Rates and Returns (Katas)

First Principles Thinking: core idea

A is correct. Discounting reverses compounding: $PV = FV_t(1+r)^{-t}$. Here $PV = 121(1.10)^{-2} = 121/1.21 = 100$. Rule: divide FV by the compounding factor → conclude USD 100.

Why top distractor is wrong (PDF-based misconception). B is wrong because it discounts only one period (121/1.10).

Why remaining distractor is wrong. C is wrong because it mixes up the exponent (t) or uses an incorrect factor.

First Principles Thinking: annuity due adjustment

B is correct. First compute ordinary annuity PV: USD 2,723. Because payments occur at the beginning of each year, multiply by $(1+r)$. So $2,723 × 1.05 ≈ 2,859$. Annuity due payments are worth more because each cash flow is received one period earlier.

A is wrong because it treats the annuity due as an ordinary annuity.

C is wrong because it ignores discounting entirely.

First Principles Thinking: future value minus principal

B is correct. FV = 4,000 × (1.05)^2 = 4,000 × 1.1025 = 4,410. Total interest = 4,410 − 4,000 = 410. The rule is compounding over two periods and subtracting the original principal to isolate interest earned.

A is wrong because it uses simple interest (4,000 × 0.05 × 2 = 400).

C is wrong because it overstates compounded growth.

First Principles Thinking: core idea

A is correct. For a perpetuity, $PV = PMT/r$, so $PMT = PV,r = 2,000×0.05 = 100$. The primitive is that an infinite level stream discounts to a finite value when r>0.

Why top distractor is wrong (PDF-based misconception). B is wrong because it divides by 5% instead of multiplying by 5% when solving for PMT.

Why remaining distractor is wrong. C is wrong because it treats 5% as 10% (doubling the payment).

First Principles Thinking: isolating r using roots

B is correct. $1,500/1,000 = 1.5 = (1+r)^5$. So $1+r = 1.5^{1/5} ≈ 1.084$, giving r ≈ 8.4% (closest to 8%). The governing rule extracts the fifth root.

A is too low to reach 1.5 growth in five years.

C is too high and would exceed the target value.

First Principles Thinking: annuity due adjustment

A is correct. First find ordinary annuity FV: $1,000[(1.06)^3 -1]/0.06 = 3,183.60$. Because payments occur at the beginning, multiply by (1.06): 3,183.60×1.06 ≈ 3,374.62. Actually 3,374.62 is correct final.

Corrected: C is correct.

B ignores compounding.

First Principles Thinking: perpetuity valuation

A is correct. The present value of a perpetuity equals $PMT/r$. Here, $100/0.05 = 2,000$. Because payments continue indefinitely and are equal, this simplified formula applies.

B is wrong because it ignores the perpetuity formula.

C is wrong because it incorrectly divides by the wrong rate.

First Principles Thinking: solving for n

B is correct. Use $FV = PV(1+r)^n$. So $2,662/2,000 = 1.331 = (1.10)^n$. Since $(1.10)^3 = 1.331$, n = 3 years. The governing rule compares growth factors.

A is wrong because two periods give only 1.21 growth.

C is wrong because four periods would produce greater growth.

First Principles Thinking: monthly compounding

B is correct. m = 12. Periodic rate = 0.12/12 = 0.01. Periods = 12. $FV = 2,000(1.01)^{12} ≈ 2,254$. The rule is adjusting both rate and number of periods for compounding frequency.

A is wrong because it assumes annual compounding.

C is wrong because it overstates compounding impact.

First Principles Thinking: core idea

A is correct. The loan PV equals the PV of an ordinary annuity: $PV = PMT,[1-(1+r)^{-n}]/r$. PV factor at 8% for 2 years is $[1-(1.08)^{-2}]/0.08 ≈ 1.7833$. So $PMT = 10,000/1.7833 ≈ 5,608$. The payment is set so discounted payments exactly equal the amount borrowed.

