Macaulay and modified duration

Q: What is Modified Duration ($D_{mod}$) and how is it defined mathematically?

Modified Duration measures the price sensitivity of an asset to changes in yield and is defined as $D_{mod} = -\frac{P'(i)}{P(i)}$.

Q: What is the mathematical relationship between Macaulay Duration and Modified Duration?

$D_{mod} = \frac{D_{mac}}{1+i} = v \cdot D_{mac}$ (or $D_{mod} = \frac{D_{mac}}{1 + \frac{i^{(m)}}{m}}$ under $m$-thly compounding).

Q: How is price change estimated using the first-order linear approximation based on Modified Duration?

$\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta i$, or in dollar terms, $P(i + \Delta i) \approx P(i)(1 - D_{mod} \cdot \Delta i)$.

Q: How do you calculate the duration of an entire portfolio of assets?

The portfolio duration is the market-value-weighted average of the individual asset durations: $D_{port} = \sum w_j D_j$, where $w_j = \frac{P_j}{P_{total}}$.

Q: What is the default SOA Exam FM convention when the term "duration" is used without a modifier?

Unless explicitly stated otherwise, "duration" on Exam FM refers to Macaulay duration .

Exam FM – Financial Mathematics · SOA (Society of Actuaries) · 6 min read · Updated: 11 Nov 2026

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 $t$ and an annual effective interest rate $i$, the Macaulay Duration is defined as:

$D_{mac} = \frac{\sum_{t} t \cdot R_t (1+i)^{-t}}{\sum_{t} R_t (1+i)^{-t}} = \frac{\sum_{t} t \cdot v^t \cdot R_t}{\sum_{t} v^t \cdot R_t}$

The denominator $\sum_{t} v^t \cdot R_t$ is just the Price (P) or present value of the cash flows. So:

$D_{mac} = \frac{\sum_{t} t \cdot PV(R_t)}{P(i)}$

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 weighted average time is just that payment 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, giving earlier payments greater relative weight in the average).

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 sensitivity. Mathematically, it is defined in terms of the first derivative of the price function with respect to the yield rate $i$:

$D_{mod} = -\frac{P'(i)}{P(i)}$

The Relationship Between Macaulay and Modified Duration

There is a direct mathematical link between the two measures. Differentiating $P(i) = \sum R_t (1+i)^{-t}$ gives $P'(i) = -\sum t \cdot R_t (1+i)^{-(t+1)} = -\frac{1}{1+i} \sum t \cdot R_t (1+i)^{-t}$. Substituting this into the definition shows:

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

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

3. First-Order Approximations of Price Change

Exam FM requires candidates to approximate the change in present value due to a change in interest rate ($\Delta i$) using two distinct 1st-order approximations:

Method A: 1st-Order Linear Approximation Based on Modified Duration

Using the first term of Taylor's expansion, the percentage change in price is approximated as:

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

Or in dollar terms:

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

$P(i + \Delta i) \approx P(i) \cdot (1 - D_{mod} \cdot \Delta i)$

Example: If a bond has a Modified Duration of 7 and interest rates increase by 0.5% ($\Delta i = +0.005$), the price of the bond changes by approximately $-7 \cdot 0.005 = -3.5\%$.

Method B: 1st-Order Approximation Based on Macaulay Duration

The annual compounding approximation based directly on Macaulay duration is given by:

$P(i + \Delta i) \approx P(i) \left(\frac{1+i}{1+i+\Delta i}\right)^{D_{mac}} = P(i) \left(1 + \frac{\Delta i}{1+i}\right)^{-D_{mac}}$

The approximate percentage change is therefore:

$\frac{\Delta P}{P} \approx \left(1 + \frac{\Delta i}{1+i}\right)^{-D_{mac}} - 1$

Both methods yield very close approximations for small changes in $i$, but make sure to use the specific version requested in an exam question.

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

Follow these steps for any Exam FM problem involving duration from a set of cash flows:

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 rate: $PV(R_t) = R_t (1+i)^{-t}$.
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 $D_{mac}$.
Step 6: If the question asks for Modified Duration, calculate $D_{mod} = \frac{D_{mac}}{1+i}$.

