Yield curves, rates of return, duration, convexity, and immunization terminology

Welcome to General Cash Flows & Asset Liability Management!

In this chapter, we are moving beyond simple loans and annuities to look at the "big picture" of how portfolios behave. Think of this as the "manager's toolkit." We will learn how to evaluate investment performance, understand why interest rates change over time, and—most importantly—how to protect a portfolio from the "rollercoaster" of interest rate fluctuations. Don't worry if these terms sound intimidating; we will break them down piece by piece!

1. Rates of Return: Measuring Performance

When you invest money, you want to know how well you're doing. In Exam FM, we look at two main ways to measure this. The key difference between them is who gets the credit (or blame) for the timing of the money moving in and out.

Dollar-Weighted Rate of Return (DWRR)

The Dollar-Weighted Rate of Return is essentially the Internal Rate of Return (IRR) for a portfolio. It accounts for both the interest earned and the timing/size of cash flows.

The Concept: If you put a lot of money into an account right before it crashes, your DWRR will be low. It measures the performance of the actual dollars in the account.

The Calculation: You set the Present Value of all inflows equal to the Present Value of all outflows (including the final balance):
\( PV(Inflows) = PV(Outflows) \)

Time-Weighted Rate of Return (TWRR)

The Time-Weighted Rate of Return measures the performance of the investment itself, ignoring the timing of deposits or withdrawals. This is how we usually evaluate mutual fund managers because they can't control when investors put money in or take it out.

How to calculate it:
1. Break the total time into sub-periods (every time a cash flow occurs).
2. Calculate the growth factor \( (1+j) \) for each sub-period: \( \frac{Value \space Before \space Cash \space Flow}{Value \space After \space Previous \space Cash \space Flow} \).
3. Multiply them all together: \( (1+i_{TW}) = (1+j_1) \times (1+j_2) \times ... \times (1+j_n) \).

Quick Review Box:
- DWRR: Sensitive to when you move money. Use the IRR method.
- TWRR: Ignores the amount of money moved. Use the "chain-multiplying" method.

2. Yield Curves and Forward Rates

In the real world, the interest rate for a 1-year loan is rarely the same as the rate for a 30-year loan. This relationship is shown on a Yield Curve.

Spot Rates

A Spot Rate (denoted as \( r_t \)) is the annual effective interest rate for a "zero-coupon bond" that matures at time \( t \). It is the rate you get if you invest money today for exactly \( t \) years.

Forward Rates

A Forward Rate is an interest rate agreed upon today for a loan that will happen in the future.

Notation: \( f_{t, t+k} \) is the rate for a loan starting at time \( t \) and ending at time \( t+k \).

The "Chain" Rule: You can think of a long-term spot rate as a series of forward rates linked together. For example, to find the 2nd-year forward rate \( f_{1,2} \):
\( (1+r_2)^2 = (1+r_1) \times (1+f_{1,2}) \)

Analogy: Imagine you are planning a trip. The 2-year Spot Rate is the price for a direct 2-year flight. The Forward Rate is like booking the second leg of a connecting flight in advance.

3. Duration: Sensitivity to Interest Rates

Interest rates go up, and bond prices go down. Duration tells us how much the price will change.

Macaulay Duration (\( D_{mac} \))

Think of Macaulay Duration as the "average time" it takes to receive your money back, weighted by the present value of the cash flows.

The Formula:
\( D_{mac} = \frac{\sum t \cdot v^t \cdot CF_t}{\sum v^t \cdot CF_t} \)
Note: The denominator is just the Price (P) of the investment!

Modified Duration (\( D_{mod} \))

Modified Duration measures the percentage change in price for a 1% change in interest rates. It is the derivative of the price with respect to the interest rate.

The Easy Connection:
\( D_{mod} = \frac{D_{mac}}{1+i} \)

Did you know? The Macaulay Duration of a zero-coupon bond is always equal to its term to maturity. So, a 10-year zero-coupon bond has a \( D_{mac} \) of exactly 10!

4. Convexity: The Curvature

Duration is like a straight-line approximation of a curve. It's pretty good for small changes in interest rates, but it gets less accurate as the change gets bigger. Convexity is the "correction factor" that accounts for the curve.

Macaulay Convexity (\( C_{mac} \)):
\( C_{mac} = \frac{\sum t^2 \cdot v^t \cdot CF_t}{\sum v^t \cdot CF_t} \)

Modified Convexity (\( C_{mod} \) or \( C \)):
This is related to the second derivative of the price. In Exam FM, remember that high convexity is generally good for an investor because it means the price increases more when rates fall and decreases less when rates rise.

5. Immunization: Protecting the Portfolio

Immunization is a strategy used by insurance companies to ensure they can pay their future liabilities even if interest rates change unexpectedly.

Redington Immunization

To achieve Redington Immunization for a small change in interest rates (\( \Delta i \)), you must meet three conditions:

1. Present Value (PV): \( PV_{Assets} = PV_{Liabilities} \)
2. Duration: \( D_{Assets} = D_{Liabilities} \)
3. Convexity: \( Convexity_{Assets} > Convexity_{Liabilities} \)

Full Immunization

Full Immunization is a stronger version that protects against any change in interest rates. To do this, you typically need to "sandwich" the liability. This means you have at least one asset cash flow before the liability and at least one asset cash flow after the liability, while still matching the PV and Duration.

Common Mistake to Avoid:
When checking for Redington Immunization, many students forget the third step (Convexity). If the Assets have lower convexity than the Liabilities, the portfolio is actually at more risk!

Summary Table: Key Terminology

Term: \( D_{mac} \)
What it is: Weighted average time until cash flows.
Why it matters: Used to calculate price sensitivity.

Term: \( D_{mod} \)
What it is: \( - \frac{P'(i)}{P(i)} \)
Why it matters: Direct measure of % price change.

Term: Spot Rate (\( r_t \))
What it is: Yield on a zero-coupon bond.
Why it matters: Foundation for all pricing.

Term: Immunization
What it is: Matching Assets and Liabilities.
Why it matters: Keeps companies solvent when rates move.

Keep practicing! Asset-Liability Management is often the "ah-ha!" moment for students where all the interest rate formulas finally come together to solve real business problems. You've got this!