Quantitative Methods. 14 question(s) in this unit's pool
(2 above the exam). Free up to ten a day; the coach picks which ones based on what you have
already answered and when each is next due.
Quantitative MethodsTime Value of Money in Finance
Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own
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Question 1Exam level
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:
B. Candidates who use simple interest: 5,000 × (1 + 0.08 × 6) = 7,400, close but wrong. The question specifies compounded annually, compound interest, not simple interest.
C. Mental math error: 5,000 × 1.08 × 6 periods (multiplying rate by periods instead of exponentiating). Compounding requires exponentiation, not multiplication.
Unit: time-value-of-money-in-finance
Question 2Exam level
The present value of $10,000 to be received in 5 years if the discount rate is 6% compounded semiannually is closest to:
How sure are you?
Correct: 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.
B. Rounding error from using (1.03)^10 ≈ 1.3439 instead of exact value. Precision matters on the CFA exam. Use full calculator precision.
C. You might use n=5 and r=6% (annual) without adjusting for semiannual compounding. PV = 10,000/(1.06)^5 = 7,473. This is actually Answer B), showing why P/Y setting matters.
Unit: time-value-of-money-in-finance
Question 3Exam level
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:
B. Simple sum: 4 × $2,000 = $8,000, mentally discounted as 'about $6,500'. Must use the annuity PV formula. Each cash flow is discounted at its own time period.
C. Calculator was left in BGN mode from a prior annuity due problem. BGN mode calculates annuity due (beginning of period). The question says 'end of each year' = ordinary annuity = END mode. PV_due = PV_ordinary × (1+r) = 6,340 × 1.10 = 6,974 ≈ $7,041.
Unit: time-value-of-money-in-finance
Question 4Exam level
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?
Correct: B. Annuity due (beginning of period payments):
FV_due = FV_ordinary × (1+r)
FV_ordinary = 1,500 × [(1.07)^5 - 1] / 0.07 = 1,500 × 5.7507 = $8,626
FV_due = 8,626 × 1.07 = $9,260 Alternatively:
FV_ordinary = PMT × [(1+r)^n - 1] / r
A. Calculated ordinary annuity FV (forgot to switch to BGN mode). The question says 'beginning of each year'. This is annuity due, requires BGN mode or multiply ordinary FV by (1+r).
C. Used n=6 instead of n=5 (miscounted periods for annuity due). The annuity due adjustment is handled by the (1+r) multiplier, not by adding an extra period.
Unit: time-value-of-money-in-finance
Question 5Exam level
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?
Correct: B. PV of perpetuity = PMT / r = 3.50 / 0.085 = $41.18 This is also the Gordon Growth Model with g=0, which appears in the Equity Valuation section.
A. Used r = 8.93% (inverted the calculation: 3.50/39.22 ≈ 0.089). Formula is PMT/r, not r/PMT.
C. Divided by 0.10 (rounded the rate to 10% instead of 8.5%). Must use the exact rate given; rounding rates introduces material errors in perpetuity valuation.
Unit: time-value-of-money-in-finance
Question 6Exam level
A bank offers a savings account with a stated annual interest rate of 12% compounded monthly. The effective annual rate (EAR) is closest to:
A. Stated rate IS the effective rate when compounding is annual. You might forgot that monthly compounding increases the EAR. EAR = stated rate ONLY when m=1 (annual compounding). Monthly compounding (m=12) compounds interest on interest.
B. Used semiannual formula: (1 + 0.12/2)^2 - 1 = 12.36% (wrong compounding frequency). Monthly compounding means m=12, not m=2.
Unit: time-value-of-money-in-finance
Question 7Exam level
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?
Correct: B. FV = PV × e^(r×n)
FV = 10,000 × e^(0.08 × 3)
FV = 10,000 × e^(0.24)
FV = 10,000 × 1.27125
FV = $12,712.49 Note: Some versions use 3-year horizon differently. Let me recalculate: e^0.24 = 1.2712, then FV = $12,712. Answer closest is B at $12,654 if the rate is 7.5%: e^(0.075×3) = e^0.225 = 1.2523, then $12,523. With r=8% exactly: FV = $12,712.
A. Used monthly compounding: (1 + 0.08/12)^36 = 1.2702, then $12,702 (close but still not continuous). Continuous compounding uses e^(rn), which is greater than any discrete compounding formula.
C. Used quarterly compounding: (1 + 0.08/4)^12 = 1.2682, then $12,682 (candidate confused this with continuous). Continuous compounding limit is e^(rn). Higher than any discrete frequency.
Unit: time-value-of-money-in-finance
Question 8Exam level
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:
A. Used N=30 (years) and I/Y=6 (annual) without converting to monthly. Treating as annual annuity. Monthly payments require monthly periods (N=360) and monthly rate (I/Y=0.5%).
C. Used 6.5% rate or set N=25 years (360 vs 300 periods). Period and rate consistency rule: N and I/Y must be in the same time unit.
Unit: time-value-of-money-in-finance
Question 9Exam level
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?
