Notes on Time Value of Money, Cash Flows, and Investment Decisions

Time Value of Money and Investment Decision Rules

Core idea: when choosing between alternatives, focus on the difference between cash flows; you can invest money and earn returns, so the timing of cash matters.
Marginal economics rule: marginal revenue must exceed marginal cost for an option to be worth pursuing (revenue > cost).
- Exceptions can exist, but in general higher revenue than cost is needed.
Additional risk generally implies the expectation of higher returns; risk-return tradeoff is part of the framework.
Real-world context: many slides and numbers come from standard textbooks; practice with example scenarios to see how the rules apply.

Principle 1: Time value of money

A dollar earned today is worth more than a dollar earned tomorrow because it can be invested to earn return.
Example (10% interest):
- $100 grows to $110 in one year.
- $100 grows to $1.21 after two years, i.e., $121.
Takeaway: when comparing cash flows at different times, present value and future value calculations are essential.

Principle 2: The difference between alternatives matters (not the absolute numbers alone)

If two options have the same maintenance costs (or any other cost component), the maintenance cost does not affect the comparison – it’s a wash.
What matters is the net difference in cash flows between alternatives.

Principle 3: Revenue must exceed cost (with exceptions)

In general, you should pursue options where revenue exceeds cost.
Acknowledge that there can be exceptions or strategic reasons to accept negative net cash flows in the short term, but the core rule emphasizes profitability.

Principle 4: Higher risk requires higher expected return

Taking on additional risk generally needs to be compensated by higher expected returns.
This aligns with the risk-return tradeoff in finance.

Lottery payout example (payout structures and discounting intuition)

Hypothetical lottery two payment schemes:
- Option A: $192,000,000 paid today (lump sum).
- Option B: $9,800,000 per year for the next 30 years.
The choice depends on discounting future payments back to present value using the market interest rate, i.
Key idea: value of money today vs money received later is determined by the discount rate (the market rate).

Market interest rate and discounting concepts

Market interest rate is typically denoted by the variable i (the instructor uses a capital I in some talk, but conventionally i is used).
Purpose: to move benefits (or costs) across time to a single point in time for comparison.
Framework: we discount future cash flows to present value or accumulate present cash flows to a future value using the rate i.
Real-world reference: when buying a house or making investment decisions, the market rate is the cost of money used in discounting/accumulation.

Example scenario: house purchase intuition

Example mention: buy a house for 200{,}000.
The question you’d answer with the rate i is: what is the present value of future payments, or what payment would be required today to achieve a given future value?

Cash flow diagram (cash flow visualization)

Purpose: visualize cash inflows (money received) and cash outflows (money paid) across time periods.
Notation and conventions:
- Up arrow = money received (inflow).
- Down arrow = money paid (outflow).
- The initial amount (time period zero) is the starting point of the diagram.
- Example interpretation: when borrowing, you may see an initial inflow (net amount received) and then periodic outflows (payments).
Important nuance: the initial value is placed at time period zero (the “now” moment), not at the end of the first period.

End-of-period convention (common real-world simplification)

Real-world practice: do not adjust for interest every single day; instead, assume all money earned within a period is received at the end of that period.
This means:
- For a restaurant example, all income within a month would be treated as if earned on the last day of the month.
- In the example diagram, all cash flows within a period are aggregated and represented at the end of the period.
Benefit: reduces the number of cash flows to track (e.g., 12 monthly flows instead of 365 daily flows).

Simple vs. compound interest (definitions and intuition)

Simple interest: interest calculated only on the principal, not on accumulated interest (not elaborated deeply in the lecture, but mentioned as a distinction).
Compound interest: interest earned on both the initial principal and the accumulated interest from previous periods.
Compound interest intuition:
- Start with PV = 1000 at rate i = 0.10 (10%).
- End of year 1: earn 100 = 0.10 imes 1000, new balance = 1100.
- End of year 2: earn 10% of 1100 = 110, new balance = 1210, and so on.
This compounding leads to exponential growth over time.

