Introduction: Putting the Pieces Together

Welcome! So far in your CM1 journey, you have learned how to move a single sum of money forward and backward in time. But in the real world, financial projects rarely involve just one payment. Think of a pension fund, a construction project, or a loan—these involve "streams" of money flowing in and out at different times.

In this chapter, we learn how to value these general cashflow streams. This is a vital skill because it forms the foundation for the "Equation of Value," which you will use to price almost every financial product in the syllabus. Whether the payments are chunky (discrete) or smooth like water (continuous), and whether interest rates are steady or jumping around, the principles remain the same.

1. Discrete Cashflows: The Sum of Parts

A discrete cashflow stream consists of individual payments made at specific points in time. Imagine receiving \$100 today, \$200 in one year, and \$500 in five years.

Present Value (PV)

To find the value of the whole stream today, we simply discount each individual payment to time \(t=0\) and add them up. If we have payments \(c_1, c_2, ..., c_n\) at times \(t_1, t_2, ..., t_n\), the Present Value is:

\(PV = \sum_{j=1}^{n} c_j v^{t_j}\)

Where \(v = (1+i)^{-1}\) is the discount factor.

Accumulated Value (AV)

To find the value at some future time \(T\), we roll every payment forward to that date:

\(AV = \sum_{j=1}^{n} c_j (1+i)^{T-t_j}\)

Quick Tip: You can also find the AV by first calculating the PV and then multiplying the total by \((1+i)^T\). This is often much faster in exams!

Example: Find the PV at 5% effective interest of receiving £100 at time 1 and £200 at time 3.
\(PV = 100(1.05)^{-1} + 200(1.05)^{-3}\)
\(PV = 95.24 + 172.77 = 268.01\)

2. Continuous Cashflows: The "Flow" of Money

Sometimes, payments happen so frequently that we model them as a continuous flow. Think of a large company’s daily sales or a person’s utility bills. We use a payment rate function, often denoted as \(\rho(t)\).

Instead of a sum (\(\sum\)), we use an integral (\(\int\)) to "add up" these infinitely small payments.

Present Value of a Continuous Stream

If money is flowing from time \(a\) to time \(b\) at a rate of \(\rho(t)\), the PV at a constant force of interest \(\delta\) is:

\(PV = \int_{a}^{b} \rho(t) e^{-\delta t} dt\)

Why \(e^{-\delta t}\)? Remember from previous chapters that \(v^t = e^{-\delta t}\). In continuous math, using the force of interest \(\delta\) is almost always easier than using the effective rate \(i\).

Common Exam Scenario: If the payment rate is constant (e.g., \(\rho(t) = 100\)), you can pull the constant out of the integral:
\(PV = 100 \int_{a}^{b} e^{-\delta t} dt\)

Quick Review:
  • Discrete: Use \(\sum\) (Summation). Best for fixed dates.
  • Continuous: Use \(\int\) (Integration). Best for "rates" of payment.

3. Dealing with Varying Interest Rates

In the real world, interest rates aren't locked in forever. The syllabus requires you to handle cases where the interest rate changes over time.

Discrete Changes

If the interest rate is \(i_1\) for the first 2 years and \(i_2\) thereafter, you must discount in stages.

Example: Value a payment of \$1,000 due at time 5. Interest is 4% for years 0-2 and 6% for years 2-5.
Step 1: Discount from time 5 to time 2: \(1,000 \times (1.06)^{-3}\)
Step 2: Discount that result from time 2 to time 0: \([1,000 \times (1.06)^{-3}] \times (1.04)^{-2}\)

Continuous Changes (The Force of Interest)

If the force of interest is a function of time, \(\delta(t)\), we cannot simply multiply by \(e^{-\delta t}\). We must use the total accumulation factor.

The PV of a payment \(C\) at time \(T\) is:
\(PV = C \times \exp\left(-\int_{0}^{T} \delta(s) ds\right)\)

And for a continuous stream \(\rho(t)\) from time \(0\) to \(n\):
\(PV = \int_{0}^{n} \rho(t) \exp\left(-\int_{0}^{t} \delta(s) ds\right) dt\)

Don't panic! This looks scary, but in CM1 exams, \(\delta(s)\) is usually a simple function (like a constant or a simple linear formula) that is easy to integrate.

4. Combined Streams and Deferrals

Often, you will face a "mixed bag." For example: a project that pays nothing for 2 years (deferred), then pays a constant amount for 3 years, followed by a final lump sum.

Strategy for Success:

  1. Draw a Timeline: This is the single most important step. Mark every payment and every interest rate change.
  2. Split the Problem: Calculate the PV of the continuous part and the discrete part separately.
  3. Check Your Dates: If a stream is deferred for \(m\) years, calculate its value at the start of the payment period (time \(m\)), then discount that single value back to time 0.

Example of a deferred stream:
A stream of \$500 p.a. starts at time 5 and ends at time 10.
1. Find the value at time 5.
2. Discount that value by \(v^5\) to get to time 0.

5. Common Pitfalls to Avoid

  • Mixing Rates: Ensure your interest rate period matches your payment period. If payments are monthly, it's often easiest to work in months.
  • Wrong Limits: When integrating, double-check your limits. If a payment starts "at the end of year 2," that is time \(t=2\).
  • Calculator Errors: When calculating \(\exp(-\int \delta dt)\), calculate the integral first, then apply the exponential.

Summary & Key Takeaways

Key Takeaway 1: The Present Value of any cashflow is the amount you would need to invest today, at the given interest rate, to exactly replicate those future payments.

Key Takeaway 2: For discrete flows, \(PV = \sum C_t v^t\). For continuous flows, \(PV = \int \rho(t) v^t dt\).

Key Takeaway 3: When interest rates vary, the discount factor \(v^t\) is replaced by the integral form \(\exp(-\int_0^t \delta(s) ds)\).

Key Takeaway 4: Always break complex problems into smaller, manageable chunks. Calculate the PV of each chunk and then sum them up at the very end.

In the next chapters, we will apply these general principles to specific, standard patterns of payments called annuities. But remember: if you ever forget an annuity formula in the exam, you can always fall back on these first principles of general cashflows!