Welcome to the Term Structure of Interest Rates!
In previous chapters, we often assumed that the interest rate stays the same regardless of how long you borrow or lend money. But in the real world, the interest rate for a 1-year loan is rarely the same as the rate for a 30-year mortgage. This "link" between the length of time and the interest rate is what we call the Term Structure of Interest Rates.
Understanding this is crucial for an actuary because we deal with cash flows that happen many years into the future. By the end of these notes, you'll understand why interest rates change over time and how to calculate them like a pro! Don't worry if this seems tricky at first—we'll break it down step-by-step.
1. What is the Term Structure?
The term structure is simply the relationship between the term (the length of time) and the yield (the interest rate). If you plot this on a graph, you get a Yield Curve.
Quick Review: Why does the time matter?
Imagine you lend $100 to a friend. If they promise to pay you back tomorrow, you might charge a small interest. But if they say they’ll pay you back in 20 years, you’d probably want a much higher rate because you’re waiting longer and taking more risk! That difference in rates is the essence of the term structure.
2. Spot Rates
A Spot Rate is the interest rate agreed upon today for a loan that starts today and lasts for a specific period.
Discrete Spot Rates
We usually denote the n-year spot rate as \( s_n \). If you invest 1 unit today for \( n \) years at the spot rate \( s_n \), your investment grows to:
\( (1 + s_n)^n \)
Continuous Spot Rates (Force of Interest)
Sometimes we use the continuous version, denoted by \( \bar{s}_n \) or \( \delta_n \). The growth factor here is:
\( e^{n \cdot \bar{s}_n} \)
Key Takeaway: Spot rates always start now. If you see \( s_5 \), it means the annual rate for a 5-year investment starting today.
3. Forward Rates
This is where students often get a bit confused, but here is a simple way to think about it: A Forward Rate is an interest rate agreed upon today for a loan that will happen in the future.
The Analogy: Imagine you are booking a hotel for next summer. If you lock in the price today, even though you aren't staying there yet, you are setting a "forward price." A forward rate is just the "forward price" of borrowing money.
Notation and Formulas
We use the notation \( f_{t, n} \) to represent the forward rate applicable between time \( t \) and time \( t+n \).
- \( t \) is when the loan starts.
- \( n \) is the duration of the loan.
Discrete Forward Rate Formula:
\( (1 + s_{t+n})^{t+n} = (1 + s_t)^t \cdot (1 + f_{t, n})^n \)
Continuous Forward Rate Formula:
\( e^{(t+n)\bar{s}_{t+n}} = e^{t\bar{s}_t} \cdot e^{n\bar{f}_{t, n}} \)
Did you know? In the exam, they might use the notation \( f_t \) for a 1-year forward rate starting at time \( t \). Always read the definitions carefully!
4. The Relationship between Spot and Forward Rates
Think of the spot rate as the "average" of all the forward rates leading up to that point. If you want to invest for 2 years, you have two choices that should cost the same (to avoid arbitrage):
- Invest at the 2-year spot rate \( s_2 \) for two years.
- Invest at the 1-year spot rate \( s_1 \) for the first year, and lock in a 1-year forward rate \( f_{1, 1} \) for the second year.
The Equation:
\( (1 + s_2)^2 = (1 + s_1) \cdot (1 + f_{1, 1}) \)
Memory Aid: The Spot is the Total Journey, and the Forward is just one leg of the trip.
5. Yield Curves and their Shapes
A yield curve is a graph of the spot rates (y-axis) against the term (x-axis). There are three main shapes you need to know:
- Normal (Upward Sloping): Long-term rates are higher than short-term rates. This is the most common shape.
- Inverted (Downward Sloping): Short-term rates are higher than long-term rates. This often signals that investors expect a recession.
- Flat: All rates are the same, regardless of the term.
Quick Review Box:
- If forward rates are higher than spot rates, the yield curve is rising.
- If forward rates are lower than spot rates, the yield curve is falling.
6. Theories of the Term Structure
Why does the yield curve change shape? Actuaries use three main theories to explain this:
A. Expectations Hypothesis
This theory suggests that the shape of the curve depends entirely on what people expect interest rates to be in the future. If everyone thinks rates will rise, the curve slopes upward.
B. Liquidity Preference Theory
Investors prefer to have cash "liquid" (available). Lending money for a long time is risky because you can't get your cash back easily. Therefore, investors demand a premium (extra interest) for long-term loans. This explains why the curve is usually upward sloping.
C. Market Segmentation Theory
This theory suggests that the market is divided into "segments." For example, pension funds like long-term bonds, while banks prefer short-term loans. The interest rate in each segment is determined by supply and demand within that specific "bucket," regardless of what's happening in other segments.
7. Common Mistakes to Avoid
1. Mixing up \( t \) and \( n \): In \( f_{t, n} \), always double-check if \( n \) is the end time or the duration. The curriculum usually defines it as duration, but exam questions can vary.
2. Forgetting to compound: When using discrete rates, remember the power of \( n \). \( (1+s_2)^2 \) is not the same as \( 1+2s_2 \)!
3. Assuming the yield curve is always rising: While "Normal" is common, don't assume. Always look at the data provided in the question.
Summary: Key Takeaways
Spot Rates: Rates for investments starting now. Notation: \( s_n \).
Forward Rates: Rates agreed now for the future. Notation: \( f_{t, n} \).
The Link: The spot rate is the geometric mean of the intervening forward rates.
Yield Curve: A visual representation of the term structure (Normal, Inverted, or Flat).
Theories: Expectations (future views), Liquidity Preference (risk premium), and Market Segmentation (supply/demand in niches).
You've reached the end of the Term Structure chapter! Take a break, try a few practice questions on calculating forward rates, and you'll be an expert in no time!