Future and present value of mixed streams

When dealing with the Time Value of Money (TVM), we can encounter situations where payments or receipts occur in multiple periods. These are called mixed streams because the payments or receipts are not uniform (i.e., not a single lump sum). In such cases, we deal with the Future Value (FV) and Present Value (PV) of a series of payments or receipts, which may occur at different times.

Mixed Streams:

A mixed stream consists of a series of uneven cash flows occurring at different points in time. These cash flows can be both positive (like incoming payments) or negative (like outgoing payments).

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Future Value of Mixed Streams

✅ Definition:

The Future Value of a Mixed Stream is the sum of the future values of individual cash flows, each grown to the future point at the given interest rate. In other words, we calculate the future value for each payment and then sum them up.

🧮 Formula:

The formula for the future value of mixed streams is:

FV = Σ (CFt × (1 + r)^(n-t))

Where:

• CFt = Cash flow at time $t$
• r = Interest rate per period
• n = Number of periods (the future period)
• t = The specific period in which the cash flow occurs

You calculate the future value of each individual cash flow separately, and then sum them up to get the total future value.

🔍 How it Works:

You take each cash flow (CF) and compound it to the future (the final period). The number of periods each cash flow is compounded depends on how far it is from the future period.

📋 Example:

Let’s consider a scenario where you receive the following mixed stream of cash flows over 3 years, and the interest rate is 5% per year:

• Year 1: ₹1,000
• Year 2: ₹1,500
• Year 3: ₹2,000

We want to calculate the future value of these cash flows at the end of Year 3.

Step-by-Step Calculation:

1. Year 1 Cash Flow (₹1,000):

We need to compound ₹1,000 for 3 periods (since the future value is calculated at the end of Year 3):

FV1 = 1,000 × (1 + 0.05)^3 = 1,000 × 1.157625 = ₹1,157.63

2. Year 2 Cash Flow (₹1,500):

We need to compound ₹1,500 for 2 periods:

FV2 = 1,500 × (1 + 0.05)^2 = 1,500 × 1.1025 = ₹1,653.75

3. Year 3 Cash Flow (₹2,000):

This cash flow is already in the final year, so we don’t need to compound it. The future value of this cash flow is simply ₹2,000.

FV3 = ₹2,000

Total Future Value:

Now, we add up all the future values:

Total FV = FV1 + FV2 + FV3 = ₹1,157.63 + ₹1,653.75 + ₹2,000 = ₹4,811.38

So, the future value of this mixed stream of cash flows at the end of Year 3 is ₹4,811.38.

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Present Value of Mixed Streams

✅ Definition:

The Present Value of a Mixed Stream is the sum of the present values of all the individual cash flows, discounted back to the present time at the given interest rate. In other words, we calculate the present value of each cash flow and then sum them up to get the total present value.

🧮 Formula:

The formula for the present value of mixed streams is:

PV = Σ (CFt / (1 + r)^(t))

Where:

• CFt = Cash flow at time $t$
• r = Interest rate per period
• t = The specific period in which the cash flow occurs

You calculate the present value of each individual cash flow, discounting it to the present, and then sum them up to get the total present value.

🔍 How it Works:

For each cash flow, you discount it to the present time using the given interest rate. The amount of discount depends on how far the cash flow is from the present period.

📋 Example:

Let’s use the same scenario where you receive the following mixed stream of cash flows over 3 years, and the interest rate is 5% per year:

• Year 1: ₹1,000
• Year 2: ₹1,500
• Year 3: ₹2,000

We want to calculate the present value of these cash flows.

Step-by-Step Calculation:

1. Year 1 Cash Flow (₹1,000):

This cash flow occurs in 1 year, so we discount it by 1 period:

PV1 = 1,000 / (1 + 0.05)^1 = 1,000 / 1.05 = ₹952.38

2. Year 2 Cash Flow (₹1,500):

This cash flow occurs in 2 years, so we discount it by 2 periods:

PV2 = 1,500 / (1 + 0.05)^2 = 1,500 / 1.1025 = ₹1,361.11

3. Year 3 Cash Flow (₹2,000):

This cash flow occurs in 3 years, so we discount it by 3 periods:

PV3 = 2,000 / (1 + 0.05)^3 = 2,000 / 1.157625 = ₹1,726.54

Total Present Value:

Now, we add up all the present values:

Total PV = PV1 + PV2 + PV3 = ₹952.38 + ₹1,361.11 + ₹1,726.54 = ₹5,039.03

So, the present value of this mixed stream of cash flows is ₹5,039.03.

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📊 Comparison: Future and Present Value of Mixed Streams

Concept	Formula	Description
Future Value of Mixed Stream	FV = Σ (CFt × (1 + r)^(n-t))	The sum of future values of all cash flows, compounded to the future period.
Present Value of Mixed Stream	PV = Σ (CFt / (1 + r)^(t))	The sum of present values of all cash flows, discounted to the present period.

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🧠 Key Takeaways:

• Future Value (FV) of Mixed Streams helps you understand what a series of uneven cash flows will be worth in the future. You compound each cash flow to the future period and then sum them up.
• Present Value (PV) of Mixed Streams helps you determine the current value of future cash flows. You discount each cash flow back to the present period and then sum them up.
• These calculations are essential when evaluating investments, loans, and cash flow analysis where the amounts or timing of payments vary over time.