{Macaulay Duration} = \frac{\sum_{t=1}^{n}\frac{t \times C}{(1 + y)^t} + \frac{n \times M}{(1 + y)^n}}{\text{Current Bond Price}} )
Where:
- t = Respective time period
- C = Periodic coupon payment
- y = Periodic yield
- n = Total number of periods
- M = Maturity value
Understanding Macaulay Duration
Named after its creator, Frederick Macaulay, this metric serves as the economic balance point of cash flows. Essentially, it’s the weighted average duration that an investor must hold a bond so that the present value of its cash flows equals the price paid for the bond.
Factors Influencing Duration
The factors affecting a bond’s duration include:
- Price
- Maturity
- Coupon payments
- Yield to maturity
All else remaining unchanged, as the bond’s maturity increases, its duration also increases. Conversely, higher coupon payments and rising interest rates tend to reduce duration, thereby making the bond less sensitive to future rate changes. Specific provisions like sinking funds and call options can further lower a bond’s duration.
Calculation Example
Consider a $1,000 face-value bond paying a 6% coupon, maturing in three years, with interest rates also at 6% per annum, compounded semiannually. The bond’s cash flows for the next three years, alongside the necessary calculations, are as follows:
Bond Cash Flows and Relevant Periods:
Period 1: $30
Period 2: $30
Period 3: $30
Period 4: $30
Period 5: $30
Period 6: $1,030
To determine discount factors for each period using the formula ( 1 ÷ (1 + r)^n ):
Period 1 Discount Factor: (1 ÷ 1.03)^1 = 0.9709
Period 2 Discount Factor: (1 ÷ 1.03)^2 = 0.9426
Period 3 Discount Factor: (1 ÷ 1.03)^3 = 0.9151
Period 4 Discount Factor: (1 ÷ 1.03)^4 = 0.8885
Period 5 Discount Factor: (1 ÷ 1.03)^5 = 0.8626
Period 6 Discount Factor: (1 ÷ 1.03)^6 = 0.8375
Next, calculate the present value (PV) of cash flows using: Cash Flow × Period × Discount Factor
Period 1 PV: 1 × $30 × 0.9709 = $29.13
Period 2 PV: 2 × $30 × 0.9426 = $56.56
Period 3 PV: 3 × $30 × 0.9151 = $82.36
Period 4 PV: 4 × $30 × 0.8885 = $106.62
Period 5 PV: 5 × $30 × 0.8626 = $129.39
Period 6 PV: 6 × $1,030 × 0.8375 = $5,175.65
Sum of PV: $5,579.71 (Numerator)
The denominator, representing the current bond price, is also provided areas: $1,000.
The final step:
( \text{Macaulay Duration} = \frac{\text{Sum of PV}}{\text{Current Bond Price}} = \frac{5,579.71}{1,000} = 5.58 )
Since there are two semiannual periods within a year, the Macaulay Duration is converted into years by dividing 5.58 by 2, arriving at 2.79 years. This value indicates the time it takes for the bond’s price to equate to the present value of its remaining cash flows.
In summation, Macaulay Duration offers a valuable metric for understanding bond investment sensitivity to interest rate changes, providing essential analysis for strategic financial planning.
Related Terms: yield to maturity, coupon payment, present value, bond price, weighted average.
