Introduction to Spot and Forward Rates
In your CM1 journey so far, you have often used a single, constant interest rate \(i\) for every cashflow, regardless of when it occurs. However, in the real world, interest rates usually vary depending on how long you intend to borrow or lend money. This is what we call the term structure of interest rates.
In this chapter, we will break down the two primary ways actuaries describe this structure: Spot Rates (the rates for "immediate" investment) and Forward Rates (the rates we can agree on today for investment "later"). Whether these are expressed discretely or continuously, the underlying logic remains the same. Don't worry if the notation looks intimidating at first—once you see the "time-line" logic, it all clicks into place!
1. Spot Rates: Investing from "Now"
A spot rate is the yield on a zero-coupon bond. In simple terms, it is the annual interest rate agreed upon today for a loan that begins today and lasts for a specific period of \(t\) years.
Discrete Spot Rates
The discrete spot rate for a term \(t\), often denoted as \(y_t\) or \(s_t\), is the annual effective rate of interest applicable over the period from time \(0\) to time \(t\).
If you invest \(1\) today for \(t\) years at the spot rate \(y_t\), your accumulated value at time \(t\) will be:
\( (1 + y_t)^t \)
Conversely, the Present Value (PV) of a payment \(C\) due at time \(t\) is calculated by discounting it using the spot rate for that specific duration:
\( PV = C \cdot (1 + y_t)^{-t} \)
Continuous Spot Rates
Sometimes, it is mathematically more convenient to work with a force of interest. The continuous spot rate \(\bar{y}_t\) is the constant force of interest acting over the period \([0, t]\) that results in the same accumulation as the discrete spot rate.
The relationship is:
\( e^{\bar{y}_t \cdot t} = (1 + y_t)^t \)
The Present Value of a payment \(C\) at time \(t\) using continuous spot rates is:
\( PV = C \cdot e^{-\bar{y}_t \cdot t} \)
Quick Tip: Always check the question to see if it asks for an "annual effective rate" (Discrete) or a "force of interest" (Continuous). If it just says "spot rate," look at the context of the formulas provided!
2. Forward Rates: Locking in the Future
Imagine you know you will have \$1,000 to invest in 2 years' time, and you want to lock in an interest rate for that future investment today. This "locked-in" rate is the forward rate.
Discrete Forward Rates
The discrete forward rate \(f_{t, n}\) is the annual effective rate of interest agreed today for an investment made at time \(t\) for a duration of \(n\) years (ending at time \(t+n\)).
The fundamental principle here is No Arbitrage. This means that investing for \(t+n\) years at the spot rate \(y_{t+n}\) should give the same result as investing for \(t\) years at the spot rate \(y_t\) and then reinvesting at the forward rate \(f_{t, n}\).
The Core Formula:
\( (1 + y_{t+n})^{t+n} = (1 + y_t)^t \cdot (1 + f_{t, n})^n \)
From this, we can solve for the forward rate:
\( (1 + f_{t, n})^n = \frac{(1 + y_{t+n})^{t+n}}{(1 + y_t)^t} \)
Continuous Forward Rates
In a continuous world, the forward rate is often expressed as an instantaneous force of interest at time \(t\), denoted as \(f(t)\) or \(\delta_t\).
The relationship between the continuous spot rate \(\bar{y}_T\) and the instantaneous forward rates is that the spot rate is the average of the forward rates over that period:
\( \bar{y}_T = \frac{1}{T} \int_0^T f(t) dt \)
Key Takeaway: Spot rates look from today forward. Forward rates look at a specific window in the future. They are linked by a simple "chain" of accumulation.
3. Comparing Spot and Forward Rates
To help you remember the difference, think of this analogy:
The Spot Rate is like the average speed of a car from the start of the journey to a specific milestone. The Forward Rate is like the speed the car is expected to travel during a specific leg of the journey later on.
Common Relationships:
- If spot rates are increasing with time (an "upward sloping yield curve"), then the forward rate for a future period will be higher than the spot rate for the total period.
- If spot rates are decreasing ("inverted yield curve"), the forward rates will be lower than the spot rates.
- If spot rates are constant for all terms, the forward rates will be equal to the spot rates.
4. Step-by-Step: Calculating a Forward Rate
If you are given the 1-year spot rate \(y_1 = 4\%\) and the 2-year spot rate \(y_2 = 5\%\), how do you find the 1-year forward rate starting in 1 year's time (\(f_{1, 1}\))?
- Set up the link: \((1 + y_2)^2 = (1 + y_1)^1 \cdot (1 + f_{1, 1})^1\)
- Plug in the numbers: \((1.05)^2 = (1.04) \cdot (1 + f_{1, 1})\)
- Rearrange: \(1 + f_{1, 1} = \frac{1.05^2}{1.04}\)
- Calculate: \(1 + f_{1, 1} = 1.0601\)
- Final Answer: \(f_{1, 1} = 6.01\%\)
Note: The 1-year forward rate is higher than both spot rates because the yield curve is upward sloping.
5. Summary and Common Pitfalls
Summary:
- Discrete Spot Rate (\(y_t\)): Annual effective rate from \(0\) to \(t\).
- Continuous Spot Rate (\(\bar{y}_t\)): Force of interest from \(0\) to \(t\).
- Discrete Forward Rate (\(f_{t, n}\)): Annual effective rate from \(t\) to \(t+n\).
- Instantaneous Forward Rate (\(f(t)\)): Force of interest at time \(t\).
Mistakes to Avoid:
- Mixing Discrete and Continuous: Ensure you don't use \(e\) when you should be using \((1+i)\), and vice versa. Read the question carefully!
- Term Errors: In the formula \((1+f_{t, n})^n\), the power is \(n\) (the length of the forward period), not \(t+n\) (the end date).
- Ignoring the "No Arbitrage" Logic: If you forget the formula, just draw a time-line. The accumulation from \(0 \to t\) followed by \(t \to t+n\) must equal the direct accumulation from \(0 \to t+n\).
In the next chapters, we will use these rates to calculate Par Yields and assess the Duration of cashflows, where the sensitivity to interest rate changes becomes our main focus.