25400 · Financial Literacy
Time Value of Money I: Single Cash Flows
Week 1 of University of Technology Sydney 25400 Financial Literacy introduces the single-cash-flow time value of money: compounding a present amount forward to a future value and discounting a future amount back to present value, with FV = PV(1 + i)^n and PV = FV(1 + i)^−n. It contrasts simple interest (linear, on principal only) with compound interest (exponential, on principal plus accumulated interest) and converts a stated APR into the Effective Annual Rate. These single-sum skills are the foundation for every later valuation and appear directly in the Week-6 in-class Quiz 1.
What this chapter covers
- 01The single-sum core: FV = PV(1 + i)^n and PV = FV(1 + i)^−n
- 02Simple interest FV = P(1 + r·T) — linear growth on the original principal only
- 03Compound interest FV = PV(1 + i)^n — exponential growth on principal plus accrued interest
- 04The three drivers of a TVM answer: time (n), rate (i) and compounding frequency (m)
- 05Periodic rate i = APR / m and number of periods n = years × m
- 06APR vs EAR: EAR = (1 + APR/m)^m − 1, and continuous limit EAR = e^APR − 1
- 07Nominal vs real rate (Fisher): 1 + r_real = (1 + r_nominal) / (1 + inflation)
- 08Excel tooling: PV, FV — with the outflow-negative / inflow-positive sign convention
Simple vs compound future value on the same lump sum
- +1Simple interest earns 7% of the original $5,000 every year and never on the interest, so FV = P(1 + r·T) = 5,000 × (1 + 0.07 × 8) = 5,000 × 1.56 = $7,800.
- +1Compound interest earns interest on interest, so FV = PV(1 + i)^n = 5,000 × 1.07^8. Since 1.07^8 = 1.71819, FV = 5,000 × 1.71819 = $8,590.93.
- +1Difference = 8,590.93 − 7,800 = $790.93. Compounding wins because each year's interest itself earns interest; the gap widens the longer the horizon, which is why compound growth is exponential and simple growth is linear.
Key terms
- Present value (PV)
- The value today of a cash flow to be received in the future, found by discounting: PV = FV(1 + i)^−n. It answers 'what is a future amount worth now?' and is the backbone of every valuation in the subject.
- Future value (FV)
- The value at a future date of an amount invested today, found by compounding: FV = PV(1 + i)^n. It answers 'what will today's money grow to?'
- Simple interest
- Interest charged only on the original principal, so the interest per period is constant and the balance grows linearly: FV = P(1 + r·T). Typically used for short-term lending.
- Compound interest
- Interest charged on the principal plus all previously accumulated interest, so the balance grows exponentially: FV = PV(1 + i)^n. Used for savings, mortgages and most investments.
- Effective Annual Rate (EAR)
- The true annual rate once intra-year compounding is counted: EAR = (1 + APR/m)^m − 1, where m is compounding periods per year. Only EAR lets you compare two rates with different compounding frequencies on a like-for-like basis.
- Periodic rate (i)
- The interest rate per compounding period, i = APR / m. Matching i with the number of periods n = years × m is the step that prevents most TVM errors.
Time Value of Money I: Single Cash Flows FAQ
What is the single most common mistake here?
Mismatching the rate and the period. If interest compounds monthly you must use i = APR/12 and n = years × 12; using an annual rate with a monthly period count (or vice versa) is the classic slip. Convert both to the same compounding basis before you compound or discount.
When do I use simple versus compound interest?
Use simple interest only when the problem explicitly says so (often short-term arrangements); assume compound interest for savings, loans and investments unless told otherwise. Over long horizons the two diverge sharply because compound interest earns interest on interest.
Why does APR differ from EAR?
APR is just the stated nominal rate and ignores how often interest is added within the year. EAR = (1 + APR/m)^m − 1 folds in that compounding, so for any APR a higher compounding frequency m gives a higher EAR. Two products are only comparable on an EAR basis.
Does this appear in the quizzes?
Yes. Single-sum PV/FV, simple-vs-compound and APR-to-EAR conversions are foundational and are examined in the Week-6 in-class Quiz 1 (which covers Weeks 1-5). Practise them until the setup is automatic, since the quizzes are timed and prohibit Generative AI.
Assessment move
Anchor everything on the two single-sum formulas FV = PV(1 + i)^n and PV = FV(1 + i)^−n, and make the rate-and-period conversion a reflex: whenever compounding is not annual, immediately write i = APR/m and n = years × m before touching the power. Build intuition for the three drivers by re-running one example with a longer horizon, a higher rate and a higher compounding frequency and watching the FV rise each time. Rehearse APR-to-EAR both ways, and practise the PV and FV Excel functions with the sign convention (outflows negative). Since Quiz 1 is timed and GenAI-free, drill until each single-sum answer takes seconds; ask Sia to set fresh single-sum problems and check your unit choices.
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