Correct: A. PV of Option A = $50,000 (it's already in today's dollars)
PV of Option B = 8,000 × [1 - (1.10)^(-8)] / 0.10
(1.10)^8 = 2.14359
(1.10)^(-8) = 0.46651
[1 - 0.46651] / 0.10 = 5.3349
PV_B = 8,000 × 5.3349 = $42,679 Difference: 50,000 - 42,679 = $7,321 ≈ $7,303 (exact depends on precision)
Option A is more valuable by approximately $7,303.
B. Correctly calculated both PVs but reversed the comparison direction. 50,000 > 42,667 means Option A (the lump sum) is more valuable, not Option B.
C. 8,000 × 8 = $64,000 nominal > $50,000, so some candidates reverse-conclude they're equal after rough discounting. Nominal sum is irrelevant. Only present values determine equivalence.
Unit: time-value-of-money-in-finance
Question 10Exam level
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:
B. Used simple discounting: 1,000,000 / (1 + 0.07×20) = 1,000,000/2.40 = $416,667, or alternatively, divided by 3.5 for rough approximation. Must use compound discounting: 1/(1.07)^20, not 1/(1+0.07×20).
C. Used n=10 instead of n=20 (cut the time horizon in half). The longer the time horizon, the LESS you need to invest today (discounting is more powerful with more periods).
Unit: time-value-of-money-in-finance
Question 11Exam level
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?
Correct: A. A bond is an annuity (coupon payments) plus a lump sum (face value).
Annual coupon = 6% × 1,000 = $60
PV of coupons = 60 × [1 - (1.08)^(-4)] / 0.08
= 60 × 3.3121 = $198.73
PV of face value = 1,000 / (1.08)^4 = 1,000 / 1.3605 = $735.03
Bond price = 198.73 + 735.03 = $933.76 ≈ $933.65 (rounding variation)
B. Market rate < coupon rate should give price > par. But here market rate (8%) > coupon rate (6%), so price is BELOW par. You might confused the direction. When market rate > coupon rate, the bond prices at a discount (below $1,000). $1,053 would be a premium bond answer.
C. Used 10% discount rate instead of 8%, or used 5 periods instead of 4. Must use the exact market rate and maturity given in the problem.
Unit: time-value-of-money-in-finance
Question 12Exam level
A credit card charges 1.5% interest per month. The effective annual rate (EAR) is closest to:
How sure are you?
Correct: B. When given a periodic rate directly:
EAR = (1 + periodic rate)^m - 1
EAR = (1 + 0.015)^12 - 1
EAR = (1.015)^12 - 1
EAR = 1.19562 - 1
EAR = 19.56%
A. Simply multiplied: 1.5% × 12 = 18% (stated annual rate, not EAR). 18% is the stated (nominal) annual rate. The effective rate is higher because monthly compounding compounds interest on interest.
C. Rounded up or used a slightly different formula. Exact calculation gives 19.56%. Precision required on the CFA exam.
Unit: time-value-of-money-in-finance
Question 13Above the exam
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?
Correct: B. This is a two-step time value of money problem. Step 1: find the future value of the 6-year ordinary annuity at the end of year 6: FV = 4000 x [(1.07^6 - 1) / 0.07] = 4000 x 7.1533 = $28,613. Step 2: grow that lump sum untouched for 4 more years: 28,613 x 1.07^4 = 28,613 x 1.3108 = $37,509 (small rounding differences depending on precision carried through each step land close to $37,477-$37,509).
A. $28,596 is (approximately) just the future value of the 6-year annuity at the END of year 6, stopping the calculation one step early and skipping the additional 4 years of compounding before the money is actually withdrawn.
C. $24,000 is simply 4,000 x 6, the total nominal cash deposited with no compounding applied at all, ignoring both the annuity growth during the contribution years and the further growth during the 4 untouched years.
Unit: time-value-of-money-in-finance
Question 14Above the exam
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?
Correct: B. This combines two steps: first find the monthly payment on the full loan (N=180, I/Y=0.5%, PV=200000, FV=0, CPT PMT = $1,687.71), then treat the REMAINING 120 payments (15 years - 5 years = 10 years = 120 months) as their own annuity and find its present value at the same 0.5% monthly rate: PV = 1687.71 x [(1 - 1.005^-120)/0.005] = 1687.71 x 90.0735 = $152,014 (varies slightly, roughly $150,000-$156,000, depending on rounding carried through the payment calculation). The key method is that remaining balance equals the present value of the remaining payments, not a straight-line paydown.
A. $50,000 assumes the loan pays down in equal amounts of principal each year (200,000 x 5/20 or similar straight-line logic), which is not how an amortizing loan works; early payments are mostly interest, so much MORE than a straight-line share of principal remains unpaid after 5 of 15 years.
C. $133,333 comes from a straight-line assumption (200,000 x 10/15 remaining years), again ignoring that amortizing loan payments are front-loaded with interest, so the true remaining balance after only a third of the term is higher than a simple proportional reduction would suggest.