Macaulay Duration

Definition

Macaulay duration was introduced in the previous learning objective. It provides an understanding of the bond’s sensitivity to interest rate fluctuations. At its core, Macaulay duration is the weighted average time until a bond’s cash flows are received. It signifies the holding period for a bond that balances both reinvestment and price risk.

Calculation

The calculation for Macaulay Duration is derived from the bond’s cash flows. Each cash flow is weighted by its share of the bond’s full price, which is its present value. The following steps outline the calculation:

1. Time to Receipt of Cash Flow: Determine the time until each cash flow is received.
2. Cash Flow: Identify the cash flow amount for each period.
3. Present Value of Cash Flow: Calculate the present value of each cash flow using the bond’s yield-to-maturity.
4. Weight of Cash Flow: Compute the weight of each cash flow by dividing its present value by the total present value of all cash flows.
5. Weighted Average Time: Multiply the time to receipt of each cash flow by its respective weight.
6. Macaulay Duration: Sum up the results from step 5 to obtain the Macaulay Duration.

Formula

The general formula to calculate Macaulay duration, represented as MacDur, is:

\[MacDur = \frac{\sum_{i = 1}^{N}\mspace{2mu}\frac{t \times CF_{t}}{(1 + r)^{t}}}{\sum_{i = 1}^{N}\mspace{2mu}\frac{CF_{t}}{(1 + r)^{t}}}\]

Where:

• \(t\) is the time (in periods) until the cash flow is received.
• \(CF_{t}\) is the cash flow at time \(t\).
• \(r\) is the yield-to-maturity per period.
• \(N\) is the total number of periods.

Example: Calculating the Macaulay Duration

Think about a bond with five years left to maturity, a 1% annual coupon, and a yield-to-maturity of 0.10%. Assume it’s 120 days into the first coupon period and follows a 30/360 day-count basis. What’s the closest estimate for the bond’s annualized Macaulay duration?

Considerations:

1. The bond has a \(1\%\) annual coupon, which means a cash flow of 1 per year for the next 4 years and 101 ( \(1 + 100\) par value) in the 5th year.
2. The yield-to-maturity is \(0.10\%\) or 0.0010 in decimal form.
3. It is 120 days into the first coupon period, so the first cash flow will be received in \(1 – \frac{120}{360}\) years or 0.6667 years.
4. Macaulay duration is calculated as:

\[MacDur = \frac{\sum_{i = 1}^{N}\frac{t \times CF_{t}}{(1 + r)^{t}}}{\sum_{i = 1}^{N}\frac{CF_{t}}{(1 + r)^{t}}}\]

Where:

• \(CF_{t} =\) Cash Flow at time \(t\).
• \(r =\) Yield to Maturity.
• \(t =\) Time to receipt of the cash flow.

$$\begin{array}{c|c|c|c|c|c}  \textbf{Period} & \textbf{Time to Receipt} & \textbf{Cashflow Amount} & \textbf{PV} & \textbf{Weight} & \textbf{Time to Receipt*Weight} \\ \hline 1 & 0.6667 & 1 & 0.9993 & 0.0096 & 0.01 \\ \hline 2 & 1.6667 & 1 & 0.9983 & 0.0096 & 0.02 \\ \hline 3 & 2.6667 & 1 & 0.9973 & 0.0095 & 0.03 \\ \hline 4 & 3.6667 & 1 & 0.9963 & 0.0095 & 0.03 \\ \hline 5 & 4.6667 & 101 & 100.5300 & 0.9618 & 4.49 \\ \hline \textbf{Total} & & & \textbf{104.5213} & \textbf{1} & \textbf{4.5712} \\ \end{array}$$

This means that an investor would, on average, wait 4.5712 years to receive the bond’s cash flows, weighted by their present value.

Interpretation of Macaulay Duration

The Macaulay Duration provides insights into the bond’s interest rate risk. A bond with a higher Macaulay Duration has greater sensitivity to interest rate changes.

For instance, if the investment horizon matches the Macaulay Duration, the bond is nearly hedged against interest rate risk. Any losses due to rising interest rates (price risk) would approximately be offset by gains from the reinvestment of coupons (reinvestment risk) and vice versa.

Furthermore, the Macaulay Duration is often annualized. For bonds with semiannual coupons, the Macaulay Duration is divided by 2 to get the annualized figure.

It’s also noteworthy that the Macaulay Duration is typically less than the bond’s time-to-maturity because it’s a present value-weighted average of the time until cash flows are received.

Question

Consider a bond that has three years remaining to maturity, a coupon of 3.5% paid semiannually, and a yield-to-maturity of 3.80%. Assuming it is 18 days into the first coupon period and using a 30/360 basis, the bond’s annualized Macaulay duration is closest to:

1. 2.81 years.
2. 2.82 years.
3. 2.84 years.

Solution

The correct answer is C.

Given:

• The bond has three years remaining to maturity.
• A coupon of 3.5% is paid semiannually.
• Yield-to-maturity is 3.80%.
• 18 days into the first coupon period.
• A 30/360 basis.

This means the bond pays 1.75% every 6 months. The yield per period is 1.90% (3.80% divided by 2).

Let’s compute MacDur and then divide by 2 to annualize it since the bond pays semiannually.

$$
\begin{array}{c|c|c|c|c|c}
\text { Period } & \begin{array}{l}
\text { Time to } \\
\text { Receipt }
\end{array} & \begin{array}{l}
\text { Cashflow } \\
\text { Amount }
\end{array} & \begin{array}{l}
\text { Present } \\
\text { Value } \\
\text { (PV) }
\end{array} & \text { Weight } & \begin{array}{l}
\text { Time to } \\
\text { Receipt } \\
\text { *Weight }
\end{array} \\
\hline & 0.95 & 1.75 & 1.718987 & 0.01732 & 0.016454 \\
\hline & 1.95 & 1.75 & 1.686935 & 0.016997 & 0.033144 \\
\hline & 2.95 & 1.75 & 1.655481 & 0.01668 & 0.049206 \\
\hline & 3.95 & 1.75 & 1.624613 & 0.016369 & 0.064657 \\
\hline & 4.95 & 1.75 & 1.594321 & 0.016064 & 0.079515 \\
\hline & 5.95 & 101.75 & 90.96996 & 0.916571 & 5.453598 \\
\hline & & & 99.2503 & 1 & 5.696573 \\
\end{array}
$$

The annualized MacDur is 5.6966/2 = 2.8483