Present value using a yield curve from spot and forward rates

Welcome to the World of Variable Interest Rates!

Up until now, you have likely been using a single interest rate (\( i \)) to value cash flows. While that makes the math simpler, the real world is a bit more complex. In the professional world of actuarial science and finance, interest rates usually depend on when a cash flow occurs. This is known as the Term Structure of Interest Rates.

In this chapter, we are going to learn how to value cash flows using Spot Rates and Forward Rates. Don't worry if these terms sound intimidating! By the end of these notes, you’ll see that they are just different ways of looking at the same "price of money." Let’s dive in!

1. Understanding Spot Rates

A Spot Rate is the annual interest rate currently available for a single payment to be made at a specific time in the future. We usually denote the spot rate for a term of \( t \) years as \( s_t \).

The Concept: Imagine you want to buy a "Zero-Coupon Bond" (a bond that pays nothing until it matures). If you buy a 1-year bond, the interest rate you get is \( s_1 \). If you buy a 5-year bond, the interest rate you get is \( s_5 \). Usually, these rates are not the same!

The Yield Curve: If you plot these spot rates on a graph where the x-axis is "Time" and the y-axis is "Interest Rate," the resulting line is called the Yield Curve.
• Normal Yield Curve: Rates go up as time increases (investors want more reward for locking money away longer).
• Inverted Yield Curve: Long-term rates are lower than short-term rates.

Calculating Present Value (PV) using Spot Rates

When you have a series of cash flows at different times, you cannot use one single \( i \). Instead, you must discount each cash flow by its specific spot rate.

The Formula:
\( PV = \frac{CF_1}{(1+s_1)^1} + \frac{CF_2}{(1+s_2)^2} + \dots + \frac{CF_n}{(1+s_n)^n} \)

Example: You are promised \$100 in one year and \$200 in two years. The spot rates are \( s_1 = 3\% \) and \( s_2 = 4\% \).
\( PV = \frac{100}{1.03} + \frac{200}{(1.04)^2} = 97.09 + 184.91 = 282.00 \)

Quick Review:
• Spot Rate (\( s_t \)): The rate for a "point-to-point" investment from today until time \( t \).
• Key Rule: Use the spot rate that matches the timing of the cash flow.

2. Understanding Forward Rates

A Forward Rate is an interest rate that is agreed upon today for a loan or investment that will start in the future and last for a specific duration.

The Analogy: Think of a spot rate like a "non-stop flight" from today to Year 3. A forward rate is like one "leg" of a connecting flight. It’s the rate you would earn during Year 3 specifically.

Notation Alert: Notation can vary, but for Exam FM, you will often see:
• \( f_t \): This usually represents the 1-year forward rate for the period from time \( t \) to \( t+1 \).
• \( f[t_1, t_2] \): The annual interest rate for the period starting at \( t_1 \) and ending at \( t_2 \).

The Relationship Between Spot and Forward Rates

There is a very important mathematical link between these two. To avoid "arbitrage" (making free money), the result of investing for 2 years at the 2-year spot rate must be the same as investing for 1 year at the 1-year spot rate and then "rolling" that money over for another year at the 1-year forward rate.

The "Golden Equation":
\( (1+s_t)^t = (1+s_{t-1})^{t-1} \times (1 + f_{t-1, t}) \)

Wait, let's simplify that!
To find the 1-year forward rate for Year 2 (\( f_1 \)):
\( (1+s_2)^2 = (1+s_1) \times (1+f_1) \)

Step-by-Step Calculation:
1. Take the total growth over the long period: \( (1+s_2)^2 \).
2. Divide by the growth of the first period: \( (1+s_1) \).
3. Subtract 1 to get the rate.

Did you know? If the spot rate \( s_2 \) is higher than \( s_1 \), the forward rate \( f_1 \) must be higher than both of them to "pull" the average up!

3. Calculating PV using Forward Rates

If you are given a string of 1-year forward rates, you can find the PV by discounting one year at a time, moving backward through the chain of rates.

The Process:
To discount a cash flow at time 3 to time 0:
\( PV = \frac{CF_3}{(1+s_1)(1+f_1)(1+f_2)} \)

Example: You have \$500 due at time 2. You are given \( s_1 = 5\% \) and the 1-year forward rate starting at time 1 is \( f_1 = 7\% \).
\( PV = \frac{500}{(1.05)(1.07)} = 445.03 \)

Key Takeaway:
• Spot rates look at the whole journey from time 0.
• Forward rates look at individual segments of the journey.
• You can build any spot rate if you have all the forward rates that come before it.

4. Common Mistakes and How to Avoid Them

Don't worry if this seems tricky at first—most students find the notation confusing. Here are the most common traps to watch out for:

1. Mixing up the periods: Always double-check if a forward rate is "the rate for year 3" (which covers time 2 to 3) or "the rate starting in 3 years."
2. Forgetting exponents: When using spot rates, you must use the exponent \( t \). When using 1-year forward rates in a chain, you are multiplying several \( (1+f) \) terms, each usually with an exponent of 1.
3. Misinterpreting "Annual Effective": In this chapter, almost all rates are quoted as annual effective rates unless stated otherwise. If you see a "nominal rate compounded semiannually," convert it to annual effective first to make your life easier!

5. Summary and Quick Formula Sheet

The Spot Rate Formula for PV:
\( PV = \sum \frac{CF_t}{(1+s_t)^t} \)

The Forward Rate Connection:
\( (1+s_n)^n = (1+f_{0,1})(1+f_{1,2})\dots(1+f_{n-1,n}) \)
(Note: \( f_{0,1} \) is just another name for \( s_1 \))

The General Link:
For a period from time \( m \) to time \( n \):
\( (1+s_n)^n = (1+s_m)^m \times (1+f_{m,n})^{n-m} \)

Final Tip for Success: On the exam, if you are given a table of spot rates and asked for a forward rate, draw a timeline! Mark the "Total Growth" (the long spot rate) and the "Known Growth" (the shorter spot rate). The "Missing Link" is your forward rate.

You've got this! Understanding the relationship between these rates is a huge step toward mastering the "General Cash Flows" section of Exam FM.