CFA Quantitative Methods: Time Value of Money and Hypothesis Testing Explained

CFA quantitative methods at Level 1 rests on two skills: moving money through time, and testing a claim about a population with sample data. Time value of money gives you present value, future value, annuities, and the effective annual rate. Hypothesis testing gives you the null and alternative hypotheses, the test statistic, the p-value, and a clear decision rule. This note works through both with numbers you can redo by hand.

Related CFA revision on this site: CFA Fixed Income in 15 Minutes: Bond Valuation and the Duration Traps That Cost Level 1 Candidates Marks. Bond pricing uses the same discounting ideas covered here.

CFA Quantitative Methods at a Glance

CFA quantitative methods cover time value of money and hypothesis testing: discount and compound cash flows with present value, future value, annuities, and the effective annual rate, then test claims with a null hypothesis, a test statistic, and a p-value decision rule.

Quantitative methods rewards a small set of habits. Draw a timeline before you touch the calculator. Match the interest rate period to the cash flow period. Write down the null hypothesis before you compute anything. State the decision as “reject” or “fail to reject” rather than claiming proof. Most lost marks here come from timing errors and rushed decisions, not hard mathematics.

Time Value of Money: The Core Formulas

Money today is worth more than the same amount in the future because money today can earn a return. Future value is FV = PV x (1 + r)^N, where PV is present value, r is the periodic interest rate, and N is the number of periods. Present value reverses it: PV = FV / (1 + r)^N. Every annuity formula below is built from these two.

Worked example, lump sum. Invest $10,000 today at 6 percent per year for 5 years. The future value is $10,000 x (1.06)^5, and (1.06)^5 equals 1.338226, giving $13,382.26. Turn it around: $13,382.26 / 1.338226 returns $10,000. If your two answers do not reconcile, check the rate period and the number of periods first.

Longer example. Save $5,000 at the end of each year for 20 years at 6 percent, an ordinary annuity, and its future value is $183,927.96. With payments at the start of each year it is $194,963.63, because every payment earns one extra year of interest. Timing alone adds more than $11,000, which is why the exam tests timing so often.

Annuities: Ordinary, Due, and Perpetuities

An annuity is a series of equal cash flows at regular intervals. With an ordinary annuity, payments arrive at the end of each period. By contrast, an annuity due pays at the start of each period, so the first payment is immediate. Present value of an ordinary annuity is PV = PMT x [1 – (1 + r)^-N] / r, and future value is FV = PMT x [(1 + r)^N – 1] / r, where PMT is the level payment. For an annuity due, multiply the ordinary result by (1 + r), because every payment shifts one period earlier.

Worked example, ordinary annuity. Receive $2,000 at the end of each year for 10 years, discounted at 8 percent. The annuity factor is [1 – (1.08)^-10] / 0.08, which equals 6.710081, giving a present value of $13,420.16. The future value factor is [(1.08)^10 – 1] / 0.08, which equals 14.486562, giving $28,973.12. Present value is measured one period before the first payment, future value at the last payment date.

Now change only the timing. If the same $2,000 is received at the start of each year for 10 years, it is an annuity due. Multiply the ordinary annuity present value by 1.08 to get $14,493.78, and the future value by 1.08 to get $31,290.97. Only the timing changed, and both values rose. On a financial calculator, this is the END versus BGN mode setting, and leaving it in the wrong mode is a common source of wrong answers.

Annuity Due and Perpetuities

A perpetuity pays forever, so there is no future value to compute. Its present value is PV = PMT / r, with the first payment at the end of period 1. A perpetuity paying $500 per year forever at 8 percent is worth $500 / 0.08, which is $6,250 today.

Effective Annual Rate and Loan Amortization

A stated annual rate is not always the rate you actually earn. The effective annual rate (EAR) converts a stated rate with intra-year compounding into a true annual growth rate: EAR = (1 + stated rate / m)^m – 1, where m is the number of compounding periods per year. A 12 percent stated rate compounded monthly gives 12.68 percent, and compounded quarterly gives 12.55 percent. More frequent compounding means a higher effective rate, so convert to EAR first, then compare.

Never divide an effective annual rate by 12 to get a monthly rate. That shortcut only works for a stated nominal rate. An EAR of 12 percent gives a monthly rate of (1.12)^(1/12) – 1, about 0.95 percent, not 1 percent.

