CFA Level I · CFA Level I Exam

The Time Value of Money in Finance: formula sheet

Full chapter guide

Key formulas

Nominal risk-free rate: Nominal risk-free rate ≈ real risk-free rate + expected inflation premium; The curriculum uses this additive form. The exact link is the Fisher relation, (1 + nominal) = (1 + real) × (1 + inflation), which gives a slightly different value.
Required rate of return on a security: Required rate = real risk-free rate + inflation premium + default risk premium + liquidity premium + maturity premium; Drop any premium that does not apply to the security. A default-free, liquid, short-term bill has none of the three risk premiums to speak of.
Exact Fisher relation: (1 + nominal rate) = (1 + real rate) × (1 + expected inflation); Use it only when the question asks for an exact or compounded relationship. Otherwise add the rates.
Opportunity cost: Opportunity cost = return on the best alternative forgone; The discount rate for a project is the return available on alternatives of equal risk.
Periodic rate: Periodic rate = stated annual rate ÷ m; m is the number of compounding periods per year.
Effective annual rate (discrete compounding): EAR = (1 + stated rate ÷ m)^m − 1; Equals the stated rate when m = 1. Rises with m.
EAR with continuous compounding: EAR = e^(stated rate) − 1; Use the stated rate as a decimal. Upper limit of EAR for a given stated rate.
Future value with m periods per year: FV = PV × (1 + stated rate ÷ m)^(m × N); N is years. Stated rate and m must match the periods in the exponent.
Future value with continuous compounding: FV = PV × e^(stated rate × N); N in years.
EAR to periodic rate: Periodic rate = (1 + EAR)^(1 ÷ m) − 1; Use a root, not EAR ÷ m.
EAR to continuously compounded stated rate: Stated rate = ln(1 + EAR); Inverse of e^r − 1.
Future value of a lump sum: FV = PV × (1 + r)^N; r is the rate per period and N is the number of periods. (1 + r)^N is the future value factor.
Present value of a lump sum: PV = FV ÷ (1 + r)^N = FV × (1 + r)^−N; Discounting is the inverse of compounding. A higher r or longer N gives a lower PV.
Non-annual compounding: FV = PV × (1 + rs ÷ m)^(m × N); rs is the stated annual rate, m is compounding periods per year, N is years. Use rs ÷ m per period and m × N periods.
Solving for the rate: r = (FV ÷ PV)^(1 ÷ N) − 1; N is the number of compounding periods. The result is the rate per period. Multiply by m to restate it as a stated annual rate if periods are shorter than a year. This is not the effective annual rate.
Solving for the number of periods: N = ln(FV ÷ PV) ÷ ln(1 + r); N is in periods, not necessarily years. ln is the natural logarithm.
Continuous compounding: FV = PV × e^(rs × N); Used when compounding is continuous. e is about 2.71828. PV = FV × e^(−rs × N).
PV of an ordinary annuity: PV = PMT × [1 − (1 + r)^−N] ÷ r; Value sits one period before the first payment. r is the rate per payment period, N is the number of payments.
FV of an ordinary annuity: FV = PMT × [(1 + r)^N − 1] ÷ r; Value sits at the date of the last payment.
Annuity due conversion: PV(due) = PV(ordinary) × (1 + r); FV(due) = FV(ordinary) × (1 + r); Each payment is one period earlier, so value is higher. Valid for the same PMT, r and N.
PV of a perpetuity: PV = PMT ÷ r; Value sits one period before the first payment. Requires r > 0.
Deferred annuity: PV today = PV of annuity at date (k) ÷ (1 + r)^k; k is the number of periods from today to the date one period before the first payment.
Solving for payment: PMT = PV × r ÷ [1 − (1 + r)^−N] (ordinary annuity); PMT = FV × r ÷ [(1 + r)^N − 1]; For an annuity due, divide the resulting PMT by (1 + r).
Present value of uneven cash flows: PV = Σ [CFt ÷ (1 + r)^t], for t = 1 to N; Use the periodic rate r. Each cash flow gets its own t. A cash flow at time 0 is not discounted.
Future value of uneven cash flows: FV at time N = Σ [CFt × (1 + r)^(N − t)]; A cash flow received at time N is not compounded. Alternatively, FV = PV × (1 + r)^N.
Periodic rate: r = stated annual rate ÷ m, and N = years × m; m is payments per year. Match the rate and the number of periods to the payment frequency.
Level loan payment: PMT = Loan ÷ [(1 − (1 + r)^−N) ÷ r]; Valid for equal payments at the end of each period (ordinary annuity).
Interest portion of a payment: Interest(t) = r × Balance(t − 1); Use the balance at the start of the period.
Principal portion and new balance: Principal(t) = PMT − Interest(t); Balance(t) = Balance(t − 1) − Principal(t); The final balance should be zero apart from rounding.
Balance as PV of remaining payments: Balance after k payments = PMT × [(1 − (1 + r)^−(N − k)) ÷ r]; A shortcut that skips the schedule.
Total interest over the loan: Total interest = (PMT × N) − Loan; Holds for a fully amortizing loan with equal payments.
Future value of a single sum: FV = PV × (1 + r)^N; The base equation. Every other formula here is a rearrangement of it.
Implied rate of return or growth rate: r = (FV ÷ PV)^(1/N) − 1; N is the number of compounding periods between the two values, not the number of data points. Use the starting value as PV and the ending value as FV.
Implied number of periods: N = ln(FV ÷ PV) ÷ ln(1 + r); Use this to find how long it takes an amount to reach a target. On the calculator, enter I/Y, PV and FV and compute N.
Moving a cash flow to another date: Value at date t = CF at date s × (1 + r)^(t − s); If t is later than s, you compound forward. If t is earlier, the exponent is negative, which means you discount.
Cash flow additivity: Value at date t of a stream = Σ [CFₛ × (1 + r)^(t − s)]; Only cash flows valued at the same date, at the same rate, can be added.
Equivalence of two payment plans: Plan A is equivalent to Plan B if value of A at date t = value of B at date t; Pick any single date. The comparison gives the same answer at every date if the rate is constant.
Rule of 72 (approximation): Years to double ≈ 72 ÷ rate in percent; A rough check for doubling-time questions only. It works best for rates around 6% to 10%. Do not use it where the question asks for an exact answer.

