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 ($) |
| 0 | -100 |
| 1 | 0 |
| 2 | 121 |
###\begin{aligned} NPV &= C_0 + \frac{C_2}{(1+r)^2} \\ &= -100 + \frac{121}{(1+0.1)^2} \\ &= -100 + 100 \\ &= 0 \\ \end{aligned}###
What is the Internal Rate of Return (IRR) of the project detailed in the table below?
Assume that the cash flows shown in the table are paid all at once at the given point in time. All answers are given as effective annual rates.
| Project Cash Flows | |
| Time (yrs) | Cash flow ($) |
| 0 | -100 |
| 1 | 0 |
| 2 | 121 |
###\begin{aligned} 0 &= C_0 + \frac{C_2}{(1+r_\text{IRR})^2} \\ &= -100 + \frac{121}{(1+r_\text{IRR})^2} \\ \end{aligned}###
###(1+r_\text{IRR})^2 = \frac{121}{100} ######\begin{aligned} r_\text{IRR} &= \left( \frac{121}{100} \right)^{1/2} - 1 \\ &= 0.1 \\ \end{aligned}###
If a project's net present value (NPV) is zero, then its internal rate of return (IRR) will be:
By definition, the Internal Rate of Return (IRR) is the particular required return that makes the project's Net Present Value (NPV) equal to zero.
###\begin{aligned} NPV &= C_0 + \frac{C_1}{(1+r_\text{required})^1} + \frac{C_2}{(1+r_\text{required})^2} + ... + \frac{C_T}{(1+r_\text{required})^T} \\ 0 &= C_0 + \frac{C_1}{(1+r_{irr})^1} + \frac{C_2}{(1+r_{irr})^2} + ... + \frac{C_T}{(1+r_{irr})^T} \\ \end{aligned} ###
Therefore if the NPV is zero then the IRR must be equal to the required return.
The below graph shows a project's net present value (NPV) against its annual discount rate.
For what discount rate or range of discount rates would you accept and commence the project?
All answer choices are given as approximations from reading off the graph.
Projects add to the firm's asset value if their net present value (NPV) of cash flows is positive, which in this case occurs between the discount rates of zero to five percent. The positive NPV can be seen on the graph where the blue line is above above zero on the vertical y-axis which represents the NPV.
All other things remaining equal, a project is worse if its:
The lower the internal rate of return (IRR), the worse the project. Note that when the IRR is lower then the project's net present value (NPV) is also lower. Higher IRR's and NPV's are better.
If positive cash flows occur sooner, then they'll be discounted by less so the NPV of them will be higher, making the project better.
If negative cash flows occur later, then they'll be discounted by more so the NPV of them will be lower, making the project better.
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 an equal amount now (t=0) and in one year (t=1) and have nothing left in the bank at the end (t=1).
How much can you consume at each time?
Let ##C_t## be consumption at time t and ##V_t## be wealth at time t.
Common sense method:
We initially have ##V_0## wealth in the bank. Then we consume or spend ##C_0## a moment later, still at time zero. The amount left in the bank accrues interest so it grows over the next year by the interest rate. To find this future value we multiply by ##(1+r)^1##. At time one, everything that's left in the bank is consumed ##(C_1)## with nothing left over at the end.
###(V_0 -C_0)(1+r)^1 - C_1 = 0 ###| Formula Building Steps | ||
| Time | Event | Formula |
| 0 | Starting wealth | ##V_0## |
| 0 | Consume | ##V_0 - C_0## |
| 1 | Lend to bank for one year | ##(V_0 - C_0)(1+r)^1## |
| 1 | Consume all so there's nothing left | ##(V_0 - C_0)(1+r)^1 - C_1 = 0## |
The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##. So we can solve simultaneously and substitute numbers (k represents thousands),
###(V_0 -C_0)(1+r)^1 - C_1 = 0 ### ###(V_0 -C_0)(1+r)^1 - C_\color{red}{0} = 0 ### ###(100k -C_0)(1+0.1)^1 - C_0 = 0 ### ###100k(1+0.1)^1 -C_0(1+0.1)^1 - C_0 = 0 ### ###C_0\left(1+(1+0.1)^1\right) = 100k(1+0.1)^1 ### ###\begin{aligned} C_0 &= \frac{100k(1+0.1)^1}{1+(1+0.1)^1} \\ &= \frac{100,000 \times 1.1}{2.1} \\ &= 52,380.9524 \\ \end{aligned}### ###C_1 = C_0 = 52,380.9524###Present value method:
This method is easier to formulate. Since all wealth will be consumed, the present value of the positive wealth and negative consumption must equal zero.
###V_0 -C_0 - \frac{C_1}{(1+r)^1} +\frac{V_1}{(1+r)^1} = 0 ###The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##. Also ##V_1 = 0## since there's no wealth left over at the end.