Loan amortization turns the annuity formula into a repayment schedule. Borrow $20,000 at 6 percent per year, repaid in 4 equal annual payments. The payment is $1,200 / 0.207906, giving $5,771.83 per year. Each payment splits into interest on the outstanding balance and principal repayment, and the interest portion falls each year as the balance falls.

Year	Payment	Interest (6 percent on balance)	Principal repaid	Balance after payment
1	$5,771.83	$1,200.00	$4,571.83	$15,428.17
2	$5,771.83	$925.69	$4,846.14	$10,582.03
3	$5,771.83	$634.92	$5,136.91	$5,445.12
4	$5,771.83	$326.71	$5,445.12	$0.00

Total interest over the four years is $3,087.32, and total principal repaid is $20,000. The final payment clears the balance exactly, a useful check on your payment calculation. If your schedule leaves a balance, the payment or the interest split is wrong.

Hypothesis Testing: Null, Alternative, and the Decision Rule

A hypothesis test starts with a claim about a population parameter, such as a population mean. The null hypothesis (H0) is the status quo, usually written with an equals sign, for example H0: population mean = 1 percent. The alternative hypothesis (Ha) is what you seek evidence for. It can be two-sided, Ha: population mean is not equal to 1 percent, or one-sided, Ha: population mean is greater than 1 percent. The direction in the alternative decides whether the test is one-tailed or two-tailed, and that choice changes the critical value and the p-value.

The test statistic measures how far the sample result sits from the null value, in standard error units. For a population mean with a known population standard deviation, z = (sample mean – hypothesized mean) / (population standard deviation / square root of sample size). The denominator is the standard error.

The P-Value and the Decision Rule

The p-value is the probability, assuming the null hypothesis is true, of observing a test statistic at least as extreme as the one computed from the sample. A small p-value means the sample result would be unusual if the null were true. The decision rule: reject H0 if the p-value is less than the significance level, alpha, and fail to reject H0 otherwise. The usual exam level is 5 percent, or 0.05.

Two errors are possible. A Type I error rejects a true null hypothesis, a false positive, and its probability is alpha. A Type II error fails to reject a false null hypothesis, a false negative. The power of a test is the probability of correctly rejecting a false null hypothesis, equal to 1 minus the probability of a Type II error. Lowering alpha reduces Type I errors but raises Type II errors for a fixed sample size, while a larger sample generally raises power.

Truth	Reject H0	Fail to reject H0
H0 is true	Type I error (probability = alpha)	Correct decision
H0 is false	Correct decision (power)	Type II error

Language matters. “Fail to reject H0” does not mean “accept H0” and does not prove the null is true. It means the evidence was not strong enough to reject it at the chosen significance level.

Worked Hypothesis Test: A Z-Test You Can Follow

For example, a fund claims its mean monthly return exceeds 1 percent. Test H0: population mean = 0.01 against Ha: population mean is greater than 0.01, a one-tailed test at 5 percent. Population standard deviation is 0.05, sample size is 25, sample mean is 0.018.

Step 1, standard error: 0.05 / square root of 25, which is 0.01. Next, the test statistic: (0.018 – 0.01) / 0.01, which is 0.80. Then the decision: the one-tailed critical value at 5 percent is 1.645, and 0.80 is less than that, so you fail to reject H0. Finally, the p-value check: the one-tailed p-value for z = 0.80 is 0.2119, greater than 0.05, confirming the same decision. The sample mean is above 1 percent, but not far enough above it, given sampling variation, to support the claim at 5 percent.

Now change only the sample mean to 0.030, keeping everything else the same. The test statistic becomes (0.030 – 0.01) / 0.01, which is 2.00. Since 2.00 exceeds 1.645, you reject H0 at 5 percent. The one-tailed p-value for z = 2.00 is 0.0228, less than 0.05, confirming rejection. The two-tailed p-value for the same statistic would be double, 0.0455, because it counts both tails. Using the wrong one can flip your decision.

Memorize these standard normal critical values: 1.645 for a one-tailed test at 5 percent, 1.96 for a two-tailed test at 5 percent, and 2.58 for a two-tailed test at 1 percent. If the test statistic beats the critical value in the stated direction, reject. If it does not, fail to reject. Statistical significance is not a measure of economic size.