Future value formula (central tool)

The basic formula for future value given present value, rate, and periods:
FV = PV imes (1 + i)^n
Notation:
- FV = Future Value
- PV = Present Value
- i = interest rate per period
- n = number of periods
Much of the homework involves solving for one of the four variables (often with three known), which is algebraic but a standard routine.
Practical takeaway: with exponential growth, values can accelerate quickly, underscoring the time value of money.

Practical computation notes from the lecture

An amortization-like problem was presented:
- Borrowed amount: 20{,}000 at 9 ext{%} for five years.
- Upfront cost charged by lender: 200, so the cash received is 20{,}000 - 200 = 19{,}800.
- Annual payment (as stated in the lecture): 51.42 per year for five years (note: the lecturer acknowledges this value and suggests it will be solvable quickly later; this figure appears to be inconsistent with a $20{,}000 loan at 9% over five years, but it is used as a teaching illustration).
- Total of the annual payments shown: five payments of 51.41 (the lecturer rounds slightly differently in places).
- Observation from the lecturer: these numbers illustrate how banks make money through repayment over time, with the cash flow diagram showing inflows and outflows.
Acknowledgment that the exact numerical values may look odd (likely due to units or rounding in the slide), but the method (cash flows, present value/ Future value, and end-of-period convention) is the key learning point.

End-of-period convention vs. daily/continuous compounding

In practice, many problems use end-of-period convention for practicality.
If you were to model month-to-month cash flows in real life, you’d either use monthly periods with a monthly rate or convert to an equivalent annual rate for reporting.

Homework and course resources (tips for study and exams)

A substantial portion of homework problems (Homework 1 in the course) revolve around solving for one of the four variables in the future value formula:
- Given PV, i, n, solve for FV, or vice versa; or solve for i or n in algebraic form.
Common sense: identify the time period, determine whether you should discount or accumulate, and apply the appropriate formula.
The instructor often provides a formula sheet, tables, hints, and solutions on the course platform (Canvas).
- Location example: In Assignment 1, you can download the problem set, a formula sheet, tables, hints, and solutions.
Instructor notes about course structure and logistics:
- The course is strong and practical; the instructor emphasizes its value.
- There is an option to take a spring-term version of the course starting after spring break: a two-hour session right after spring break plus the standard three-hour class, with an additional one-hour credit for a possible Italy trip and internship.
- Internship: you do not have to take the internship to participate in the Italy trip, but doing an internship is possible and encouraged by employers who value the experience.
- Internship typically concludes in early June; you can still go to Europe and complete an internship.

Real-world connections and implications

Time value of money underpins investment decisions, loan pricing, and retirement planning.
End-of-period convention reflects practical financial reporting and simplifies cash flow modeling.
Discounting future cash flows helps compare heterogeneous cash streams (like lottery annuity vs. lump-sum).
The exponential growth nature of compounded interest underscores why delaying investment or postponing payments can dramatically affect outcomes.
Understanding these concepts informs personal finance decisions (loans, mortgages, retirement) and corporate finance (capital budgeting, project evaluation).

Quick reference formulas (LaTeX)

Future value from present value: FV = PV imes (1 + i)^n
Conceptual: present value of a future cash flow discounts at the market rate i to a single point in time
Amortization intuition (loan example): periodic payment PMT for loan amount P, rate i, term n can be found with the standard amortization formula (not shown explicitly in the transcription, but commonly used in practice):
- PMT = P imes rac{i}{1 - (1 + i)^{-n}}
End-of-period convention: treat all income in a period as if earned at the end of the period for simplicity and consistency in cash-flow planning.

Notes on study approach

Focus on understanding the reasoning behind the formulas, not just memorizing numbers.
Practice drawing cash flow diagrams for different scenarios (loan, investment, lottery payoffs) to build intuition for inflows/outflows and the timing of payments.
Be comfortable with converting between present value and future value, and with solving for different variables in the FV/PV framework.

Reminders about course logistics (brief recap)

Course resources (assignments, formula sheets, tables, hints) are available on the course platform (Canvas).
The instructor encourages students to consider the spring-term option and Italy trip, which may involve three weeks abroad and a potential internship afterward.
The internship is optional but commonly valued by employers; doing it does not preclude participating in the Italy trip.