The required return of a project is 10%, given as an effective annual rate. Assume that the cash flows shown in the table are paid all at once at the given point in time.
What is the Net Present Value (NPV) of the project?
| Project Cash Flows | |
| Time (yrs) | Cash flow ($) |
| 0 | -100 |
| 1 | 11 |
| 2 | 121 |
###\begin{aligned} NPV &= C_0 + \frac{C_1}{(1+r)^1} + \frac{C_2}{(1+r)^2} \\ &= -100 + \frac{11}{(1+0.1)^1} + \frac{121}{(1+0.1)^2} \\ &= -100 + 10 + 100 \\ &= 10 \\ \end{aligned}###
A project's NPV is positive. Select the most correct statement:
When the NPV of a project is positive, the project's IRR must be more than its required return.
Question 542 price gains and returns over time, IRR, NPV, income and capital returns, effective return
For an asset price to double every 10 years, what must be the expected future capital return, given as an effective annual rate?
There's no mention of income cash flows such as dividends or rent so we'll ignore them. The capital return ##(r_\text{cap})## leading to the price rise can be calculated using the 'present value of a single cash flow' formula.
###P_0 = \dfrac{P_{10}}{(1+r_\text{cap})^{10}} ###For the price to double in 10 years, then ##P_{10}## will be twice ##P_0##, so ##P_{10} = 2P_0##. Substitute this into the above equation and solve for the capital return.
###P_0 = \dfrac{2P_{0}}{(1+r_\text{cap})^{10}} ### ###\begin{aligned} (1+r_\text{cap})^{10} &= \dfrac{2P_{0}}{P_0} \\ &= 2 \\ \end{aligned}### ###{\left( (1+r_\text{cap})^{10} \right)}^{1/10} = {2}^{1/10}### ###1+r_\text{cap} = {2}^{1/10}### ###\begin{aligned} r_\text{cap} &= 2^{1/10} - 1 \\ &= 0.071773463 \\ \end{aligned}###Question 543 price gains and returns over time, IRR, NPV, income and capital returns, effective return
For an asset price to triple every 5 years, what must be the expected future capital return, given as an effective annual rate?
There's no mention of income cash flows such as dividends or rent so we'll ignore them. The capital return ##(r_\text{cap})## leading to the price rise can be calculated using the 'present value of a single cash flow' formula.
###P_0 = \dfrac{P_{5}}{(1+r_\text{cap})^{5}} ###For the price to triple in 5 years, then ##P_{5}## will be triple ##P_0##, so ##P_{5} = 3P_0##. Substitute this into the above equation and solve for the capital return.
###P_0 = \dfrac{3P_{0}}{(1+r_\text{cap})^{5}} ### ###\begin{aligned} (1+r_\text{cap})^{5} &= \dfrac{3P_{0}}{P_0} \\ &= 3 \\ \end{aligned}### ###1+r_\text{cap} = 3^{1/5} ### ###\begin{aligned} r_\text{cap} &= 3^{1/5} - 1 \\ &= 0.24573094 \\ \end{aligned}###You're considering a business project which costs $11m now and is expected to pay a single cash flow of $11m in one year. So you pay $11m now, then one year later you receive $11m.
Assume that the initial $11m cost is funded using the your firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa.
Which of the following statements about the net present value (NPV), internal rate of return (IRR) and payback period is NOT correct?
If the project is accepted, then the market value of the firm's assets will fall by $1m, since the net present value (NPV) of the project is -$1m. It's true that the firm's cash assets will fall by $11m when it buys into the project, but the positive $11m that will be received in one year adds a $10m present value to the market value of assets, giving an NPV of -$1m.
For the NPV:
###\begin{aligned} V_0 &= C_0 + \dfrac{C_1}{(1+r)^1} \\ &= -11m + \dfrac{11m}{(1+0.1)^1} \\ &= -11m + 10m \\ &= -1m \\ \end{aligned}###The project's NPV is negative so it should be rejected.
For the IRR:
###V_0 = C_0 + \dfrac{C_1}{(1+r)^1} ### ###0 = -11m + \dfrac{11m}{(1+r_{IRR})^1} ### ###\begin{aligned} r_{IRR} &= \dfrac{11m}{11m} - 1 \\ &= 0 \\ \end{aligned}###The project's IRR is less than the cost of capital (10%) so again, the project should be rejected.
You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume twice as much now (t=0) as in one year (t=1) and have nothing left in the bank at the end.
How much can you consume at time zero and one? The answer choices are given in the same order.
No explanation provided.
You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume half as much now (t=0) as in one year (t=1) and have nothing left in the bank at the end.
How much can you consume at time zero and one? The answer choices are given in the same order.
No explanation provided.
A project to build a toll road will take 3 years to complete, costing three payments of $50 million, paid at the start of each year (at times 0, 1, and 2).
After completion, the toll road will yield a constant $10 million at the end of each year forever with no costs. So the first payment will be at t=4.
The required return of the project is 10% pa given as an effective nominal rate. All cash flows are nominal.
What is the payback period?
This project is interesting because it has a negative NPV, but since its positive cash flows continue forever, it must eventually pay itself off. So it has a finite payback period. Note that inflation is a red herring since the pay back period doesn't account for inflation. Usually only nominal cash flows are used in pay back period calculations and this is what is given.
Algebraic method: The cumulative cash flow ##C_\text{sum,T}## at time T is:
###\begin{aligned} C_\text{sum,T} &= -50 \times 3 + 10 \times (T - 3) \\ \end{aligned}###
The payback period ##T_\text{payback}## occurs when the cumulative cash flow is zero, so:
###\begin{aligned} C_\text{sum,T} &= -50 \times 3 + 10 \times (T - 3) \\ 0 &= -50 \times 3 + 10 \times (T_\text{payback} - 3) \\ 10 \times (T_\text{payback} - 3) &= 50 \times 3 \\ T_\text{payback} &= 3 + \frac{50 \times 3}{10} \\ &= 3 + 15 \\ &= 18 \\ \end{aligned}###
Table method:
| Payback Period Calculation | ||
| Time (yrs) |
Cash flow ($) |
Cumulative cash flow ($) |
| 0 | -50 | -50 |
| 1 | -50 | -100 |
| 2 | -50 | -150 |
| 3 | 0 | -150 |
| 4 | 10 | -140 |
| 5 | 10 | -130 |
| ... | ... | ... |
| 16 | 10 | -20 |
| 17 | 10 | -10 |
| 18 | 10 | 0 |
A firm is considering a business project which costs $10m now and is expected to pay a single cash flow of $12.1m in two years.
Assume that the initial $10m cost is funded using the firm's existing cash so no new equity or debt will be raised. The cost of capital is 10% pa.
Which of the following statements about net present value (NPV), internal rate of return (IRR) and payback period is NOT correct?
If the project is accepted then there will be a gain of $2.1m since the $10m cash spent on the project will turn into $12.1m cash 2 years later. This gives a 10% pa internal rate of return (IRR). The calculation is detailed at the end.
However, if the project is rejected then the $10m cash not spent on the project will sit in the bank and will also increase at 10% pa which is the bank interest rate, so that the cash at bank will be worth $12.1m in 2 years. Hence, the project will not make the firm worth $2.1m more if it is accepted.
The fact that the project is zero NPV and has an IRR equal to the required return indicates that the project is not very good, but not bad either. The firm's managers would be indifferent to accepting or rejecting it.
To show that the NPV of accepting the project is zero:
###\begin{aligned}NPV &= -V_0 + \dfrac{V_2}{(1+r_\text{required return})^2} \\ &= -10m + \dfrac{12.1m}{(1+0.1)^2} = -10m + 10m = 0 \end{aligned}###To show that the project's IRR is equal to its required return (also known as the cost of capital):
###NPV = -V_0 + \dfrac{V_2}{(1+r_\text{required return})^2}### ###0 = -V_0 + \dfrac{V_2}{(1+r_\text{IRR})^2}### ###0 = -10m + \dfrac{12.1m}{(1+r_\text{IRR})^2}### ###(1+r_\text{IRR})^2 = \dfrac{12.1m}{10m}### ###1+r_\text{IRR} = \left( \dfrac{12.1m}{10m} \right)^{1/2}### ###\begin{aligned} r_\text{IRR} &= \left( \dfrac{12.1m}{10m} \right)^{1/2}-1 = 1.21^{(1/2)}-1 = 0.1 \\ \end{aligned}###