Duration and convexity of a set of cash flows

Q: What is the conceptual definition of Macaulay Duration ($ D_{mac} $)?

Macaulay Duration is the weighted average time until a set of cash flows is received, where the weights are the present values of each cash flow.

Q: How does the Macaulay Duration of a zero-coupon bond compare to its time to maturity?

The Macaulay Duration of a zero-coupon bond is exactly equal to its time to maturity.

Q: What does Modified Duration ($ D_{mod} $) measure?

Modified Duration measures the price sensitivity of a portfolio, representing the approximate percentage change in price for a 1% change in the interest rate.

Q: How is Modified Duration used to approximate a change in price ($ \Delta P $)?

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

Q: What is the formula for Macaulay Convexity ($ C_{mac} $)?

$ C_{mac} = \frac{\sum t^2 \cdot R_t \cdot v^t}{\sum R_t \cdot v^t} $

Q: What is the relationship between Modified Convexity ($ C_{mod} $), Macaulay Duration, and Macaulay Convexity?

$ C_{mod} = v^2 (C_{mac} + D_{mac}) = \frac{C_{mac} + D_{mac}}{(1+i)^2} $

Q: What is the benefit of positive convexity for an investor?

Positive convexity means that for a given interest rate change, the price will rise more than duration predicts when rates fall, and fall less than duration predicts when rates rise.

Q: What is the full second-order price approximation formula using both Duration and Convexity?

$ P(i_{new}) \approx P(i_{old}) \cdot [1 - D_{mod} \cdot \Delta i + \frac{1}{2} C_{mod} \cdot (\Delta i)^2] $

Exam FM – Financial Mathematics · SOA (Society of Actuaries) · 7 min read

Welcome to the World of Duration and Convexity!

Hello future actuaries! Today, we are diving into one of the most practical and important parts of the Exam FM syllabus: Duration and Convexity. These concepts are the tools we use to measure how "sensitive" a set of cash flows (like a bond or a portfolio) is to changes in interest rates.

Why does this matter? Imagine you own a bond. If interest rates in the market go up, the value of your bond goes down. But by how much? Duration gives us a first-level estimate, and Convexity makes that estimate even more accurate. Think of Duration as your "speed" and Convexity as your "acceleration." Together, they tell the whole story of how your investments move!

Don't worry if this seems a bit math-heavy at first. We will break it down step-by-step with simple analogies and clear formulas. Let's get started!

1. Macaulay Duration: The "Balance Point" of Time

The Macaulay Duration (denoted as $ D_{mac} $) is essentially the weighted average time until you receive your cash flows. Instead of just looking at the final maturity date, we look at when every single dollar comes in and weight it by its present value.

The Formula:
If you have cash flows $ R_t $ at times $ t $, the Macaulay Duration is:
$ D_{mac} = \frac{\sum t \cdot R_t \cdot v^t}{\sum R_t \cdot v^t} $

What does this actually mean?
The denominator ($ \sum R_t \cdot v^t $) is simply the Price (P) or Present Value of the cash flows. So, we are taking each time $ t $, multiplying it by the value of the money we get at that time, and dividing by the total value.

An Analogy: The Seesaw
Imagine a seesaw where the board represents time. You place "weights" on the board at different times. The weights are the Present Values of your cash flows. The Macaulay Duration is the exact spot where you would put the fulcrum (the triangle) to make the seesaw balance perfectly. Cash flows further in the future "weigh" more in terms of time, while larger cash flows "weigh" more in terms of value.

Quick Tips for Macaulay Duration:

The Macaulay Duration of a Zero-Coupon Bond is simply its time to maturity. (If you only get one payment at year 10, the "average time" you wait is exactly 10 years!)
For any bond that pays coupons, the Macaulay Duration will always be less than its time to maturity.

Key Takeaway: Macaulay Duration tells you the "average" time you have to wait to get your money back, measured in years.

2. Modified Duration: Measuring Price Sensitivity

While Macaulay Duration is measured in years, Modified Duration (denoted as $ D_{mod} $) is a measure of how the price of a portfolio changes when the interest rate ($ i $) changes. It is the percentage decrease in price for a 1% increase in interest rates.

The Formula:
$ D_{mod} = -\frac{P'(i)}{P(i)} $
Where $ P'(i) $ is the derivative of the price with respect to the interest rate.

The Easy Shortcut:
You don't always need calculus! If you already have the Macaulay Duration, you can find Modified Duration easily:
$ D_{mod} = v \cdot D_{mac} = \frac{D_{mac}}{1+i} $

Wait, why the minus sign?
Prices and interest rates have an inverse relationship. When rates go up ($ \Delta i > 0 $), prices go down ($ \Delta P < 0 $). The Modified Duration is usually written as a positive number, so we use it to calculate the price drop like this:
$ \frac{\Delta P}{P} \approx -D_{mod} \cdot \Delta i $

Did you know?
Financial analysts use Modified Duration every day to stress-test portfolios. If a portfolio has a Modified Duration of 7, and interest rates rise by 0.5% (50 basis points), the portfolio value will drop by approximately $ 7 \cdot 0.005 = 3.5\% $.

Key Takeaway: Modified Duration is a percentage change. It tells you how "risky" a bond is when interest rates move.

3. Convexity: Adding the "Curve"

Duration is a great tool, but it assumes that the relationship between price and interest rates is a straight line. In reality, that relationship is a curve. This is where Convexity comes in. Convexity measures how much the "slope" (Duration) changes as interest rates change.

