FNCE90065 Chap.3 Present Value, Future Value and Rates
Present Value, Future Value and Rates
Time value of money makes cash flows comparable across dates. This chapter shows compounding forward, discounting backward, single-cash-flow valuation and rate conversion. It distinguishes periodic rates, nominal annual percentage rates, effective annual rates and real returns.
The recurring discipline is to choose one comparison date, align the rate with the cash-flow period, preserve signs and predict whether the answer should rise or fall before calculating.
What this chapter covers
- 01
Future value compounds present cash by a growth factor
- 02
Present value discounts future cash to a chosen comparison date
- 03
The rate and number of periods must use the same time unit
- 04
APR reports a periodic rate multiplied by periods per year
- 05
EAR reports actual one-year growth after compounding
- 06
The Fisher relationship links nominal return, real return and inflation
Worked example · free
Find the deposit needed for a future target
- 1Place $28,000 at year four and the unknown deposit at time zero.
- 1Use PV equals FV divided by the growth factor because value moves backward.
- 1Compute 28,000 divided by 1.055 to the power of four.
- 1Present value is approximately $22,602 and must be below the future target.
Key terms
- Future value
- Future value is the amount a present cash flow becomes after earning a return for specified periods.
- Discount factor
- A discount factor converts a future cash flow into value at an earlier date.
- Periodic rate
- A periodic rate is the interest rate applied during one compounding or payment interval.
- Annual percentage rate
- Annual percentage rate is a nominal annual quote formed from a periodic rate and compounding frequency.
- Effective annual rate
- Effective annual rate is the actual annual growth produced after within-year compounding.
Present Value, Future Value and Rates FAQ
How do I decide between present value and future value?
Choose the comparison date first. If a current amount must be moved to a later date, compound forward. If a future amount must be expressed at an earlier date, discount backward. The two operations use the same growth factor in opposite directions.
Why must rate and period use the same unit?
The exponent counts applications of the periodic rate. Monthly cash flows and a monthly period count therefore need a monthly rate. Using an annual rate with monthly periods applies the annual growth factor too many times and produces a large but sometimes unnoticed error.
What is the difference between APR and EAR?
APR is a nominal quote based on periodic rate times periods per year. EAR measures actual one-year growth after compounding. For a positive rate with more than annual compounding, EAR exceeds APR. Convert alternatives to the same basis before comparing their cost or return.
Exam move
Draw a timeline for every question, even when there is only one future cash flow. Label time zero, the future date, the cash-flow direction and the rate unit. Predict whether present value must be below future value at a positive rate. Practise converting APR to a periodic rate and EAR, then reverse the relationship to recover a periodic rate from EAR. Keep full precision in growth factors until the final line.
When an answer fails the direction check, inspect the exponent and whether you multiplied or divided before repeating calculator entry. Build a small matrix with present value or future value on one axis and known rate or unknown rate on the other. Solve one original question in each cell so rearranging the same relationship becomes familiar.
Use annual, quarterly and monthly versions of one cash flow to practise translating years into total periods and annual quotes into periodic rates. For rate comparisons, write the conversion path explicitly: APR to periodic rate to EAR, or EAR to periodic rate through a root. Do not divide an EAR by the number of periods.
Add a Fisher-effect drill in which you compute the exact real return and compare it with the nominal-minus-inflation approximation, then explain why the interaction term creates the difference. Test sensitivity by increasing the horizon while keeping the rate fixed and by increasing the rate while keeping the horizon fixed. State the expected direction before calculation.
Maintain an error list with three headings: wrong date, wrong rate unit and wrong operation. Most mistakes will fit one of them. On the last pass, cover the formula and rebuild it from the question: moving forward means repeated growth, while moving backward means dividing by that same accumulated factor. Include one unknown-horizon problem and one unknown-rate problem.
After solving with logarithms or roots, substitute the answer into the original accumulation equation; this catches rearrangement errors that a plausible-looking percentage or period count can conceal.
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