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An investor has an endowment of 300 in t=0t=0 and can invest in the following di...
Jul 13, 2024
Solution
a
Calculate Returns: The return for each investment project is calculated as follows: Return=Cash Inflow at t=1−Cash Outflow at t=0Cash Outflow at t=0 \text{Return} = \frac{\text{Cash Inflow at } t=1 - \text{Cash Outflow at } t=0}{\text{Cash Outflow at } t=0} For each project: ReturnIP1=80−6060=2060=13≈0.333 \text{Return}_{IP_1} = \frac{80 - 60}{60} = \frac{20}{60} = \frac{1}{3} \approx 0.333 ReturnIP2=200−100100=100100=1 \text{Return}_{IP_2} = \frac{200 - 100}{100} = \frac{100}{100} = 1 ReturnIP3=105−100100=5100=0.05 \text{Return}_{IP_3} = \frac{105 - 100}{100} = \frac{5}{100} = 0.05 ReturnIP4=60−4040=2040=12=0.5 \text{Return}_{IP_4} = \frac{60 - 40}{40} = \frac{20}{40} = \frac{1}{2} = 0.5 Sorted in descending order: IP2,IP4,IP1,IP3 IP_2, IP_4, IP_1, IP_3
b
Transformation Curve TIT_I: The transformation curve TIT_I is drawn on a C0−C1C_0 - C_1 graph. The points on the curve represent the combinations of consumption at t=0t=0 (C0C_0) and t=1t=1 (C1C_1) based on the investment opportunities. Without a capital market, the curve is determined by the endowment and the returns of the projects
c
Optimal Investment Program: The optimal investment program depends on the investor's consumption preferences. Graphically, this is shown by the tangency point between the transformation curve TIT_I and the highest possible indifference curve representing the investor's preferences
d
Capital Market Introduction: With a perfect and complete capital market at a 20% interest rate, the slope of the new transformation curve TMT_M is given by: Slope=−(1+r)=−(1+0.2)=−1.2 \text{Slope} = -(1 + r) = -(1 + 0.2) = -1.2 This extends the set of feasible consumption allocations, allowing for borrowing and lending at the same rate
e
Separation of Decisions: Investment and consumption decisions can now be separated because the capital market allows investors to achieve their desired consumption pattern independently of their investment choices. This implies that the net present value (NPV) rule can be used to make investment decisions
f
NPV Calculation: The NPVs of the investment projects are calculated as follows: NPV=Cash Inflow at t=11+r−Cash Outflow at t=0 \text{NPV} = \frac{\text{Cash Inflow at } t=1}{1 + r} - \text{Cash Outflow at } t=0 For each project: NPVIP1=801.2−60=6.67 \text{NPV}_{IP_1} = \frac{80}{1.2} - 60 = 6.67 NPVIP2=2001.2−100=66.67 \text{NPV}_{IP_2} = \frac{200}{1.2} - 100 = 66.67 NPVIP3=1051.2−100=−12.5 \text{NPV}_{IP_3} = \frac{105}{1.2} - 100 = -12.5 NPVIP4=601.2−40=10 \text{NPV}_{IP_4} = \frac{60}{1.2} - 40 = 10 The intercepts of TMT_M represent the maximum possible consumption at t=0t=0 and t=1t=1 if all resources are allocated to consumption at one time period
g
Indifference Curve Slope: The utility function is given by: U=C0×C1 U = C_0 \times C_1 The slope of the indifference curve is: dC1dC0=−C1C0 \frac{dC_1}{dC_0} = -\frac{C_1}{C_0}
h
Optimal Consumption Allocation: To find the optimal consumption allocation, we maximize the utility function subject to the budget constraint. The optimal allocation occurs where the slope of the indifference curve equals the slope of the transformation curve: −C1C0=−1.2 -\frac{C_1}{C_0} = -1.2 Solving for C1C_1 and C0C_0 gives the optimal consumption points. Financial transactions involve borrowing or lending to achieve these points
i
Imperfect Capital Market: With different rates for deposits (10%) and borrowing (40%), the transformation curve becomes kinked. The slope for deposits is: −(1+0.1)=−1.1 -(1 + 0.1) = -1.1 The slope for borrowing is: −(1+0.4)=−1.4 -(1 + 0.4) = -1.4 This affects investment and consumption decisions by creating different marginal rates of substitution for borrowing and lending
Answer
The returns for the investment projects are: IP2=1 IP_2 = 1 , IP4=0.5 IP_4 = 0.5 , IP1=0.333 IP_1 = 0.333 , IP3=0.05 IP_3 = 0.05 . The transformation curve TIT_I is based on the endowment and investment returns. With a perfect capital market, the slope of TMT_M is -1.2, and investment and consumption decisions can be separated. The NPVs are calculated, and the optimal consumption allocation is derived. An imperfect capital market results in a kinked transformation curve.
Key Concept
Investment Returns and Transformation Curves
Explanation
The returns of investment projects determine their attractiveness. The transformation curve represents feasible consumption allocations. A perfect capital market allows separation of investment and consumption decisions, while an imperfect market introduces different borrowing and lending rates, affecting decisions.
