Mr Tony Stark wishes to spend $20 billion in three years’ time to build a new headquarters for the Avengers. He has been given the choice of five investment alternatives. These are listed below:

Investment alternative One: pays 5.4% p.a. simple interest
Investment alternative Two: pays 5.2% p.a. compounded annually
Investment alternative Three: pays 5.15% p.a. compounded quarterly
Investment alternative Four: pays 5.05% p.a. compounded monthly
Investment alternative Five: pays 5.02% p.a. continuously compounded

Which of the above investment alternatives would require the lowest initial payment to reach Mr Stark's goal?

Question

Mr Tony Stark wishes to spend $20 billion in three years’ time to build a new headquarters for the Avengers. He has been given the choice of five investment alternatives. These are listed below:

Investment alternative One: pays 5.4% p.a. simple interest
Investment alternative Two: pays 5.2% p.a. compounded annually
Investment alternative Three: pays 5.15% p.a. compounded quarterly
Investment alternative Four: pays 5.05% p.a. compounded monthly
Investment alternative Five: pays 5.02% p.a. continuously compounded

Which of the above investment alternatives would require the lowest initial payment to reach Mr Stark's goal?

BlackTom AI · Accepted Answer

We start by restating the scenario and listing the investment options to keep them in view as we evaluate them.

Question: Mr Tony Stark wants to accumulate 20 billion in 3 years. Among five investment alternatives with different interest conventions, which would require the lowest initial payment to reach 20 billion in 3 years?
Options:
- One: 5.4% p.a. simple interest
- Two: 5.2% p.a. compounded annually
- Three: 5.15% p.a. compounded quarterly
- Four: 5.05% p.a. compounded monthly
- Five: 5.02% p.a. continuously compounded

For each option, the initial payment P needed is P = FV / accumulation_factor, where FV = 20 and accumulation_factor is determined by the respective interest convention and compounding.

Option Five (continuous 5.02%): The accumulation factor is exp(r t) = exp(0.0502 × 3) = exp(0.1506) ≈ 1.1626. Therefore P ≈ 20 / 1.1626 ≈ 17.21 billion.

Option One (simple interest 5.4% for 3 years): The accumulation factor is 1 + r t = 1 + 0.054 × 3 = 1.162. So P ≈ 20 / 1.162 ≈ 17.21 billion. This is very close to the result for Option Five, though the underlying calculation is different because one is simple interest and the other is continuous growth.

Option Two (5.2% p.a. compounded annually for 3 years): The accumulation factor is (1 + 0.052)^3. Calculating: 1.052^3 ≈ 1.16425. Thus P ≈ 20 / 1.16425 ≈ 17.18 billion. This is slightly lower than the P for Option One and Option Five due to the higher effective growth from annual compounding over the 3-year period.

Option Three (5.15% p.a. compounded quarterly for 3 years): The quarterly rate is 0.0515 / 4 = 0.012875. The accumulation factor is (1 + 0.012875)^{12} ≈ 1.1658. Hence P ≈ 20 / 1.1658 ≈ 17.15 billion. Here, the combination of a relatively high nominal rate with frequent quarterly compounding yields a higher accumulation than the other options, driving the required initial payment down further.

Option Four (5.05% p.a. compounded monthly for 3 years): The monthly rate is 0.0505 / 12 ≈ 0.0042083. The accumulation factor is (1 + 0.0042083)^{36} ≈ 1.1632. So P ≈ 20 / 1.1632 ≈ 17.20 billion. This is close to Option Five and Option One, with a slightly different value due to the monthly compounding frequency and rate.

Now, comparing the accumulation factors and corresponding initial payments:
- Option Five: ≈ 1.1626 → P ≈ 17.21
- Option One: ≈ 1.1620 → P ≈ 17.21
- Option Two: ≈ 1.16425 → P ≈ 17.18
- Option Three: ≈ 1.1658 → P ≈ 17.15
- Option Four: ≈ 1.1632 → P ≈ 17.20

From these approximations, the smallest initial payment arises from the option with the largest accumulation factor over the 3-year horizon, since P = FV / accumulation_factor. In this set, Option Three (5.15% p.a. compounded quarterly) provides the highest accumulation factor among the five, leading to the lowest required initial outlay.

If a student is not convinced by the numbers, it helps to think in terms of effective yield over 3 years: even though all rates are around 5%, more frequent compounding (quarterly) at a slightly higher nominal rate can push the effective growth just a bit higher than the other options, reducing the required starting amount. Conversely, the purely continuous or simple-interest options tend to have lower effective growth over the same horizon, resulting in a larger initial payment than the best-compounded option in this set.

Note: The precise numerical values depend on the rounding approach used in calculations, but the qualitative ordering here shows Option Three generally yields the lowest P, with Options Two and Four typically close behind, and Options One and Five yielding slightly larger P values.

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