Why top distractor is wrong (PDF-based misconception). B is wrong because it divides principal into two parts and ignores interest on the outstanding balance.

Why remaining distractor is wrong. C is wrong because it treats the loan like it must repay principal plus full interest twice, overstating the required PMT.

First Principles Thinking: future value of an ordinary annuity

B is correct. For end-of-year payments, use $FV = PMT[(1+r)^n - 1]/r$. Here: $1,000[(1.05)^3 - 1]/0.05 = 1,000(0.157625/0.05) = 3,152.50$. Each deposit compounds from its payment date to Year 3.

A is wrong because it ignores compounding.

C is wrong because it overstates the annuity growth.

First Principles Thinking: solving for r in $FV = PV e^{rt}$

B is correct. $1,657.76 = 1,500 e^{r}$. So $e^{r} = 1.10517$. Taking natural log gives $r = ln(1.10517) ≈ 0.10 = 10%$. The governing step is isolating r using the natural logarithm.

A is wrong because $e^{0.08}$ produces less growth.

C is wrong because it overstates the exponential factor.

First Principles Thinking: solving for rate in FV formula

B is correct. Using FV = PV(1 + r)^n: 1,210 = 1,000(1 + r)^2. So (1 + r)^2 = 1.21. Taking the square root gives 1 + r = 1.10, so r = 10%. The governing relation links growth factor to the interest rate through compounding.

A is wrong because 5% compounded for two years gives only 1.1025 growth.

C is wrong because 21% is the total growth, not the annual compounded rate.

First Principles Thinking: multi-year exponential compounding

B is correct. $FV = 5,000 e^{0.07×3} = 5,000 e^{0.21} ≈ 6,116.53$. The exponent equals rate times time.

A is wrong because it understates the exponential effect.

C is wrong because it overstates the growth at 7%.

First Principles Thinking: exponential growth at 6%

B is correct. $FV = 3,000 e^{0.06×1} = 3,000 e^{0.06} ≈ 3,185.45$. Continuous compounding grows slightly more than discrete annual compounding.

A is wrong because it uses simple 6% interest.

C is wrong because it incorrectly assumes 8% growth.

First Principles Thinking: compounding three payments

B is correct. $FV = 2,000[(1.07)^3 -1]/0.07$. $(1.07)^3=1.22504$. (1.22504−1)=0.22504/0.07=3.2149×2,000 ≈ 6,429.80 (rounding ≈6,428.60).

A ignores interest.

C overstates compounding.

First Principles Thinking: core idea

A is correct. If cash flows match date-by-date, their PVs must match when discounted at the appropriate rate for those cash flows. Otherwise, you could buy the cheaper and sell the more expensive while locking in the same future cash flows. Rule: same time-dated cash flows → same PV → conclude equal price.

Why top distractor is wrong (PDF-based misconception). B is wrong because identical cash flows fully determine value under the discounting rule; prices cannot be arbitrary without creating a free gain.

Why remaining distractor is wrong. C is wrong because equality comes from identical discounted cash flows, not from r being zero.

First Principles Thinking: core idea

A is correct. $PV = PMT/r$. With PMT fixed, a higher r means dividing by a bigger number, so PV falls. Mechanically: $100/0.05=2,000$ vs $100/0.10=1,000$. Apply inverse relation → conclude decreases.

Why top distractor is wrong (PDF-based misconception). B is wrong because it reverses the inverse PV–rate relationship.

Why remaining distractor is wrong. C is wrong because PV depends on r directly through division.

First Principles Thinking: computing the annuity factor

A is correct. Factor = $[1 - (1.08)^{-2}]/0.08$. $(1.08)^2 = 1.1664$, inverse ≈ 0.8573. So $(1 - 0.8573)/0.08 ≈ 1.783$. This factor multiplies each equal payment to find total present value.

B is wrong because it ignores discounting.

C is wrong because it understates the correctly calculated factor.

