💳Principles of Finance Unit 9 – Time Value of Money: Unequal Payments

Key Concepts and Terminology

• Time value of money (TVM) fundamental concept in finance that money available now is worth more than an identical sum in the future due to its potential earning capacity
• Present value (PV) current worth of a future sum of money or stream of cash flows given a specified rate of return
• Future value (FV) value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today
• Discount rate rate used to calculate the present value of future cash flows; represents the opportunity cost of capital
• Annuity series of equal payments or receipts that occur at evenly spaced intervals over a fixed period of time
• Perpetuity annuity that has no end, or a stream of cash payments that continues forever
• Compounding process of earning interest on previously earned interest
  • Compound interest calculated on the initial principal and also on the accumulated interest of previous periods

Time Value of Money Basics

• Money has a time value because of the opportunity to earn a return on invested funds
• A dollar today is worth more than a dollar in the future because of its earning potential
• The time value of money is affected by factors such as inflation, risk, and liquidity
• Inflation erodes the purchasing power of money over time
• Risk refers to the uncertainty of future cash flows; riskier investments require a higher rate of return
• Liquidity refers to how easily an asset can be converted into cash without affecting its market price
• The time value of money is an important consideration in financial decision-making, such as investment analysis and capital budgeting

Unequal Payment Scenarios

• Unequal payments refer to a series of cash flows that are not the same amount in each period
• Examples of unequal payment scenarios include:
  • Loans with variable interest rates
  • Investments with irregular dividend payments
  • Rental agreements with escalating lease payments
• Calculating the present or future value of unequal payments requires treating each payment separately and summing the individual results
• The process involves discounting each cash flow to its present value (for PV calculations) or compounding each cash flow to its future value (for FV calculations)
• The discount rate or interest rate used in the calculations should reflect the riskiness of the cash flows and the time value of money
• Unequal payment scenarios are common in real-world financial situations, such as corporate budgeting, investment analysis, and personal financial planning

Calculating Present Value of Unequal Payments

• To calculate the present value of unequal payments, each cash flow is discounted back to its present value using the appropriate discount rate
• The formula for the present value of a single cash flow is: $PV = \frac{C}{(1+r)^n}$
  • $C$ = cash flow
  • $r$ = discount rate per period
  • $n$ = number of periods until the cash flow occurs
• The total present value of a series of unequal payments is the sum of the present values of each individual cash flow
• Example: Calculate the PV of the following cash flows at a 5% discount rate:
  • Year 1: $1,000
  • Year 2: $1,500
  • Year 3: $2,000
  • $PV_{Year 1} = \frac{1,000}{(1+0.05)^1} =$952.38
  • $PV_{Year 2} = \frac{1,500}{(1+0.05)^2} =$1,360.54
  • $PV_{Year 3} = \frac{2,000}{(1+0.05)^3} =$1,727.65
  • Total PV = $952.38 +$1,360.54 + $1,727.65 =$4,040.57

Calculating Future Value of Unequal Payments

• To calculate the future value of unequal payments, each cash flow is compounded forward to its future value using the appropriate interest rate
• The formula for the future value of a single cash flow is: $FV = C(1+r)^n$
  • $C$ = cash flow
  • $r$ = interest rate per period
  • $n$ = number of periods until the future value is calculated
• The total future value of a series of unequal payments is the sum of the future values of each individual cash flow
• Example: Calculate the FV of the following cash flows at a 4% interest rate after 5 years:
  • Year 1: $2,000
  • Year 2: $3,000
  • Year 3: $4,000
  • $FV_{Year 1} = 2,000(1+0.04)^4 =$2,332.99
  • $FV_{Year 2} = 3,000(1+0.04)^3 =$3,373.44
  • $FV_{Year 3} = 4,000(1+0.04)^2 =$4,326.40
  • Total FV = $2,332.99 +$3,373.44 + $4,326.40 =$10,032.83

Real-World Applications

• Retirement planning: Determining how much to save each year to achieve a desired retirement income stream
• Investment analysis: Evaluating the profitability of investments with irregular cash flows, such as real estate or venture capital
• Loan amortization: Calculating the periodic payments required to pay off a loan with a variable interest rate
• Capital budgeting: Assessing the feasibility and profitability of long-term projects with uneven cash flows
• Lease agreements: Comparing the costs and benefits of leasing versus buying equipment or property with escalating payments
• Valuation of financial instruments: Pricing bonds, stocks, and derivatives with irregular payment schedules
• Personal financial planning: Making decisions about saving, investing, and spending based on the time value of money principles

Common Pitfalls and Mistakes

• Using the wrong discount rate or interest rate for the given scenario
• Failing to account for the timing of cash flows (e.g., assuming all cash flows occur at the end of a period)
• Ignoring the impact of compounding or discounting when dealing with long time horizons
• Mismatching the frequency of cash flows with the discount rate or interest rate (e.g., using a monthly rate for annual cash flows)
• Double-counting cash flows or omitting relevant cash flows from the analysis
• Neglecting to consider the riskiness of cash flows when selecting an appropriate discount rate
• Misinterpreting the results of present value or future value calculations (e.g., confusing PV with FV)
• Failing to conduct sensitivity analysis to assess the impact of changes in key assumptions (e.g., discount rates, growth rates)

Practice Problems and Examples

1. Calculate the present value of the following cash flows at a 6% discount rate:

   • Year 1: $5,000
   • Year 2: $7,000
   • Year 3: $10,000
   • Year 4: $8,000

2. Determine the future value of a series of unequal annual deposits made into an account earning 5% interest per year:

   • Year 1: $2,000
   • Year 2: $3,000
   • Year 3: $4,000
   • Year 4: $5,000
   • Calculate the future value at the end of Year 5.

3. An investor is considering purchasing a rental property that generates the following annual cash flows:

   • Year 1: $12,000
   • Year 2: $15,000
   • Year 3: $18,000
   • Year 4: $20,000
   • Year 5: $25,000
   • If the investor requires a 10% return on their investment, what is the maximum price they should pay for the property?

4. A company is evaluating a project that requires an initial investment of $500,000 and generates the following cash inflows:

   • Year 1: $100,000
   • Year 2: $150,000
   • Year 3: $200,000
   • Year 4: $250,000
   • If the company's cost of capital is 8%, calculate the net present value (NPV) of the project.