Quick revision

Required rate of return is built from the real risk-free rate, expected inflation and risk premiums.
FV = PV × (1 + r)^N; PV = FV ÷ (1 + r)^N.
Match the rate and N to the same period: monthly rate with number of months.
Periodic rate = stated annual rate ÷ m, where m is compounding periods per year.
EAR = (1 + stated rate ÷ m)^m − 1; EAR rises as compounding becomes more frequent for a given stated rate.
Continuous compounding: EAR = e^(stated rate) − 1.
An ordinary annuity pays at the end of each period; an annuity due pays at the start, so its value is the ordinary annuity value × (1 + r).
Perpetuity PV = PMT ÷ r, valued one period before the first payment.
For uneven cash flows, discount each flow separately and add the results.
A level loan payment is fixed; interest falls and principal repaid rises over time.
Implied growth rate = (FV ÷ PV)^(1 ÷ N) − 1.
Two cash flows are equivalent at a given rate when they have the same value at the same date.

Common mistakes

Treating the nominal risk-free rate as the real risk-free rate. Fix: Real risk-free rate has no inflation. Nominal risk-free equals real plus inflation premium.
Assigning the liquidity premium to credit risk. Fix: Default premium is about the chance of non-payment. Liquidity premium is about difficulty of selling at fair value.
Using the stated rate as if it were the effective rate. Fix: Always ask how often interest compounds. Convert to EAR before comparing or discounting.
Forgetting to divide the stated rate by m, or dividing but forgetting to raise to the power m. Fix: Write (1 + r ÷ m)^m − 1 in full each time. Check that both steps appear.
Using the annual rate with a periodic number of periods, such as 6% with 12 quarters. Fix: Always convert both. Rate per period = annual rate ÷ m and periods = years × m. Write both on your scratch paper first.
Getting an error or a wrong sign on the calculator because PV and FV have the same sign. Fix: Enter PV as negative if you invest it, and FV as positive. If the answer sign looks odd, it is usually a sign convention, not a wrong result.
Treating an annuity due as an ordinary annuity Fix: Draw the timeline. If a payment sits at time 0, it is a due. Multiply by (1 + r) or use BGN mode.
Forgetting to switch the calculator back to END mode Fix: Check that BGN is not shown on the display at the start of every question.
Discounting a cash flow by the wrong number of periods Fix: Write the period number under each cash flow on a timeline before you calculate. Use that number as the exponent.
Using the annuity formula for unequal payments Fix: Use the annuity formula only when every payment is the same. Otherwise discount each flow, or use the CF and NPV keys.

Exam tips

Questions are three-option and standalone. Often you can eliminate two options by asking which risk the security actually carries.
Read for the words approximate versus exact. They decide between adding and the Fisher relation.
When two bonds are compared, find the one feature that differs and name the premium tied to it.
Keep percentages in one form. Mixing 1.5% and 0.015 causes careless errors.
Questions often give the same stated rate with different frequencies and ask which has the highest EAR. Rank by frequency first, then confirm by calculation only if needed.
Watch for 'effective' versus 'stated' in the stem. When a question gives an EAR and a monthly cash flow, convert with a root, not a division.
Expect traps built from common errors: the stated rate, simple interest, and annual compounding all appear as wrong options. Compute the correct value and eliminate the other two.
Be fast with the e^x key on the BA II Plus (press 2nd, LN (e^x)). Practise continuous-compounding questions until the keystrokes are automatic.