Macaulay and modified duration

Welcome to the World of Duration!

If you have been studying for Exam FM, you already know that when interest rates go up, bond prices go down. But the big question for actuaries and investment managers is: How much will the price change? That is exactly what Duration helps us measure.

Think of duration as a way to measure the "sensitivity" or "volatility" of a set of cash flows. Whether you are managing a pension fund or a simple bond portfolio, understanding duration is the key to managing interest rate risk. Don't worry if this seems a bit abstract right now—we are going to break it down into simple, manageable pieces!

1. Macaulay Duration: The "Center of Gravity"

Imagine you are balancing a long wooden board on your finger. If there are heavy weights at both ends, the "balance point" is the center of gravity. Macaulay Duration (\(D_{mac}\)) is essentially the "center of gravity" for a series of cash flows in terms of time.

It is the weighted average time until the cash flows are received. Instead of weighting the times by the dollar amount, we weight them by the Present Value (PV) of those dollars.

The Formula

If we have payments \(R_t\) at times \(t\), the Macaulay Duration is:

\(D_{mac} = \frac{\sum t \cdot v^t \cdot R_t}{\sum v^t \cdot R_t}\)

Wait! Let's simplify that. The denominator \(\sum v^t \cdot R_t\) is just the Price (P) of the asset. So:

\(D_{mac} = \frac{\sum t \cdot PV(R_t)}{Price}\)

A Real-World Analogy

Think of a 10-year bond that pays coupons every year. You don't wait exactly 10 years to get your money; you get some after 1 year, some after 2, and so on. The Macaulay Duration tells you that, on average, you are waiting (for example) 8.2 years to get your value back.

Important Properties of Macaulay Duration

1. Zero-Coupon Bonds: For a zero-coupon bond paying at time \(n\), the Macaulay Duration is exactly \(n\). This is because there is only one payment, so the "average" time is just that one date!
2. Coupon Bonds: For any bond that pays coupons before maturity, the Macaulay Duration will always be less than the time to maturity.
3. Interest Rates: As the interest rate \(i\) increases, the Macaulay Duration decreases (because future payments are discounted more heavily, making earlier payments "heavier" in the weighting).

Quick Review: Macaulay Duration = Weighted Average Time. It is measured in years (or whatever your time unit is).

2. Modified Duration: The Sensitivity Gauge

While Macaulay Duration tells us about time, Modified Duration (\(D_{mod}\)) tells us about price change. It measures the percentage change in price for a 1% change in interest rates.

The Relationship

There is a very simple mathematical link between the two. If \(i\) is the annual effective interest rate:

\(D_{mod} = \frac{D_{mac}}{1+i} = v \cdot D_{mac}\)

Note: If you are given a nominal rate compounded \(m\) times per year, the formula becomes \(D_{mod} = \frac{D_{mac}}{1 + \frac{i^{(m)}}{m}}\).

Why do we use it?

The primary use of Modified Duration is the Linear Approximation of price change:

\(\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta i\)

Example: If a bond has a Modified Duration of 7 and interest rates increase by 0.5% (0.005), the price of the bond will decrease by approximately \(7 \cdot 0.005 = 3.5\%\).

Did you know? The negative sign in the formula is there because price and interest rates move in opposite directions. When rates go UP, prices go DOWN!

3. Step-by-Step: How to Calculate Duration

Don't let the long formulas scare you. Follow these steps for any Exam FM problem involving duration:

Step 1: List out the times \(t\) and the cash flows \(R_t\).
Step 2: Calculate the Present Value (PV) of each cash flow using the given interest rate.
Step 3: Sum these PVs to find the total Price (P).
Step 4: For each payment, multiply the PV by the time \(t\). Sum these up to get \(\sum t \cdot PV(R_t)\).
Step 5: Divide the result from Step 4 by the Price. This is your Macaulay Duration.
Step 6: If the question asks for Modified Duration, divide your answer by \((1+i)\).

Common Mistake: Many students forget to divide by the Price at the end. Remember, duration is an average, so you must divide by the total "weight" (the Price).

4. Duration of a Portfolio

One of the coolest things about duration is that it's easy to combine. If you have a portfolio of different bonds, you don't need to recalculate everything from scratch.

The duration of a portfolio is the weighted average of the durations of the individual assets, where the weights are the Market Values of those assets.

\(D_{port} = \sum w_j \cdot D_j\)

Where \(w_j = \frac{Price_j}{Total Price}\).

Example: You have \$400 in Bond A (Duration = 3) and \$600 in Bond B (Duration = 8).
Total Value = \$1,000.
\(D_{port} = (\frac{400}{1000} \cdot 3) + (\frac{600}{1000} \cdot 8) = 1.2 + 4.8 = 6.0\).

5. Summary and Key Takeaways

We’ve covered the core pillars of duration for Exam FM. Here is what you need to remember:

1. Macaulay Duration (\(D_{mac}\)): The time-weighted average of cash flows. Think "Balance Point."
2. Modified Duration (\(D_{mod}\)): The percentage sensitivity of price to interest rate changes. Think "Volatility."
3. The Link: \(D_{mod} = \frac{D_{mac}}{1+i}\).
4. Price Prediction: \(\Delta P \approx -P \cdot D_{mod} \cdot \Delta i\).
5. Zero-Coupon Rule: The Macaulay Duration of a zero-coupon bond is just its term to maturity.

Final Tip for the Exam: If a question doesn't specify which duration to use, look for the units. If it says "percentage change," it's likely Modified Duration. If it asks for the "average time," it's Macaulay Duration. You've got this!