Macaulay Convexity ($ C_{mac} $):
$ C_{mac} = \frac{\sum t^2 \cdot R_t \cdot v^t}{\sum R_t \cdot v^t} $
Notice it's almost the same as Macaulay Duration, but we use $ t^2 $ instead of $ t $.

Modified Convexity ($ C_{mod} $):
This is related to the second derivative of the price:
$ C_{mod} = \frac{P''(i)}{P(i)} $

The Relationship:
Similar to duration, there is a shortcut to link the two if you are using an annual effective interest rate $ i $:
$ C_{mod} = \frac{\sum t(t+1) R_t v^{t+2}}{P} $
Don't panic! Most Exam FM questions focus on the relationship: $ C_{mod} = v^2 (C_{mac} + D_{mac}) $.

Why is Convexity "Good"?
For most bonds, convexity is positive. This means that when interest rates drop, the price increases more than duration predicts. When interest rates rise, the price decreases less than duration predicts. It’s a win-win for the investor!

Key Takeaway: Convexity is a "correction factor" that accounts for the fact that the Price-Interest Rate relationship is a curve, not a straight line.

4. Putting It All Together: The Price Approximation Formula

If you want to be super accurate about how a price changes when rates move, you use both Duration and Convexity. This is a very common exam question!

The Full Approximation Formula:
$ P(i_{new}) \approx P(i_{old}) \cdot [1 - D_{mod} \cdot \Delta i + \frac{1}{2} C_{mod} \cdot (\Delta i)^2] $

Step-by-Step Process:
1. Calculate the initial Price $ P $.
2. Calculate $ D_{mod} $ at the current rate.
3. Calculate $ C_{mod} $ at the current rate.
4. Identify $ \Delta i $ (New Rate - Old Rate).
5. Plug everything into the formula above.

Common Mistake to Avoid:
Forgetting the $ \frac{1}{2} $ in the convexity term! The Taylor Series expansion requires that $ 1/2 $ in front of the second derivative term. If you leave it out, your answer will be way off.

Quick Review Box:
- Macaulay Duration: Weighted average time ($ \sum t \cdot PV / P $).
- Modified Duration: Percentage price sensitivity ($ D_{mac} / (1+i) $).
- Convexity: The "curvature" ($ P'' / P $).
- Price Change: $ \Delta P / P \approx -D_{mod} \Delta i + 0.5 C_{mod} (\Delta i)^2 $.

5. Duration of a Portfolio

Good news! Calculating the duration of a portfolio is very straightforward. It is simply the weighted average of the durations of the individual assets in the portfolio.

If Portfolio $ X $ is made of Asset A and Asset B:
$ D_{port} = w_A D_A + w_B D_B $
Where $ w_A $ is the market value of Asset A divided by the total market value of the portfolio.

Example:
If you have $400 in a bond with duration 5, and $600 in a bond with duration 10:
Total Value = $1,000.
$ w_A = 400/1000 = 0.4 $
$ w_B = 600/1000 = 0.6 $
$ D_{port} = (0.4 \cdot 5) + (0.6 \cdot 10) = 2 + 6 = 8 $.

Key Takeaway: Portfolios follow a "weighted average" rule for both Duration and Convexity. Just make sure you use the market values (Present Values) to determine the weights!

Summary and Final Encouragement

You’ve just covered the essentials of Duration and Convexity! These concepts are the foundation for Immunization, which is how insurance companies and pension funds protect themselves from interest rate swings.

Remember:
- Duration is about Time and First-order Sensitivity.
- Convexity is about Curvature and Second-order Sensitivity.
- Always check if the question asks for Macaulay or Modified – they are different!

Keep practicing those calculations. Once you get the hang of the "Seesaw" analogy and the "Price Approximation" formula, you’ll be picking up easy points on the FM exam. You've got this!

Quick check

Can you answer these now?

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

What is the conceptual definition of Macaulay Duration ($ D_{mac} $)?

Macaulay Duration is the weighted average time until a set of cash flows is received, where the weights are the present values of each cash flow.

What is the mathematical formula for Macaulay Duration?

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

How does the Macaulay Duration of a zero-coupon bond compare to its time to maturity?

The Macaulay Duration of a zero-coupon bond is exactly equal to its time to maturity.

What does Modified Duration ($ D_{mod} $) measure?

Modified Duration measures the price sensitivity of a portfolio, representing the approximate percentage change in price for a 1% change in the interest rate.

What is the shortcut formula to convert Macaulay Duration to Modified Duration?

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

How is Modified Duration used to approximate a change in price ($ \Delta P $)?

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

What is the formula for Macaulay Convexity ($ C_{mac} $)?

$ C_{mac} = \frac{\sum t^2 \cdot R_t \cdot v^t}{\sum R_t \cdot v^t} $

What is the relationship between Modified Convexity ($ C_{mod} $), Macaulay Duration, and Macaulay Convexity?

$ C_{mod} = v^2 (C_{mac} + D_{mac}) = \frac{C_{mac} + D_{mac}}{(1+i)^2} $

What is the benefit of positive convexity for an investor?

Positive convexity means that for a given interest rate change, the price will rise more than duration predicts when rates fall, and fall less than duration predicts when rates rise.

What is the full second-order price approximation formula using both Duration and Convexity?

$ P(i_{new}) \approx P(i_{old}) \cdot [1 - D_{mod} \cdot \Delta i + \frac{1}{2} C_{mod} \cdot (\Delta i)^2] $

How do you calculate the duration of a portfolio containing multiple assets?

The duration of a portfolio is the weighted average of the durations of the individual assets, where the weights are based on the market values (Present Values) of each asset.

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