Time Value of Money: Valuing Cash Flow Streams

Value a series of many cash flows.
Value a perpetual series of regular cash flows known as a perpetuity.
Value a common set of regular cash flows termed an annuity.
Understand the notation used in financial calculations:
- c: Cash flow
- Cn: Cash flow at date n
- FV: Future value
- FVn: Future value on date n
- g: Growth rate
- N: Date of the last cash flow in a stream of cash flows
- P: Initial principal or deposit, or equivalent present value
- PV: Present value
- r: Interest rate or rate of return
Value both perpetuities and annuities when cash flows grow at a constant rate.
Compute the number of periods, cash flow, or rate of return of a loan or investment.
Value cash flow streams with non-annual payments.

Financial managers compare costs and benefits of projects, typically involving multiple future periods.
Engage in trading off a known upfront cost against uncertain future benefits.
Evaluating these financial investments requires computing present values of future cash flows.
This chapter builds on previously developed tools to value any series of cash flows, including shortcuts for valuing annuities, perpetuities, and special cases.
Future chapters will address how interest rates are quoted and determined, crucial for cash flows occurring more frequently than annually.

Definition: A stream of cash flows refers to a series of cash flows that occur over multiple periods.
Use timelines to represent cash flow streams, allowing for clearer financial problem-solving.

Rule 1: Only values at the same point in time can be compared or combined.
Rule 2: To calculate a cash flow's future value, compound it using the following formula:
- FV_n = C imes (1 + r)^n where:
- C = cash flow
- r = interest rate
- n = number of periods
Rule 3: To calculate the present value of a future cash flow, discount it using:
- PV = rac{C}{(1 + r)^n} where:
- C = cash flow
- r = interest rate
- n = number of periods

Scenario: Planning to save $1000 today and $1000 at the end of each of the next two years, with a 10% interest rate.
- Timeline illustration:
- Year 0: $1000
- Year 1: $1000
- Year 2: $1000
Calculation:
- Calculate future values:
1. For first deposit (Year 0): $1000 × 1.10² = $1210
2. For second deposit (Year 1): $1000 × 1.10¹ = $1100
3. Combine all future values by summing:
  - Total = $1210 + $1100 + $1000 = $3641
Alternative Approach: Calculate the future value separately, yielding the same future value of $3641.

For cash flows occurring at different times, the present value calculation is crucial:
- General formula:
- PV = C0 + rac{C1}{(1 + r)} + rac{C2}{(1 + r)^{2}} + ext{…} + rac{CN}{(1 + r)^N} where:
- C0, C1, …, CN are the cash flows at respective times 0, 1, …, N.

Problem Statement: Determine how much a lender (Uncle Henry) should lend based on future payments of $5000 in year 1 and $8000 for the subsequent three years at an interest rate of 6%.
Calculation of Present Value:
- PV = rac{5000}{(1.06)^{1}} + rac{8000}{(1.06)^{2}} + rac{8000}{(1.06)^{3}} + rac{8000}{(1.06)^{4}}
- Calculating each term yields:
- $5000/(1.06) = 4716.98$
- $8000/(1.06)^{2} = 7119.97$
- $8000/(1.06)^{3} = 6716.95$
- $8000/(1.06)^{4} = 6336.75$
- Therefore, total PV = $24,890.65

Calculate future value of annual payments to uncle Henry as he keeps them in the bank:
- Yearly record:
- Year 0: $5000
- Year 1 to Year 4 with deposits and compounding yielding $31,423.88 after four years.
Drop back to verify by calculating the FV of the present value given in the bank:
- FV = PV imes (1 + r)^{n}
- FV = $24,890.65 imes (1.06)^{4} = $31,423.87
This shows concordance between two calculation methods.

Financial calculators streamline the process of calculating cash flows with functions:
- Variables include:
- N: Number of periods (NPER)
- PV: Present value
- PMT: Payment/cash flow
- FV: Future value
- l/Y: Interest rate
Example Problem:
- Invest $20,000 with an extra $1000 annual contribution for 15 years at 8% interest.

Result would be -$90,595.50 indicating the total future value available based on contributed amounts.
Note: Sign conventions matter; distinguish between cash inflows and outflows to ensure accuracy in calculations.