Quantitative Methods. Worth 6 to 9 percent of the exam. One session: the lesson, the rules, the method, then the questions.
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Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.
The exam wants you to calculate the present value and future value of a single sum, an ordinary annuity, and an annuity due, to compare a stated rate against its effective annual rate under different compounding frequencies, and to calculate the value of a perpetuity. Nearly every question hands you three of four variables and asks for the fourth.
Every time value of money problem, however it is dressed up, is the same four numbers in different combinations: PV, what a cash flow is worth today; FV, what it becomes later; r, the rate per period, written as a decimal; and n, the number of periods. The future value formula, FV = PV x (1 + r)^n, says a sum grows by that rate, compounded, for that many periods. The present value formula, PV = FV / (1 + r)^n, runs the same relationship backward: discounting a known future amount down to what it is worth now. Everything else in this module is a variation on those two lines.
Compounding frequency is where the first real trap sits. A stated annual rate that compounds more often than once a year understates the true return, because interest starts earning interest sooner. The effective annual rate, EAR = (1 + stated rate / m)^m - 1, converts any compounding frequency to a single comparable annual figure, where m is the number of compounding periods per year. A 12 percent rate compounded annually stays exactly 12 percent; the same 12 percent compounded monthly becomes 12.68 percent once converted, because interest is credited, and starts earning its own interest, twelve times a year instead of once. Whenever two rates with different compounding frequencies are being compared, convert both to EAR before comparing them; comparing stated rates directly is comparing two different units as if they were the same one.
An annuity is a series of equal payments. An ordinary annuity pays at the end of each period, the shape of a typical loan or bond coupon; an annuity due pays at the beginning. The two formulas differ by exactly one factor: an annuity due is worth (1 + r) times the equivalent ordinary annuity, because every payment has shifted one period earlier and is therefore discounted, or compounded, one period less. This is not one extra payment tacked on, it is the entire stream moved earlier. On the calculator this is the BGN versus END setting, and it is worth resetting to END immediately after any annuity-due problem, since the setting persists until changed.
A perpetuity is an annuity that never ends, and letting n approach infinity collapses the annuity formula down to the simplest expression in the whole module: PV = PMT / r. A preferred share paying a fixed dividend forever is the standard example. Continuous compounding, FV = PV x e^(rn), is the mathematical limit at the other extreme, interest credited at every instant rather than at discrete intervals; it is the ceiling compounding approaches, not a separate family of problems.
An investor deposits $2,000 at the end of each year for 5 years into an account earning 6 percent annually. What is the future value at the end of year 5? This is an ordinary annuity (payments at the end of each period), so FV = PMT x [(1 + r)^n - 1] / r. With PMT = 2,000, r = 0.06, n = 5: (1.06)^5 = 1.338226. (1.338226 - 1) / 0.06 = 5.637093. FV = 2,000 x 5.637093 = $11,274.19.
Same deposits: $2,000 at the end of each year for 5 years at 6 percent annually. Set up the annuity future value formula, FV = PMT x [(1 + r)^n - 1] / r, and compute (1.06)^5 yourself before finishing the calculation.
PMT = $2,000, end of year, n = 5, r = 6% annually. Find FV.
Leaving a stale BGN setting active after an annuity-due question, or a leftover value in a TVM register from the previous problem, is the single most common source of a wrong TVM answer on the exam; clear the worksheet and confirm END mode before every new problem unless the problem explicitly states payments occur at the start of the period.
Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.
FV = PV x (1 + r)^n and PV = FV / (1 + r)^n use the same rate r and the same number of periods n. An interest rate that grows money forward is, read backward, a discount rate that shrinks a future amount to what it is worth today. The exam calls the same number a required rate of return, a discount rate, or a cost of capital depending on context; all three sit on the same calculator key.
Shifting every payment in a level stream one period earlier does not add a payment; it multiplies the whole stream's present or future value by (1 + r), because each payment is now discounted, or compounded, one period less. Ordinary annuity payments arrive at the end of each period; an annuity due's payments arrive at the beginning.
A level cash flow with no maturity date is the limiting case of an annuity as the number of periods grows without bound. There is no year count to enter; the whole value collapses to payment divided by rate. The Gordon Growth Model used later in Equity Valuation is this same perpetuity formula with a growth rate subtracted from the discount rate.
A stated annual rate compounded m times a year is converted before it enters any formula: divide the rate by m, multiply the number of years by m. A rate left annual while the period count is monthly, or the reverse, produces an answer that is wrong by a full order of the mismatch, not just slightly off.
EAR = (1 + stated rate / m)^m - 1. Two accounts with different stated rates and different compounding frequencies cannot be compared until both are converted to EAR. The stated rate understates the true annual return whenever compounding happens more than once a year; only when m = 1 do stated and effective rates coincide.
The cash flow additivity principle says that the present value of a set of cash flows equals the sum of the present values of each cash flow taken separately, and that this holds only once every flow has been discounted to the same point in time. It is what makes a no-arbitrage forward rate, forward exchange rate, or option value derivable from spot rates and cash flows: any two ways of reaching the same future date must be worth the same today, or a riskless profit would exist.
An annuity due needs BGN mode; an ordinary annuity needs END mode. B comes before E, and beginning-of-period payments come before end-of-period payments in the same alphabetical order.
The order from smallest to largest future value, holding the stated rate fixed, is: simple interest, annual, semiannual, quarterly, monthly, daily, continuous. Continuous compounding, FV = PV x e^(rn), is the mathematical limit as compounding frequency grows without bound, never an unusual special case.
A 12% rate compounded monthly has an EAR of 12.68%, not 12%. Any answer choice that just repeats the stated rate as the effective rate is testing whether the candidate remembers that compounding frequency matters.
The order to work a question of this type in, every time, before you touch the numbers.
The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.
An investor deposits $5,000 today in an account earning 8% per year compounded annually. The value of the account at the end of 6 years is closest to:
Answer A. FV = PV × (1 + r)^n = 5,000 × (1.08)^6 = 5,000 × 1.5869 = $7,934
The present value of $10,000 to be received in 5 years if the discount rate is 6% compounded semiannually is closest to:
Identify the cash flow shape: a single sum, a level annuity, a perpetuity, or an unequal series.
Answer A. With semiannual compounding: r = 6%/2 = 3% per period, n = 5 × 2 = 10 periods PV = 10,000 / (1.03)^10 = 10,000 / 1.3439 = $7,441 Recheck: (1.03)^10 = 1.34392. PV = 10,000/1.34392 = $7,440.94, then Answer A) However, some versions of this question use 6% stated rate with semiannual periods differently. Using BA II Plus with P/Y=2: N=10, I/Y=3, FV=10000, PMT=0, then PV = $7,441.
Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.
An investor will receive $2,000 at the end of each year for the next 4 years. If the required rate of return is 10%, the present value of this annuity is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
An investor deposits $1,500 at the BEGINNING of each year for 5 years in an account earning 7% annually. The future value of this annuity at the end of year 5 is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A stock is expected to pay a constant dividend of $3.50 per share per year forever. If the required rate of return on this stock is 8.5%, its present value is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A bank offers a savings account with a stated annual interest rate of 12% compounded monthly. The effective annual rate (EAR) is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
An investment of $10,000 earns a continuously compounded annual rate of 8%. The value of the investment after 3 years is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
You take out a 30-year mortgage for $300,000 at 6% annual interest, compounded monthly (0.5% per month). Your monthly payment is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A company is choosing between two investment options: Option A, a lump sum of $50,000 received today, and Option B, $8,000 received at the end of each year for 8 years. At a 10% discount rate, the option with the higher present value, and the amount by which its present value exceeds the other's, is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
An investor needs $1,000,000 in 20 years for retirement. The amount she must invest today in a single lump sum, if she can earn 7% annually compounded, is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A bond pays a 6% annual coupon on a $1,000 face value with 4 years to maturity. If market interest rates are 8%, the bond's present value (price) is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A credit card charges 1.5% interest per month. The effective annual rate (EAR) is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
An investor deposits $4,000 at the end of each year for 6 years into an account earning 7% annually. Starting in year 7, she stops contributing and lets the account grow untouched for 4 more years before withdrawing the full balance. The amount she withdraws is closest to:
How sure are you?
Unit: time-value-of-money-in-finance
A borrower takes a $200,000 loan at 6% annual interest, compounded monthly, to be repaid in equal monthly payments over 15 years. Immediately after making the 60th payment (5 years in), the remaining loan balance is closest to:
How sure are you?
Unit: time-value-of-money-in-finance