Investment X and Investment Y are both growing perpetuities with initial cash flow of $100. Both investments have the same interest rate (r). The present value of Investment X is $5 000, while the present value of Investment Y is $4 000. Which of the following is true?

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We start by clearly restating the scenario and the options to ensure understanding before evaluating each choice.

Question: Investment X and Investment Y are both growing perpetuities with initial cash flow of $100. Both investments have the same interest rate (r). The present value of Investment X is $5,000, while the present value of Investment Y is $4,000. Which of the following is true?

Answer options:
A) The answer cannot be determined without knowing the interest rate for both investments.
B) This makes no sense—with the same initial cash flow and the same interest rate Investment X and Investment Y should have the same present value.
C) Investment X has a lower growth rate than Investment Y.
D) Investment X has a higher growth rate than Investment Y.

Now, let's evaluate each option in turn:

Option A: The answer cannot be determined without knowing the interest rate for both investments.
- Since the problem states that both investments share the same interest rate r, and provides their present values, you might wonder if the specific value of r matters. However, for perpetuities with growth, the present value formula involves both the initial cash flow, the growth rate g, and r in a particular relationship. Given that the initial cash flow is the same for both investments and r is the same, any difference in present value must be due to a difference in the growth rate g. Therefore, you do not need the numeric value of r to deduce which has the higher or lower growth rate. This means option A is not correct because the information given suffices to compare growth rates.

Option B: This makes no sense—with the same initial cash flow and the same interest rate Investment X and Investment Y should have the same present value.
- This is a common misconception: even with the same initial cash flow and the same discount rate, two growing perpetuities can have different present values if their growth rates differ. The PV formula for a growing perpetuity is PV = C1 / (r - g) where C1 is next-period cash flow (or, more precisely, C0 * (1+g) depending on indexing). If r is fixed and C0 is fixed, a different g changes the denominator and thus the present value. Therefore, differing PVs do not violate the model; they simply reflect different growth rates. Hence this option is incorrect.

Option C: Investment X has a lower growth rate than Investment Y.
- Here, we consider that PV_X = 5000 and PV_Y = 4000 with the same initial cash flow and same r. Since the PV formula for a growing perpetuity is PV = C0 * (1+g) / (r - g) for some conventions, or more simply PV ∝ 1 / (r - g) when initial cash flow structure is aligned, a larger PV corresponds to a larger denominator effect when g is smaller relative to r, depending on the exact formulation. The key implication is that a higher PV indicates a smaller (r - g) in the denominator, which corresponds to a larger g (closer to r) or equivalent change depending on the exact interpretation. In typical formulations with the same C0, a higher PV implies a higher effective growth rate g (assuming r fixed and g < r). Therefore, Investment X having a higher PV (5000) suggests it has a growth rate g_X that is closer to r than g_Y, i.e., g_X > g_Y. Consequently, option C (X has a lower growth rate than Y) is inconsistent with the given PVs and is incorrect.

Option D: Investment X has a higher growth rate than Investment Y.
- Aligning with the reasoning above, since PV_X is greater than PV_Y while initial cash flows and the discount rate are the same, Investment X must have a higher growth rate g_X than g_Y in order to yield a larger present value under the growing perpetuity framework. This makes option D plausible and consistent with standard growing perpetuity PV relationships. Therefore, this option is correct under the typical interpretation of the growing perpetuity formula, and it is the statement that matches the given PVs.

Summary of reasoning across options:
- A is incorrect because the given information (same r and same initial C0) suffices to compare g through PV values.
- B is incorrect because equal PV is not required when g differs; different PVs can arise from different growth rates even with the same C0 and r.
- C is incorrect because the higher PV for X implies a higher, not lower, growth rate relative to Y (assuming standard PV = C0*(1+g)/(r-g) type sensitivity).
- D is consistent with X having the higher PV, implying a higher growth rate g_X than g_Y under the same r and C0.

Note: The reasoning assumes the standard growing perpetuity relationship where PV is sensitive to the difference between r and g, and that higher PV with identical initial cash flows indicates a higher growth rate for that investment. If one adopts a different cash flow timing convention, the qualitative comparison would still rely on how g shifts PV when r is fixed, leading to the same conclusion about the relative growth rates.

MCD2170 - T1 - 2025 Key Concept 6 Video quiz

Investment X and Investment Y are both growing perpetuities with initial cash flow of $100. Both investments have the same interest rate (r). The present value of Investment X is $5 000, while the present value of Investment Y is $4 000. Which of the following is true?

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