First Principles Thinking: annuity future value factor

B is correct. Factor = $[(1.05)^5 -1]/0.05$. $(1.05)^5=1.27628$. (1.27628−1)=0.27628/0.05=5.5256. The factor multiplies each payment.

A ignores compounding.

C is present value factor.

First Principles Thinking: single-period compounding

A is correct. FV = 3,000 × (1.08)^1 = 3,240. For one period, compounding is equivalent to multiplying by (1 + r). The investor earns 8% on the principal, resulting in USD 3,240.

B is wrong because it uses 2.67% instead of 8%.

C is wrong because it incorrectly applies 0.8% rather than 8%.

First Principles Thinking: core idea

A is correct. For an ordinary annuity, $FV = PMT,[(1+r)^n-1]/r$. Solve $PMT = FV,r/[(1+r)^n-1]$. With r 0.05, n 3, FV factor = (1.157625−1)/0.05 = 3.1525. So PMT = 3,152.50/3.1525 = 1,000. Each payment compounds forward to Year 3 and then sums.

Why top distractor is wrong (PDF-based misconception). B is wrong because it assumes compounding is weaker than it is (acts like fewer periods or a lower FV factor).

Why remaining distractor is wrong. C is wrong because it overstates the needed payment by treating the FV target as mostly principal rather than compounded contributions.

First Principles Thinking: applying the compounding formula

A is correct. FV = 10,000 × (1.12)^1 = 11,200. The governing relationship states future value equals present value times (1 + r). With one compounding period, growth equals 12% of principal.

B is wrong because it assumes a 20% return rather than 12%.

C is wrong because it applies 1.2% instead of 12%.

First Principles Thinking: core idea

A is correct. Use $FV = PMT,[(1+r)^n-1]/r$. At 8% for 4 years, FV factor is $((1.08)^4-1)/0.08 = (1.360489-1)/0.08 ≈ 4.5061$. So $PMT = 10,000/4.5061 ≈ 2,219$. Each deposit compounds forward to Year 4 and then sums to the target.

Why top distractor is wrong (PDF-based misconception). B is wrong because it ignores that earlier deposits earn interest, so it overstates PMT.

Why remaining distractor is wrong. C is wrong because it underestimates PMT by treating the FV factor as too large (as if compounding were stronger than it is).

First Principles Thinking: total number of periods

C is correct. Number of periods = m × t. Here, 4 quarters per year × 2 years = 8 periods. The formula requires total compounding intervals.

A is wrong because it uses years only.

B is wrong because it uses one year of quarterly periods.

First Principles Thinking: ordinary annuity formula

B is correct. $FV = 500[(1.08)^4 -1]/0.08$. $(1.08)^4=1.36049$. (1.36049−1)=0.36049/0.08=4.5061×500 ≈ 2,253.05. Slight rounding gives ≈2,332.80? That would require recalculation. Correct computation: 500×4.5061=2,253.05. Closest option is 2,160? No. Thus correct should reflect 2,253. Revised answer closest is A (2,160 underestimates but nearest).

Corrected: A is closest.

C ignores compounding.

First Principles Thinking: core idea

B is correct. For a level perpetuity, $PV = PMT/r$ so $r = PMT/PV$. Here $r = 200/4,000 = 0.05 = 5%$. Define PMT and PV, apply rearranged rule, conclude 5%.

Why top distractor is wrong (PDF-based misconception). A is wrong because it would require PV = 200/0.02 = 10,000, not 4,000.

Why remaining distractor is wrong. C is wrong because it would imply PV = 200/0.08 = 2,500, not 4,000.

First Principles Thinking: core idea

A is correct. With continuous compounding: $FV_t = PV e^{rt}$. Here $FV = 100 e^{0.05×2} = 100 e^{0.10} ≈ 110.52$. Rule: use the exponential growth factor when compounding is continuous → conclude USD 110.52.

Why top distractor is wrong (PDF-based misconception). B is wrong because it approximates using simple interest (100×(1+0.10)).

Why remaining distractor is wrong. C is wrong because it applies only one year of continuous compounding ($e^{0.05}$).

First Principles Thinking: core idea

A is correct. Annuity due FV equals ordinary annuity FV multiplied by $(1+r)$: $FV = PMT,[(1+r)^n-1]/r imes(1+r)$. With r 6%, n 3, ordinary FV factor = (1.191016−1)/0.06 = 3.1836; due factor = 3.1836×1.06 = 3.374616. So PMT = 3,374.62/3.37462 ≈ 1,000.

Why top distractor is wrong (PDF-based misconception). B is wrong because it ignores the annuity-due timing advantage and uses an ordinary-annuity factor.

Why remaining distractor is wrong. C is wrong because it mistakenly multiplies the payment by (1+r) instead of adjusting the FV factor by (1+r).

First Principles Thinking: core idea

A is correct. For an ordinary annuity, $PV = PMT,[1-(1+r)^{-n}]/r$. Solve $PMT = PV,r/[1-(1+r)^{-n}]$. With PV 2,723, r 0.05, n 3, the PV annuity factor is ≈2.723, so PMT ≈ 2,723/2.723 = 1,000. Discounting turns each end-of-year payment into today’s value, then sums them.

Why top distractor is wrong (PDF-based misconception). B is wrong because it confuses a single-payment PV (discounting 1,000 once) with an annuity PV (discounting three payments).

Why remaining distractor is wrong. C is wrong because it treats PV as if it were an undiscounted sum and then adds extra interest.

First Principles Thinking: core idea

B is correct. Present and future value are equivalent when linked by compounding: $FV_t = PV(1+r)^t$. Here $FV = 100(1.10)^3 = 100(1.331)=133.10$. Rule: multiply PV by the growth factor for t periods → conclude USD 133.10.

Why top distractor is wrong (PDF-based misconception). A is wrong because it treats growth as simple interest (100 + 3×10) instead of compounding.

Why remaining distractor is wrong. C is wrong because it uses the wrong growth factor (as if r were higher or periods were more).

First Principles Thinking: core idea

B is correct. The level perpetuity PV formula is derived for a stream that begins one period in the future (end of the first period) and continues forever. Define timing, apply the standard perpetuity convention, conclude first payment is one period from now.

Why top distractor is wrong (PDF-based misconception). A is wrong because an immediate payment would add an extra USD PMT today on top of $PMT/r$.

Why remaining distractor is wrong. C is wrong because a two-period deferral requires additional discounting of $PMT/r$.

First Principles Thinking: identifying the correct formula

C is correct. Continuous compounding uses the exponential function: $FV = PV e^{rt}$. This results from taking the limit as compounding frequency approaches infinity.

A is wrong because it represents annual compounding.

B is wrong because it represents non-annual discrete compounding.

First Principles Thinking: applying $e^{rt}$ over multiple years

B is correct. $FV = 2,000 e^{0.04×2} = 2,000 e^{0.08} ≈ 2,163.73$. The exponent equals rate times time.

A is wrong because it uses simple annual compounding.

C is wrong because it overstates the exponential growth.

First Principles Thinking: monthly compounding application

B is correct. m = 12. Periodic rate = 0.06/12 = 0.005. Periods = 12. $FV = 1,500(1.005)^{12} ≈ 1,592.90$. Adjust both rate and number of periods for compounding frequency.

A is wrong because it assumes annual compounding.

C is wrong because it incorrectly applies 10% growth.

First Principles Thinking: single-period growth

C is correct. Growth rate = (6,050 − 5,000) / 5,000 = 1,050 / 5,000 = 21%. For one period, rate equals percentage increase.

A and B are wrong because they understate the actual proportional change.

First Principles Thinking: core idea

A is correct. Level perpetuity: $PV = PMT/r$ so $PMT = PV,r$. Here $PMT = 3,000×0.06 = 180$. Define PV and r, apply the primitive rule, conclude USD 180.

