How ROE and Discount Rates Affect the Value of Growth Opportunities

Summary

The document raises a valuation question about why a new investment increases common stock value only when its return on equity exceeds the discount rate. It relates earnings per share and share price to ROE, and expresses price as the value of earnings without growth plus the present value of growth opportunities. From this setup, it asks whether the difference between ROE and the discount rate is determined by the present value of growth opportunities.

The text contains the question and the investor’s algebra, but no answer or supporting analysis. It therefore does not establish a complete method for comparing ROE with the required return or show how growth opportunity value is estimated. Its useful takeaway is the distinction the question frames: ROE describes returns on equity, the discount rate reflects the return used to value cash flows, and growth value depends on investment returns relative to that hurdle. Further assumptions would be needed to apply the equations to a particular company.

Key ideas

• ROE compares earnings with equity or price, while the discount rate is used to value future cash flows.
• The document relates share price to earnings without growth and the present value of growth opportunities.
• It asks whether growth opportunity value explains the gap between ROE and the discount rate, but gives no answer.
• The value added by a new investment depends on its return relative to the discount rate.

Tags

# Relationship between ROE and IRR


# Relationship between ROE and IRR












In the textbook I read the following:

> We can increase the present value of a share of common stock with a new investment only if $ROE > r$ , where $r$ is a discount rate (capitalization rate). If a new investment results in $ROE < r$, the price of the stock will decline even though earnings could be higher.

I have a trouble to understand the mathematical relationship bertween $ROE$ and $r$. In order to understand that I did the following operations. Since $$ROE=\frac{EPS}{P},$$ where $EPS$ is earnings per share, $P$ is stock price. From another hand, we have $$P=\frac{EPS}{r}+PVGO,$$ where $PVGO$ is present value of growth opportunities. Doing simple algebra the later yields: $$r=\frac{EPS}{P-PVGO}.$$ Does it mean that the difference between $ROE$ and $r$ is determined by $PVGO$? Please kindly explain the difference between those concepts.

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