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?

Question

BlackTom AI · Accepted Answer

The present value formula for a growing perpetuity is PV = C / (r − g), where C is the first-period cash flow, r is the discount rate, and g is the growth rate (with r > g so PV is finite).

1) Relevant knowledge points and concepts
- Growing perpetuity PV formula: PV = C/(r − g).
- For a fixed C and r, PV is inversely proportional to (r − g): a larger PV implies a smaller (r − g), which (with r fixed) implies a larger g.
- The formula requires r > g to produce a finite PV; since PVs given are finite, that condition holds.

2) Step-by-step reasoning
- Both investments have C = $100 and the same r. Let gX and gY be their growth rates.
- Use the formula for each investment and solve for (r − g):

3) Calculations and intermediate steps
- For Investment X: 100/(r − gX) = 5,000 ⇒ r − gX = 100/5,000 = 0.02.
  Thus gX = r − 0.02.
- For Investment Y: 100/(r − gY) = 4,000 ⇒ r − gY = 100/4,000 = 0.025.
  Thus gY = r − 0.025.
- Compare: gX − gY = (r − 0.02) − (r − 0.025) = 0.005 > 0, so gX > gY.

4) Why the correct answer is correct
Because with identical initial cash flow and identical discount rate, the only remaining parameter that can change PV is the growth rate. A larger PV (5,000 vs 4,000) corresponds to a smaller denominator (r − g), hence a larger g. Algebra above shows gX exceeds gY by 0.5 percentage points.

5) Why the other options are incorrect
- (a) says X has a lower growth rate: contradicted by algebra (gX = r − 0.02 vs gY = r − 0.025 → gX > gY).
- (c) claims we need r: not true because r cancels in the comparison; PVs and C suffice to compare g.
- (d) claims PVs must be equal: false because growth rates can differ; same C and r do not guarantee same PV if g differs.

6) Conclusion
Therefore Investment X has a higher growth rate than Investment Y, matching answer b.

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?Single choice

Similar Questions

Expansionary fiscal policy in an open economy with a floating exchange rate will likely:

Contractionary monetary policy in an open economy with a floating exchange rate will likely:

Contractionary monetary policy in an open economy with a fixed exchange rate will likely ____ a trade surplus.

Which is expansionary monetary policy?

Which is contractionary fiscal policy?

Which is expansionary fiscal policy?

What would raise aggregate supply (AS)?

What would shift aggregate supply (AS) to the right?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler