Perpetuities & Growing Perpetuities - Finance

1. An investment project is expected to generate cash flows of $50,000 per year for the first 10 years (from year 1 to year 10). Starting in year 11, the cash flow is expected to be $30,000 and remain at that level in perpetuity. Given a discount rate of 9%, what is the present value of this entire project?
   Answer: $461,682 This problem involves two parts: a 10-year annuity and a deferred perpetuity. First, calculate the PV of the annuity: $PV_{annuity} = \$50,000 \times [\frac{1 - (1.09)^{-10}}{0.09}] = \$320,883$. Second, calculate the value of the perpetuity. The perpetuity of $30,000 starts in Year 11, so its value at Year 10 is $PV_{10} = \frac{\$30,000}{0.09} = \$333,333$. This value must be discounted back 10 years to find its present value: $PV_{perp} = \frac{\$333,333}{(1.09)^{10}} = \$140,799$. Finally, add the present values of both parts: $PV_{total} = \$320,883 + \$140,799 = \$461,682$.
2. A trust is established to fund a university professorship. The first grant of $120,000 will be awarded 5 years from today. Subsequent annual grants are expected to increase by 3% each year indefinitely. If the appropriate discount rate is 8%, what is the present value of this commitment today?
   Answer: $1,763,942 This is a deferred growing perpetuity. The first step is to find the value of the perpetuity one period before the first cash flow, which is at Year 4. The formula for a growing perpetuity is $PV = C_1 / (r - g)$. Here, the first cash flow (C₁) occurs at Year 5. Thus, the formula gives the value at Year 4: $PV_4 = \frac{\$120,000}{0.08 - 0.03} = \frac{\$120,000}{0.05} = \$2,400,000$. The second step is to discount this value back to today (Year 0): $PV_0 = \frac{PV_4}{(1 + r)^4} = \frac{\$2,400,000}{(1.08)^4} \approx \$1,763,942$.
3. A foundation needs to fund an annual scholarship of $50,000 in perpetuity, starting one year from today. The foundation's investment portfolio is expected to yield 8% per year. The founder plans to fund this endowment with two equal payments: one today and one in a year. What is the amount of each payment?
   Answer: $324,519 First, calculate the total present value required today to fund the perpetuity. The scholarship payments start in one year. $PV_{total} = \frac{C}{r} = \frac{\$50,000}{0.08} = \$625,000$. This total amount is funded by two equal payments, 'P'. One payment is made today (its present value is P) and the other is made in one year (its present value is $P / (1.08)$). The sum of the present values of these two payments must equal the total PV required: $P + \frac{P}{1.08} = \$625,000$. Factor out P: $P(1 + \frac{1}{1.08}) = \$625,000$. $P(1.925926) = \$625,000$. Solve for P: $P = \frac{\$625,000}{1.925926} \approx \$324,519$.
4. A stock is priced at $80.00 per share. It is expected to pay a dividend of $3.00 one year from now. After that, the dividend is expected to grow at a constant rate, 'g', forever. If the required rate of return for the stock is 10%, what is the implied annual growth rate 'g'?
   Answer: 6.25% The formula for the price of a stock with constant dividend growth is $P_0 = \frac{D_1}{r - g}$. We are given $P_0 = \$80$, $D_1 = \$3.00$, and $r = 10\%$. We need to solve for 'g'. $\$80 = \frac{\$3.00}{0.10 - g}$. Rearranging the formula: $0.10 - g = \frac{\$3.00}{\$80} = 0.0375$. Solving for 'g': $g = 0.10 - 0.0375 = 0.0625$, or 6.25%.
5. A share of preferred stock pays a dividend of $1.50 quarterly, with the first dividend due in three months. The stated annual required rate of return is 6%, compounded quarterly. What is the price of the stock?
   Answer: $100.00 This is a perpetuity with payments and compounding occurring on a quarterly basis. The simplest approach is to use the periodic rate and periodic cash flow. The annual rate is 6% compounded quarterly, so the quarterly rate is $r_{quarterly} = \frac{6\%}{4} = 1.5\%$. The quarterly dividend is $1.50. The present value is calculated as: $PV = \frac{C_{period}}{r_{period}} = \frac{\$1.50}{0.015} = \$100.00$.
6. A philanthropist wants to endow a research grant that pays out $25,000 annually in perpetuity. The first payment is scheduled to be made three years from today. The philanthropist plans to fund this endowment with a single lump-sum donation today. The endowment fund is expected to earn 5% annually, but a 1% annual management fee is deducted from the fund balance at the end of each year. What is the required donation amount?
   Answer: $577,848 First, determine the net discount rate after fees: $r_{net} = 5\% - 1\% = 4\%$. Second, determine the value of the perpetuity. Since the first payment is at Year 3, the perpetuity formula $PV = C/r$ will give its value at Year 2: $PV_2 = \frac{\$25,000}{0.04} = \$625,000$. Finally, discount this value back two years to find the present value today: $PV_0 = \frac{\$625,000}{(1.04)^2} = \frac{\$625,000}{1.0816} \approx \$577,848$.
7. An investor is choosing between two stocks. Stock A is expected to pay a dividend of $2.00 next year, growing at 5% annually. Stock B is expected to pay a dividend of $2.50 next year, growing at 3% annually. At what discount rate would an investor be indifferent between these two stocks?
   Answer: 13.0% An investor would be indifferent if the present values of the two stocks are equal. Set the growing perpetuity formulas for both stocks equal to each other and solve for 'r'. $PV_A = \frac{\$2.00}{r - 0.05}$ and $PV_B = \frac{\$2.50}{r - 0.03}$. Set $PV_A = PV_B$: $\frac{\$2.00}{r - 0.05} = \frac{\$2.50}{r - 0.03}$. Cross-multiply: $2.00(r - 0.03) = 2.50(r - 0.05)$. $2r - 0.06 = 2.5r - 0.125$. $0.065 = 0.5r$. $r = \frac{0.065}{0.5} = 0.13$, or 13.0%.
8. A company has a perpetual liability that grows at 2% per year. The first payment of $1,000,000 is due in 5 years. The company wants to set aside a lump sum today in an account that earns 7% annually to cover this liability. What is the required lump sum?
   Answer: $15,228,843 This is a deferred growing perpetuity. First, calculate the value of the liability one period before the first payment. The first payment is at Year 5, so we find the value at Year 4: $PV_4 = \frac{C_5}{r-g} = \frac{\$1,000,000}{0.07 - 0.02} = \frac{\$1,000,000}{0.05} = \$20,000,000$. This is the value at Year 4. To find the required lump sum today, this amount must be discounted back 4 years: $PV_0 = \frac{PV_4}{(1+r)^4} = \frac{\$20,000,000}{(1.07)^4} = \frac{\$20,000,000}{1.31079601} \approx \$15,228,843$.
9. An investor is considering a security that promises to pay $10,000 at the end of each year, forever. The investor's required return is stated as an 8% annual percentage rate (APR) compounded semi-annually. What is the maximum price the investor should be willing to pay for this security?
   Answer: $122,549 Since the cash flows are annual, the discount rate must be an effective annual rate (EAR). The given rate is an APR compounded semi-annually. First, convert the APR to an EAR: $EAR = (1 + \frac{APR}{m})^m - 1 = (1 + \frac{0.08}{2})^2 - 1 = (1.04)^2 - 1 = 0.0816$, or 8.16%. Then, use this EAR to find the present value of the perpetuity: $PV = \frac{C}{EAR} = \frac{\$10,000}{0.0816} \approx \$122,549$.
10. A company is valued using a growing perpetuity model. Its next year's free cash flow ($FCF_1$) is projected to be $10 million. The weighted average cost of capital (WACC) is 12%, and the constant growth rate is 4%. The company currently has $50 million in debt and 5 million shares outstanding. What is the estimated stock price per share?
    Answer: $15.00 This question requires calculating the firm value, then equity value, then price per share. Step 1: Calculate the total value of the firm (enterprise value) using the growing perpetuity formula. $V_{firm} = \frac{FCF_1}{WACC - g} = \frac{\$10,000,000}{0.12 - 0.04} = \frac{\$10,000,000}{0.08} = \$125,000,000$. Step 2: Calculate the value of the equity by subtracting the value of debt. $V_{equity} = V_{firm} - V_{debt} = \$125,000,000 - \$50,000,000 = \$75,000,000$. Step 3: Calculate the price per share by dividing the equity value by the number of shares. $Price = \frac{V_{equity}}{\text{Shares outstanding}} = \frac{\$75,000,000}{5,000,000} = \$15.00$.
11. A company's stock has just paid an annual dividend of $2.00 per share ($D_0$). An analyst expects this dividend to grow at a constant rate of 4% per year forever. If the required rate of return for this stock is 11%, what is the estimated value of one share today?
    Answer: $29.71 The Gordon Growth Model (a growing perpetuity formula) requires the next period's dividend ($D_1$). The dividend just paid ($D_0$) is $2.00. First, calculate $D_1$: $D_1 = D_0 \times (1 + g) = \$2.00 \times (1.04) = \$2.08$. Next, apply the formula $P_0 = \frac{D_1}{r - g}$: $P_0 = \frac{\$2.08}{0.11 - 0.04} = \frac{\$2.08}{0.07} \approx \$29.71$.
12. A consol bond is priced at $1,200 and pays an annual coupon. A preferred stock is priced at $90 and is expected to pay a constant annual dividend of $4.50 starting next year. Assuming the two securities have the same risk and are priced efficiently, what is the annual coupon payment of the consol bond?
    Answer: $60.00 Both a consol bond and a preferred stock can be valued as perpetuities. If they have the same risk and are priced efficiently, they must have the same discount rate 'r'. First, find the discount rate implied by the preferred stock: $r = \frac{D}{P} = \frac{\$4.50}{\$90} = 0.05$, or 5%. Second, use this discount rate to find the coupon payment 'C' for the consol bond: $P = \frac{C}{r} \Rightarrow C = P \times r = \$1,200 \times 0.05 = \$60.00$.
13. A special type of preferred stock will pay its first annual dividend of $5.00 two years from today. The dividend will remain at $5.00 for the following year (i.e., in year 3). Thereafter, starting in year 4, the dividend will grow at a constant rate of 2% per year. If the required rate of return is 10%, what is the value of this stock today?
    Answer: $55.79 This problem involves two distinct cash flow streams. First, a single payment of $5.00 at Year 2. Second, a growing perpetuity starting at Year 3. The cash flow at Year 3 ($C_3$) is $5.00, and it grows at 2% thereafter. The value of this growing perpetuity at Year 2 (one period before its first payment) is $PV_2 = \frac{C_3}{r-g} = \frac{\$5.00}{0.10 - 0.02} = \frac{\$5.00}{0.08} = \$62.50$. So, at Year 2, there is a cash flow of $5.00 and the perpetuity is worth $62.50. The total value at Year 2 is $5.00 + $62.50 = $67.50\). To find the value today, discount this amount back two years: $PV_0 = \frac{\$67.50}{(1.10)^2} = \frac{\$67.50}{1.21} = \$55.79$.
14. A company that owns a patent for a declining technology expects to receive royalties of $500,000 one year from today. The royalty payments are expected to decline by 4% per year in perpetuity. If the appropriate discount rate is 10%, what is the present value of these royalty payments?
    Answer: $3,571,429 This is a growing perpetuity with a negative growth rate (g = -4%). The formula is $PV = \frac{C_1}{r - g}$. Plugging in the values: $PV = \frac{\$500,000}{0.10 - (-0.04)} = \frac{\$500,000}{0.10 + 0.04} = \frac{\$500,000}{0.14} \approx \$3,571,429$.
15. A real estate investment is expected to generate a net rental income of $80,000 at the end of this year. This income is expected to grow at a constant 3% per year. However, the property requires a major renovation at the end of year 5, costing $200,000. If the discount rate is 11%, what is the value of this investment?
    Answer: $881,171 The value of the investment is the present value of the growing perpetuity of rental income minus the present value of the renovation cost. Step 1: Calculate the PV of the growing perpetuity of income. $PV_{income} = \frac{C_1}{r - g} = \frac{\$80,000}{0.11 - 0.03} = \frac{\$80,000}{0.08} = \$1,000,000$. Step 2: Calculate the PV of the single renovation cost. $PV_{cost} = \frac{\$200,000}{(1.11)^5} = \frac{\$200,000}{1.68506} = \$118,829$. Step 3: Subtract the PV of the cost from the PV of the income. $Value = \$1,000,000 - \$118,829 = \$881,171$.
16. A mining operation will produce net cash flows of $5 million next year. The cash flows are expected to decline at a rate of 10% per year for the following 4 years (i.e., through year 5). After year 5, the cash flows are expected to be zero. If the discount rate is 15%, what is the present value of the operation's cash flows?
    Answer: $14.11 million This problem describes a 5-year growing annuity with a negative growth rate, not a perpetuity. The cash flows must be discounted individually or by using the growing annuity formula. The cash flows are: C₁=$5M, C₂=$4.5M, C₃=$4.05M, C₄=$3.645M, C₅=$3.2805M. Discounting each at 15%: $PV = \frac{5}{1.15} + \frac{4.5}{1.15^2} + \frac{4.05}{1.15^3} + \frac{3.645}{1.15^4} + \frac{3.2805}{1.15^5}$ $= 4.348 + 3.403 + 2.663 + 2.084 + 1.631 = \$14.13M$. Using the formula $PV = C_1 [\frac{1 - (\frac{1+g}{1+r})^n}{r-g}]$: $PV = 5 [\frac{1 - (\frac{0.90}{1.15})^5}{0.15 - (-0.10)}] = 5 [\frac{1 - 0.2945}{0.25}] = \$14.11M$.
17. The present value of a growing perpetuity is $1,500. The discount rate is 9% and the growth rate is 3%. What is the cash flow expected in Year 5 ($C_5$)?
    Answer: $101.30 This requires working backward from the present value to find the future cash flow. Step 1: Use the growing perpetuity formula to find the cash flow in Year 1 ($C_1$). $PV = \frac{C_1}{r - g} \Rightarrow \$1,500 = \frac{C_1}{0.09 - 0.03} \Rightarrow \$1,500 = \frac{C_1}{0.06}$. Therefore, $C_1 = \$1,500 \times 0.06 = \$90.00$. Step 2: Calculate the cash flow for Year 5 using the growth formula. $C_5 = C_1 \times (1 + g)^{5-1} = \$90.00 \times (1.03)^4 = \$90.00 \times 1.1255 = \$101.30$.
18. A perpetuity-due makes its first payment of $X today, with subsequent payments occurring at the beginning of each year and growing at 2% annually. If the effective annual discount rate is 7%, and the present value of the perpetuity-due is $1,070, what is the value of the first payment, $X?
    Answer: $50.00 A perpetuity-due can be valued as an ordinary perpetuity with its value compounded forward one period, or as an immediate payment plus an ordinary perpetuity. Using the latter approach, the PV of a growing perpetuity-due is $PV = C_0 + \frac{C_1}{r-g}$, where $C_0$ is the payment today and $C_1 = C_0(1+g)$. The question uses X for the first payment, which is today (C0). So, $PV = X + \frac{X(1+g)}{r-g}$. Plugging in the values: $\$1,070 = X + \frac{X(1.02)}{0.07-0.02} = X + \frac{1.02X}{0.05} = X + 20.4X = 21.4X$. Solving for X: $X = \frac{\$1,070}{21.4} = \$50.00$.

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