CFA Quantitative Methods Formula Sheet

Concept	Formula	Watch out
Future value, single sum	FV = PV x (1 + r)^N	Match r and N to the same period
Present value, single sum	PV = FV / (1 + r)^N	Discount, do not just subtract interest
PV, ordinary annuity	PV = PMT x [1 – (1 + r)^-N] / r	Valued one period before the first payment
FV, ordinary annuity	FV = PMT x [(1 + r)^N – 1] / r	Valued at the last payment date
Annuity due	Multiply the ordinary annuity result by (1 + r)	Check END versus BGN mode
Perpetuity	PV = PMT / r	First payment at end of period 1
Effective annual rate	EAR = (1 + stated rate / m)^m – 1	Convert before comparing rates
Loan payment	PMT = PV x r / [1 – (1 + r)^-N]	Interest falls as the balance falls
Standard error of the mean	Population standard deviation / square root of n	Use the population value only when it is known
Z-test statistic	z = (sample mean – hypothesized mean) / standard error	State H0 and Ha before computing
Decision rule	Reject H0 if p-value is less than alpha	Otherwise fail to reject, never “accept”
Power of a test	1 – probability of a Type II error	Larger samples generally raise power

Exam Traps in CFA Quantitative Methods

First, annuity due timing. If the first payment is today, or at the start of a period, it is an annuity due, even when the question never uses that phrase. Draw the timeline and count the first payment at time zero. Second, EAR versus nominal rate. A loan quoted at 12 percent compounded monthly costs 12.68 percent per year, not 12 percent, and comparing it with a loan quoted at 12 percent compounded annually without converting is an error. Third, one-tailed p-value halving. Halve a two-tailed p-value only when the sample result lies in the direction of the alternative hypothesis. Fourth, “fail to reject” is not “accept”. Fifth, a test statistic without a stated alternative is unfinished work, because the alternative decides the tail, the critical value, and the p-value.

One-Page Revision Box: CFA Quantitative Methods

• FV = PV x (1 + r)^N and PV = FV / (1 + r)^N. Match the rate period to the cash flow period.
• Ordinary annuity: payments at period end. Annuity due: payments at period start, multiply the ordinary result by (1 + r).
• Perpetuity: PV = PMT / r, first payment at the end of period 1.
• EAR = (1 + stated rate / m)^m – 1. Convert to EAR before comparing rates with different compounding.
• Loan payment: PMT = PV x r / [1 – (1 + r)^-N]. Interest is charged on the outstanding balance, so it falls each year.
• H0 is the status quo, Ha is what you seek evidence for. One-sided Ha means a one-tailed test, two-sided Ha means a two-tailed test.
• z = (sample mean – hypothesized mean) / standard error. Reject H0 if the p-value is less than alpha, otherwise fail to reject.
• Type I error: reject a true H0. Type II error: fail to reject a false H0. Power = 1 – probability of a Type II error.
• Memorize: 1.645 one-tailed at 5 percent, 1.96 two-tailed at 5 percent, 2.58 two-tailed at 1 percent.
• Calculator check: END versus BGN mode, payments per year, and rate period, before you trust any TVM answer.

CFA Quantitative Methods FAQ

How do I tell an ordinary annuity from an annuity due?

Payments at the end of each period form an ordinary annuity. Payments at the start of each period, including a payment today, form an annuity due. Draw the timeline, place the first payment, and the answer follows.

Why is the effective annual rate higher than the stated rate?

Because interest earned within the year itself earns interest. A 12 percent rate compounded monthly gives 12.68 percent per year, since each monthly credit starts earning its own return.

What does a p-value of 0.21 actually mean?

It means that, if the null hypothesis were true, you would see a test statistic at least as extreme as the one observed about 21 percent of the time. That is not unusual, so you fail to reject the null. It does not mean there is a 21 percent chance the null is true.

Can I halve a two-tailed p-value for a one-tailed test?

Yes, but only when the sample result lies in the direction stated by the alternative hypothesis. If it points the other way, the one-tailed p-value is 1 minus half the two-tailed p-value, and the test will not reject in the stated direction.

What is the difference between statistical significance and importance?

Significance says the result is unlikely under the null hypothesis. Importance asks how large the effect is in practical terms. A tiny effect can be significant with a large sample, and a large effect can miss significance with a small sample.

Sources

AnalystPrep, Present and Future Values, Annuities, and Cash Flows (CFA Level 1 Quantitative Methods): annuity, perpetuity, and uneven cash flow treatment. LibreTexts Statistics, Hypothesis Testing with One Sample: Type I and Type II errors, p-value interpretation, and full test examples. All worked numbers here were computed from the formulas stated above, not copied from either source.