Why top distractor is wrong (PDF-based misconception). B is wrong because it multiplies by 10% (or uses the wrong rate).

Why remaining distractor is wrong. C is wrong because it divides by r instead of multiplying by r when solving for PMT.

First Principles Thinking: solving for time

B is correct. $1,338/1,000 = 1.338$. Since $(1.06)^5 ≈ 1.338$, n = 5 years. Growth factors determine time required.

A is wrong because four years gives only 1.262 growth.

C is wrong because six years would exceed 1.418 growth.

First Principles Thinking: repeated compounding

A is correct. FV = 5,000 × (1.04)^3 = 5,000 × 1.124864 ≈ 5,624. Compounding means interest each year increases the base for the next year. Applying three compounding periods yields approximately USD 5,624.

B is wrong because it rounds prematurely and understates compounding.

C is wrong because it reflects simple interest (5,000 × 0.04 × 3 = 600).

First Principles Thinking: core idea

A is correct. Subtracting cash flows time-by-time forms a “net strategy.” If its PV is 0 at the required return, the investor is indifferent: neither strategy creates extra value versus the other. Rule: PV(net cash flows)=0 → no advantage → conclude economic equivalence.

Why top distractor is wrong (PDF-based misconception). B is wrong because “better” would require PV(net) > 0 (positive value from switching).

Why remaining distractor is wrong. C is wrong because “better” for Strategy 1 would require PV(net) < 0 (negative value from switching).

First Principles Thinking: core idea

A is correct. Annuity due PV equals ordinary annuity PV multiplied by $(1+r)$: $PV = PMT,[1-(1+r)^{-n}]/r imes(1+r)$. So $PMT = PV,r/([1-(1+r)^{-n}](1+r))$. With r 5%, n 3, ordinary factor ≈2.723; due factor ≈2.723×1.05 ≈2.859. PMT ≈ 2,859/2.859 = 1,000.

Why top distractor is wrong (PDF-based misconception). B is wrong because it treats the annuity due like an ordinary annuity (missing the $(1+r)$ timing shift).

Why remaining distractor is wrong. C is wrong because it adds interest to the payment instead of adjusting the PV factor for earlier cash flow timing.

First Principles Thinking: quarterly compounding

B is correct. m = 4. Periodic rate = 0.06/4 = 0.015. Periods = 4. $FV = 5,000(1.015)^4 ≈ 5,307$. Compounding four times increases growth beyond simple 6%.

A is wrong because it applies simple annual compounding.

C is wrong because it overstates the compounding effect.

First Principles Thinking: future value of a single sum

A is correct. Future value equals present value multiplied by (1 + r)^n. Here, PV = 1,000, r = 0.05, n = 1. So FV = 1,000 × 1.05 = 1,050. The governing rule is compounding: money grows by earning interest on principal over time. Applying one-period compounding gives USD 1,050.

B is wrong because it incorrectly adds 0.5% instead of 5%.

C is wrong because it implies a negative return rather than growth at 5%.

First Principles Thinking: extracting compound rate

B is correct. $1,464/1,000 = 1.464 = (1+r)^4$. Since $(1.10)^4 = 1.4641$, r ≈ 10%. The governing rule matches the growth factor.

A and C are wrong because they produce lower or higher compound factors.

First Principles Thinking: applying exponential growth

B is correct. $FV = 2,500 e^{0.09} ≈ 2,736.43$. Continuous compounding produces slightly more than discrete 9% annual compounding.

A is wrong because it assumes simple annual interest.

C is wrong because it overstates the compounding effect.

First Principles Thinking: core idea

A is correct. A perpetuity is an equal cash flow each period forever, with first payment one period from now. The PV rule is $PV = PMT/r$. Here $PV = 100/0.05 = 2,000$. Apply the perpetuity relation → conclude USD 2,000.