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

5. Duration of a Portfolio

The duration of a portfolio is the weighted average of the durations of the individual assets, where the weights are the Market Values (prices) of those assets:

$D_{port} = \sum_{j} w_j \cdot D_j$

Where $w_j = \frac{P_j}{P_{total}}$ and $\sum w_j = 1$. This formula applies identically to both Macaulay and Modified durations.

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

6. Summary and Key SOA Exam FM Conventions

Here are the core takeaways you must know on exam day:

1. Default SOA Convention: Unless otherwise stated in an Exam FM problem, "duration" means Macaulay duration (and "convexity" means modified convexity).
2. Macaulay Duration ($D_{mac}$): Weighted average time until cash flows: $D_{mac} = \frac{\sum t \cdot v^t R_t}{P}$. Think "Center of Gravity."
3. Modified Duration ($D_{mod}$): Defined as $-\frac{P'(i)}{P(i)}$. Measures price sensitivity.
4. The Link: $D_{mod} = \frac{D_{mac}}{1+i} = v \cdot D_{mac}$.
5. Linear Approximation: $\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta i$.
6. Macaulay Approximation: $P(i + \Delta i) \approx P(i) \left(1 + \frac{\Delta i}{1+i}\right)^{-D_{mac}}$.
7. Zero-Coupon Rule: The Macaulay Duration of an $n$-year zero-coupon bond is exactly $n$.

References

Notation and terminology used for Exam FM — Society of Actuaries
Financial Mathematics Exam—December 2026 — Society of Actuaries

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the conceptual definition and formula for Macaulay Duration ($D_{mac}$)?

Macaulay Duration is the weighted average time until cash flows are received, weighted by their present values: $D_{mac} = \frac{\sum_t t \cdot v^t R_t}{\sum_t v^t R_t} = \frac{\sum_t t \cdot PV(R_t)}{P(i)}$.

How does Macaulay Duration behave for zero-coupon bonds, coupon-bearing bonds, and changes in interest rates?

For an $n$-year zero-coupon bond, $D_{mac} = n$; for coupon bonds, $D_{mac} < n$; and as interest rates increase, $D_{mac}$ decreases.

What is Modified Duration ($D_{mod}$) and how is it defined mathematically?

Modified Duration measures the price sensitivity of an asset to changes in yield and is defined as $D_{mod} = -\frac{P'(i)}{P(i)}$.

What is the mathematical relationship between Macaulay Duration and Modified Duration?

$D_{mod} = \frac{D_{mac}}{1+i} = v \cdot D_{mac}$ (or $D_{mod} = \frac{D_{mac}}{1 + \frac{i^{(m)}}{m}}$ under $m$-thly compounding).

How is price change estimated using the first-order linear approximation based on Modified Duration?

$\frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta i$, or in dollar terms, $P(i + \Delta i) \approx P(i)(1 - D_{mod} \cdot \Delta i)$.

How is the price change estimated using the first-order approximation based on Macaulay Duration?

$P(i + \Delta i) \approx P(i) \left(1 + \frac{\Delta i}{1+i}\right)^{-D_{mac}}$ (or $\frac{\Delta P}{P} \approx \left(1 + \frac{\Delta i}{1+i}\right)^{-D_{mac}} - 1$).

How do you calculate the duration of an entire portfolio of assets?

The portfolio duration is the market-value-weighted average of the individual asset durations: $D_{port} = \sum w_j D_j$, where $w_j = \frac{P_j}{P_{total}}$.

What is the default SOA Exam FM convention when the term "duration" is used without a modifier?

Unless explicitly stated otherwise, "duration" on Exam FM refers to Macaulay duration.

Ready to test yourself?

Turn these notes into exam-style practice. Get unlimited AI questions on this topic with instant marking and explanations.

Practise This Topic

More Exam FM – Financial Mathematics chapters

* The content provided by thinka is generated by AI and may not always be accurate or up-to-date. Please use it as a supplementary resource and verify with official materials.

Put These Notes into Practice

Reading the notes is a great start. Now practise with unlimited AI-generated questions and get instant feedback. 100,000+ students are already improving their grades.

Start Practising Now View Pricing

Done reading? Test yourself with AI practice questions

Practise This Topic Now