Time Value of Money
Understand compounding, discounting and how to compare cash flows on the same date.
Original bilingual pilot · About 30–40 minutes · No prerequisites beyond basic arithmetic.
What you will be able to do
- Distinguish present value (PV) and future value (FV).
- Use a rate and number of periods with consistent units.
- Calculate annual and more frequent compounding.
- Explain the assumptions behind a numerical answer.
This is a standalone IBBC Capital foundation lesson. It is not an official CFA Institute lesson, examination or credential. The examples and questions were written for this pilot; they do not reproduce an official question bank.
Original content drafted with AI assistance. IBBC Capital is responsible for editorial review and content. Foundation self-study lesson.
1. Put cash flows on a timeline
Present value is an amount at the valuation date, called time 0. Future value is an equivalent amount at a specified later date, given an assumed rate. With a positive rate, money available today can grow before that later date.
In this example, the rate is 6% per year, interest is reinvested annually and there are no additional deposits or withdrawals. Amounts are hypothetical currency units, not a forecast of returns.
2. Grow forward; discount backward
For a single cash flow with a constant effective rate per period:
FV = PV × (1 + r)n
PV = FV ÷ (1 + r)n
Here r is the rate per period as a decimal, and n is the number of those same periods. A 6% annual rate is 0.06; two years means n = 2 when compounding annually.
Worked example A · Future value
PV = 10,000; r = 0.06; n = 2. After year 1: 10,000 × 1.06 = 10,600. After year 2: 10,600 × 1.06 = 11,236. The second year earns 636, including 36 of interest on the first year’s interest.
Worked example B · Present value
An amount of 12,100 is due in two years. At a 10% annual discount rate, PV = 12,100 ÷ 1.10² = 10,000. This is a value under the stated discount-rate assumption; it does not establish whether a real investment is suitable.
3. Keep the rate and the periods consistent
If j is a nominal annual rate compounded m times per year and t is the number of years, the periodic rate is j/m and the number of periods is m × t.
FV = PV × (1 + j/m)m × t
EAR = (1 + j/m)m − 1
For 5,000 at a nominal 8% annual rate compounded quarterly for three years: m = 4, periodic rate = 0.08/4 = 0.02, and n = 12. FV = 5,000 × 1.02¹² ≈ 6,341.21. The effective annual rate (EAR) is 1.02⁴ − 1 ≈ 8.2432%.
Do not divide an effective annual rate by 12 to obtain an exact monthly equivalent. For an effective annual rate R, the equivalent monthly rate is (1 + R)^(1/12) − 1. State whether a quoted annual rate is nominal or effective before calculating.
4. Check the assumptions
- The rate stays constant and interest is reinvested at that rate.
- There is one initial amount, with no intervening deposits or withdrawals.
- Fees, taxes, inflation, default risk and exchange-rate changes are excluded.
- Use the same currency and valuation date when comparing amounts.
- Keep full precision during calculation; round only the displayed final amount.
Before looking at a solution, ask: Which date am I valuing? Is this rate annual or periodic? How many periods apply? What cash flows are missing from my model?
Practice calculator
Single amount; fixed nominal annual rate; whole years. No deposits, withdrawals, fees, taxes or inflation. The example values are editable.
Knowledge check · 6 original questions
Answer every question, then check your work. At least 5 out of 6 correct answers marks this sample lesson complete on this browser. You may retry. This is a learning checkpoint, not an examination result or credential.
Prepared questions and answers
These are fixed explanations written for the lesson. There is no live AI tutor on this page.
Is money today always worth more?
The usual statement assumes a positive rate for comparable cash flows. At zero, PV and FV are equal. With a negative rate, FV can be less than PV. Risk and purchasing power need separate analysis.
How is compound interest different from simple interest?
Simple interest on principal P uses P × (1 + r × n). Compound interest uses P × (1 + r)^n. For 1,000 at 10% for three annual periods, the totals are 1,300 and 1,331 respectively.
Can I compare amounts received on different dates directly?
First move them to one valuation date using consistent assumptions. For example, 3,000 received at the end of year 1 grows to 3,307.50 at the end of year 3 at 5% annually. Compare that with 3,000 received at the end of year 3.
Bilingual glossary
- Present value (PV)
- Giá trị hiện tại
- Future value (FV)
- Giá trị tương lai
- Compounding
- Ghép lãi / tích lũy lãi kép
- Discounting
- Chiết khấu
- Nominal annual rate
- Lãi suất danh nghĩa năm
- Effective annual rate (EAR)
- Lãi suất hiệu dụng năm
Next suggested topic: multiple cash flows and annuities. Not included in this sample lesson.