ING Stock Valuation Exam Study Notes

University of Amsterdam Exam Question: ING Stock Valuation

• The objective is to determine the current value ($PV_0$) of a share of ING stock based on a projected dividend schedule and a specified equity cost of capital.
• The problem provides specific growth phases and a discount rate to apply to the future cash flows.

Problem Parameters

• Dividend Year 1 ($Div_1$): The company is expected to pay a dividend of $\$2$ at the end of the current year ($t=1$).
• Dividend Horizon (Years 1–5): The dividend will remain constant at $\$2$ for the first 5 years.
• Long-term Growth ($g$): From year 6 onwards, the dividend is expected to grow at a rate of $2\%$ per year forever.
• Equity Cost of Capital ($r_e$): The required rate of return or discount rate is $7\%$.
• Goal: Calculate the current price of the share ($PV_0$).

Identified Cash Flows

• $t=1$: $\$2$
• $t=2$: $\$2$
• $t=3$: $\$2$
• $t=4$: $\$2$
• $t=5$: $\$2$
• $t=6$: \2 \times (1 + 0.02) = \$2.04$
• $t=7$: Growing by $2\%$ annually (e.g., \2.04 \times 1.02 = \$2.0808$)

Valuation Method 1: Four-Year Annuity and Year 5 Perpetuity

This approach splits the valuation into a constant annuity for the first four years and a growing perpetuity that begins using the year 5 dividend.

Stage 1: First 4 Dividends (Annuity)

• Cash Flow ($C$): $\$2$
• Rate ($r$): $7\%$
• Time Periods ($N$): $4$
• Formula: $PV_{\text{annuity}} = \frac{C}{r} \times \left(1 - \frac{1}{(1+r)^N}\right)$
• Calculation: $PV_{\text{annuity}} = \frac{2}{0.07} \times \left(1 - \frac{1}{(1.07)^4}\right) = \$6.77$

Stage 2: Dividends from Year 5 Onward (Growing Perpetuity)

• Terminal Value Calculation at $t=4$: To capture all dividends from year 5 to infinity, we use the growing perpetuity formula where the next expected dividend is $Div_5$.
• Calculation of $PV_4$: $PV_4 = \frac{Div_5}{r_e - g} = \frac{2}{0.07 - 0.02} = \frac{2}{0.05} = \$40$
• Discounting to Present Value ($PV_0$): $PV_0 = \frac{PV_4}{(1+r_e)^4} = \frac{40}{(1.07)^4} = \$30.52$

Total Value (Method 1)

• Combine Stages: \6.77 (\text{Stage 1}) + \$30.52 (\text{Stage 2}) = \$37.29$

Alternative Valuation Method 2: Five-Year Annuity and Year 6 Perpetuity

This approach treats the first five years as a constant annuity and the growing perpetuity as starting with the dividend at year 6 ($Div_6$).

Stage A: First 5 Dividends (Annuity)

• Cash Flow ($C$): $\$2$
• Rate ($r$): $7\%$
• Time Periods ($N$): $5$
• Calculation: $PV_{\text{annuity}} = \frac{2}{0.07} \times \left(1 - \frac{1}{(1.07)^5}\right) = \$8.20$

Stage B: Dividends from Year 6 Onward (Growing Perpetuity)

• Dividend at Year 6 ($Div_6$): $Dividend \, at \, Year \, 5 \times (1 + g) = 2 \times 1.02 = \$2.04$
• Continuation Value at Time 5 ($PV_5$): $PV_5 = \frac{Div_6}{r_e - g} = \frac{2.04}{0.07 - 0.02} = \frac{2.04}{0.05} = \$40.80$
• Discounting to Present Value ($PV_0$): $PV_0 = \frac{PV_5}{(1+r_e)^5} = \frac{40.80}{(1.07)^5} = \$29.09$

Total Value (Method 2)

• Combine Stages: \8.20 (\text{Stage A}) + \$29.09 (\text{Stage B}) = \$37.29$

Final Exam Answer Results

• Comparison of Options provided in the exam question:
  • A. $\$31.84$
  • B. $\$32.62$
  • C. $\$34.56$
  • D. $\$37.28$
• Conclusion: The calculated share price is $\$37.29$. The value closest to the calculated price is option D ($\$37.28$), making it the correct choice.