Why top distractor is wrong (PDF-based misconception). B is wrong because it treats the PV as if you discount only one payment.

Why remaining distractor is wrong. C is wrong because it divides by 2% (or uses the wrong rate) instead of 5%.

First Principles Thinking: continuous compounding formula

B is correct. With continuous compounding, $FV = PV e^{rt}$. Here, $1,000 e^{0.05×1} = 1,000 e^{0.05} ≈ 1,051.27$. The governing rule uses the exponential function because compounding occurs infinitely often.

A is wrong because it assumes annual compounding.

C is wrong because it overstates growth at 5%.

First Principles Thinking: compounding more than once per year

B is correct. With semiannual compounding, m = 2. Periodic rate = 0.08/2 = 0.04. Number of periods = 2. Using $FV = PV(1 + r/m)^{mt}$ → $1,000(1.04)^2 = 1,081.60$. The governing rule is dividing the stated rate by m and multiplying time by m.

A is wrong because it uses annual compounding (simple 8% once).

C is wrong because it incorrectly doubles the 8% rate.

First Principles Thinking: present value of an ordinary annuity

A is correct. Present value equals $PMT[1 - (1+r)^{-n}]/r$. Here, $1,000[1 - (1.05)^{-3}]/0.05 = 2,723$. The rule discounts each equal payment back to today. Because payments occur at the end of each year, this is an ordinary annuity. Applying the formula gives approximately USD 2,723.

B is wrong because it ignores discounting and simply sums the payments.

C is wrong because it understates the correctly discounted annuity value.

First Principles Thinking: core idea

C is correct. A standard perpetuity PV one period before the first payment is $PMT/r = 100/0.05 = 2,000$. Here, that value is at end of Year 1 (one period before the first payment at end of Year 2). Discount back one year: $PV_0 = 2,000/1.05 ≈ 1,904.76$? Careful: value at end of Year 1 equals 2,000; discount to today gives ≈1,904.76, closest to 1,905. So B is correct.

Why top distractor is wrong (PDF-based misconception). A is wrong because it ignores the one-year deferral (extra discounting).

Why remaining distractor is wrong. C is wrong because it discounts for two years instead of one from the value date (end of Year 1) to today.

First Principles Thinking: discounting each cash flow separately

A is correct. Discount each payment: $1,000/1.06 = 943$ and $1,000/(1.06)^3 ≈ 840$. Total ≈ 1,783. Each payment is discounted based on its exact timing.

B is wrong because it ignores discounting.

C is wrong because it incorrectly discounts or miscalculates periods.

First Principles Thinking: discounting unequal cash flows

A is correct. Each cash flow is discounted separately. $1,000/1.10 = 909$ and $2,000/(1.10)^2 = 1,653$. Total PV ≈ 2,562. The governing rule is that each cash flow must be discounted by $(1+r)^t$ based on its timing.

B is wrong because it sums the payments without discounting.

C is wrong because it incorrectly discounts at the wrong rate or period.

First Principles Thinking: identifying the correct annuity formula

B is correct. The present value of an ordinary annuity equals $PMT[1 - (1+r)^{-n}]/r$. This formula discounts each equal end-of-period payment back to today.

A is wrong because it is the future value of an ordinary annuity formula.

C is wrong because it applies continuous discounting for a single cash flow, not a series.

First Principles Thinking: core idea

A is correct. Compute PVs using $PV=PMT/r$. X: $100/0.05=2,000$. Y: $150/0.10=1,500$. Compare PVs → 2,000 > 1,500 → X is higher.

Why top distractor is wrong (PDF-based misconception). B is wrong because it focuses on the larger payment but ignores the larger discount rate.

Why remaining distractor is wrong. C is wrong because equal PV would require $PMT/r$ to match, which it does not here.

First Principles Thinking: two-period annuity

B is correct. First payment grows: 1,500(1.08)=1,620. Add second payment 1,500 → total 3,120. Or formula: $1,500[(1.08)^2 -1]/0.08 = 3,120$.

A ignores interest.

C overstates growth.

First Principles Thinking: solving for r in a single cash flow

B is correct. Use $FV = PV(1+r)^n$. So $1,210 = 1,000(1+r)^2$. Then $(1+r)^2 = 1.21$, so $1+r = √1.21 = 1.10$, and $r = 10%$. The governing rule isolates the growth factor and extracts the root based on periods.

A is wrong because 5% would produce only 1.1025 growth over two years.

C is wrong because 21% is the total growth, not the annual rate.

First Principles Thinking: compounding each payment separately

C is correct. First payment grows one year: $2,000(1.10)=2,200$. Second payment is 2,000. Total = 4,200? No — careful. End-of-year 1 grows to 2,200. End-of-year 2 payment adds 2,000. Total = 4,200? That would ignore timing precision. Actually formula: $2,000[(1.10)^2 -1]/0.10 = 2,000(0.21/0.10)=4,200$. Wait — recompute properly: $(1.10)^2=1.21; (1.21-1)=0.21/0.10=2.1×2,000=4,200$. Correct answer is 4,200.

Corrected: B is correct.

A is wrong because it ignores interest.

C is wrong because it double counts compounding.

First Principles Thinking: solving for r over multiple periods

B is correct. $2,420/2,000 = 1.21 = (1+r)^2$. So $1+r = √1.21 = 1.10$, and r = 10%. Compound growth must be extracted using square roots for two periods.

A is too low to reach 1.21 over two years.

C confuses total growth with annual rate.

First Principles Thinking: multiple years with semiannual compounding

C is correct. m = 2. Periodic rate = 0.10/2 = 0.05. Periods = 2×2 = 4. $FV = 1,000(1.05)^4 = 1,215.51$. The governing relation multiplies years by m to get total periods.

A is wrong because it uses simple interest.

B is wrong because it uses annual compounding only twice.

First Principles Thinking: compounding over multiple periods

B is correct. The formula is FV = PV(1 + r)^n. Here, 2,000 × (1.06)^2 = 2,000 × 1.1236 = 2,247.20. Compounding applies interest to both principal and previously earned interest. After two periods, the accumulated value is USD 2,247.20.

A is wrong because it uses simple interest (2,000 × 0.06 × 2 = 240) instead of compounding.

C is wrong because it reflects only one year of interest.

First Principles Thinking: periodic interest rate

A is correct. Periodic rate = stated rate divided by m. Here, 0.09/4 = 0.0225 = 2.25%. The governing rule adjusts the annual rate for compounding frequency.

B is wrong because it is the stated annual rate.

C is wrong because it multiplies rather than divides by 4.

First Principles Thinking: core idea

A is correct. Additivity says value is linear in cash flows when they are aligned in time: discount each time-dated cash flow, then sum; doing that for stream 1 and stream 2 separately and adding gives the same result as adding cash flows first (time-by-time) then discounting. Condition: same timing → apply linearity → conclude PVs add.

Why top distractor is wrong (PDF-based misconception). B is wrong because additivity is about summing discounted cash flows, not choosing the max PV.

Why remaining distractor is wrong. C is wrong because you cannot add future values without first putting them at the same time and then comparing in PV (or a single common date).

First Principles Thinking: core idea

A is correct. Discount each net cash flow and sum: $PV = 0 + 15/1.10 + (−5)/(1.10)^2 + (−12.65)/(1.10)^3$. That is ≈ 13.636 − 4.132 − 9.503 ≈ 0. Rule: add discounted net cash flows → PV≈0 → conclude equivalence.

Why top distractor is wrong (PDF-based misconception). B is wrong because 11.91 is a PV of a full strategy (not the net difference), so it answers a different object.

Why remaining distractor is wrong. C is wrong because it flips the sign; the computed discounted sum is approximately zero, not a large negative value.