Solution
a
Calculate the NPV of each project for discount rates of 0%, 10%, and 20%: - For Project A: - NPV at 0%: NPVA=−100+60(1+0)1+60(1+0)2+0(1+0)3=20NPV_A = -100 + \frac{60}{(1+0)^1} + \frac{60}{(1+0)^2} + \frac{0}{(1+0)^3} = 20 - NPV at 10%: NPVA=−100+60(1+0.1)1+60(1+0.1)2+0(1+0.1)3=4.13NPV_A = -100 + \frac{60}{(1+0.1)^1} + \frac{60}{(1+0.1)^2} + \frac{0}{(1+0.1)^3} = 4.13 - NPV at 20%: NPVA=−100+60(1+0.2)1+60(1+0.2)2+0(1+0.2)3=−11.11NPV_A = -100 + \frac{60}{(1+0.2)^1} + \frac{60}{(1+0.2)^2} + \frac{0}{(1+0.2)^3} = -11.11 - For Project B: - NPV at 0%: NPVB=−100+0(1+0)1+0(1+0)2+140(1+0)3=40NPV_B = -100 + \frac{0}{(1+0)^1} + \frac{0}{(1+0)^2} + \frac{140}{(1+0)^3} = 40 - NPV at 10%: NPVB=−100+0(1+0.1)1+0(1+0.1)2+140(1+0.1)3=5.78NPV_B = -100 + \frac{0}{(1+0.1)^1} + \frac{0}{(1+0.1)^2} + \frac{140}{(1+0.1)^3} = 5.78 - NPV at 20%: NPVB=−100+0(1+0.2)1+0(1+0.2)2+140(1+0.2)3=−19.44NPV_B = -100 + \frac{0}{(1+0.2)^1} + \frac{0}{(1+0.2)^2} + \frac{140}{(1+0.2)^3} = -19.44
b
Calculate the IRR for each project: - For Project A, solve for rr in 0=−100+60(1+r)1+60(1+r)20 = -100 + \frac{60}{(1+r)^1} + \frac{60}{(1+r)^2}: - IRR for Project A: r≈18.92%r \approx 18.92\% - For Project B, solve for rr in 0=−100+140(1+r)30 = -100 + \frac{140}{(1+r)^3}: - IRR for Project B: r≈11.53%r \approx 11.53\% - Plot the NPV functions with NPV on the vertical axis and discount rate on the horizontal axis
c
Circumstances for accepting projects: - Accept Project A if the discount rate is less than the IRR of Project A (18.92%). - Accept Project B if the discount rate is less than the IRR of Project B (11.53%)
d
Calculate the NPV of the incremental investment (B - A) for discount rates of 0%, 10%, and 20%: - Incremental cash flows: CF0=0CF_0 = 0, CF1=−60CF_1 = -60, CF2=−60CF_2 = -60, CF3=140CF_3 = 140 - NPV at 0%: NPVB−A=0+−60(1+0)1+−60(1+0)2+140(1+0)3=20NPV_{B-A} = 0 + \frac{-60}{(1+0)^1} + \frac{-60}{(1+0)^2} + \frac{140}{(1+0)^3} = 20 - NPV at 10%: NPVB−A=0+−60(1+0.1)1+−60(1+0.1)2+140(1+0.1)3=1.65NPV_{B-A} = 0 + \frac{-60}{(1+0.1)^1} + \frac{-60}{(1+0.1)^2} + \frac{140}{(1+0.1)^3} = 1.65 - NPV at 20%: NPVB−A=0+−60(1+0.2)1+−60(1+0.2)2+140(1+0.2)3=−8.33NPV_{B-A} = 0 + \frac{-60}{(1+0.2)^1} + \frac{-60}{(1+0.2)^2} + \frac{140}{(1+0.2)^3} = -8.33 - Plot the NPV function of the incremental investment on the graph. - Accept Project A if the IRR on the incremental investment is less than the opportunity cost of capital
2 Solution
a
Estimate of ABC's intrinsic value per share: - Calculate the value of dividends starting in year 6: - D6=0.4×EPS6=0.4×10×(1+0.2)5=24.88D_6 = 0.4 \times EPS_6 = 0.4 \times 10 \times (1+0.2)^5 = 24.88 - Calculate the present value of dividends from year 6 onwards: - P5=D6r−g=24.880.15−0.15=165.87P_5 = \frac{D_6}{r - g} = \frac{24.88}{0.15 - 0.15} = 165.87 - Calculate the present value of P5P_5: - P0=165.87(1+0.15)5=82.19P_0 = \frac{165.87}{(1+0.15)^5} = 82.19 - Intrinsic value per share: V0=P0=82.19V_0 = P_0 = 82.19
b
Effect of changing dividend payout to 20%: - Calculate the value of dividends starting in year 6: - D6=0.2×EPS6=0.2×10×(1+0.2)5=12.44D_6 = 0.2 \times EPS_6 = 0.2 \times 10 \times (1+0.2)^5 = 12.44 - Calculate the present value of dividends from year 6 onwards: - P5=D6r−g=12.440.15−0.15=82.94P_5 = \frac{D_6}{r - g} = \frac{12.44}{0.15 - 0.15} = 82.94 - Calculate the present value of P5P_5: - P0=82.94(1+0.15)5=41.10P_0 = \frac{82.94}{(1+0.15)^5} = 41.10 - Intrinsic value per share: V0=P0=41.10V_0 = P_0 = 41.10
3 Solution
Answer
- NPV of Project A at 0%, 10%, 20%: 20,4.13,−11.1120, 4.13, -11.11 - NPV of Project B at 0%, 10%, 20%: 40,5.78,−19.4440, 5.78, -19.44 - IRR of Project A: 18.92%18.92\% - IRR of Project B: 11.53%11.53\% - Accept Project A if discount rate < 18.92%18.92\% - Accept Project B if discount rate < 11.53%11.53\% - NPV of incremental investment at 0%, 10%, 20%: 20,1.65,−8.3320, 1.65, -8.33 - Intrinsic value per share with 40% payout: 82.1982.19 - Intrinsic value per share with 20% payout: 41.1041.10
Key Concept
NPV and IRR calculations for investment projects
Explanation
NPV and IRR are crucial for evaluating the profitability of investment projects. NPV measures the difference between the present value of cash inflows and outflows, while IRR is the discount rate that makes the
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