Solving simultaneously and substituting numbers (k represents thousands),
###100k -C_0 - \frac{C_0}{(1+0.1)^1} +\frac{0}{(1+0.1)^1} = 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1}\right) = 100k ### ###\begin{aligned} C_0 &= \frac{100k}{\left(1+ \frac{1}{(1+0.1)^1}\right)} \\ &= 52,380.9524 = C_1 \\ \end{aligned}###Future value method:
Similarly to the present value method, this method is easy to formulate. Since all wealth will be consumed, the future value of the positive wealth and negative consumption must equal zero.
###V_0(1+r)^1 -C_0(1+r)^1 - C_1 = 0 ###The question stated that consumption at t=0 and t=1 are equal, so ##C_0 = C_1 ##.
Solving simultaneously and substituting numbers (k represents thousands),
###100k(1+0.1)^1 -C_0(1+0.1)^1 - C_0 = 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1}\right) = 100k ### ###C_0 = 52,380.9524 = C_1 ###You have $100,000 in the bank. The bank pays interest at 10% pa, given as an effective annual rate.
You wish to consume an equal amount now (t=0), in one year (t=1) and in two years (t=2), and still have $50,000 in the bank after that (t=2).
How much can you consume at each time?
Let ##C_t## be consumption at time t and ##V_t## be wealth at time t.
Common sense method:
We have ##V_0## in the bank then we consume ##C_0## of that at time zero. The amount left in the bank accrues interest over the year so we grow it by ##(1+r)^1##. Again at time one we consume ##C_1##, and the amount remaining in the bank accrues more interest. At time two we consume ##C_2## and the amount left after this is ##V_2##.
###\left( (V_0 -C_0)(1+r)^1 - C_1 \right)(1+r)^1-C_2 = V_2 ###| Formula Building Steps | ||
| Time | Event | Formula |
| 0 | Starting wealth | ##V_0## |
| 0 | Consume | ##V_0 - C_0## |
| 1 | Lend to bank for one year | ##(V_0 - C_0)(1+r)^1## |
| 1 | Consume more | ##(V_0 - C_0)(1+r)^1 - C_1## |
| 2 | Lend to bank for another year | ##((V_0 - C_0)(1+r)^1 - C_1)(1+r)^1## |
| 2 | Consume again but leave some wealth aside | ##((V_0 - C_0)(1+r)^1 - C_1)(1+r)^1 - C_2 = V_2## |
The question stated that consumption at t=0,1 and 2 are equal, so ##C_0 = C_1 = C_2##.
Solving simultaneously and substituting numbers (k represents thousands),
###\left( (V_0 -C_0)(1+r)^1 - C_1 \right)(1+r)^1-C_2 = V_2 ### ###\left( (100k -C_0)(1+0.1)^1 - C_0 \right)(1+0.1)^1-C_0 = 50k ### ###\begin{aligned} C_0 &= \dfrac{100k(1+0.1)^2 - 50k }{1+(1+0.1)^1 + (1+0.1)^2} \\ &= 21,450.1511 = C_1 = C_2 \\ \end{aligned}###Present value method:
###V_0 -C_0 - \frac{C_1}{(1+r)^1} -\frac{C_2}{(1+r)^2}- \frac{V_2}{(1+r)^2}= 0 ###Also, consumption at t=0, 1 and 2 are all equal, so
###C_0 = C_1 = C_2 ###Solving simultaneously and substituting numbers (k represents thousands),
###100k -C_0 - \frac{C_0}{(1+r)^1} -\frac{C_0}{(1+r)^2} - \frac{50k}{(1+r)^2}= 0 ### ###C_0\left(1+ \frac{1}{(1+0.1)^1} + \frac{1}{(1+0.1)^2}\right) = 100k - \frac{50k}{(1+0.1)^2} ### ###\begin{aligned} C_0 &= \frac{100k - \dfrac{50k}{(1+0.1)^2}}{\left(1+ \dfrac{1}{(1+0.1)^1} + \dfrac{1}{(1+0.1)^2}\right)} \\ &= 21,450.1511 = C_1 = C_2 \\ \end{aligned}###Your neighbour asks you for a loan of $100 and offers to pay you back $120 in one year.
You don't actually have any money right now, but you can borrow and lend from the bank at a rate of 10% pa. Rates are given as effective annual rates.
Assume that your neighbour will definitely pay you back. Ignore interest tax shields and transaction costs.
The Net Present Value (NPV) of lending to your neighbour is $9.09. Describe what you would do to actually receive a $9.09 cash flow right now with zero net cash flows in the future.
Right now (at t=0), borrow $109.09 from the bank and lend $100 of it to your neighbour now. This leaves us with a positive cash flow of $9.09 at t=0.
In on year (t=1), take the $120 from your neighbour and use it to pay back the bank the $120 owed ##\left(109.09\times(1+0.1)^1\right)##. This leaves us with a cash flow of zero at t=1.
To work out the amounts, use an arbitrage table.
| Arbitrage Table of Cash Flows | ||
| Instrument | Time 0 | Time 1 |
| Buy debt from (lend to) neighbour now, and wait for repayment in one year. | -100 | 120 |
| Sell debt to (borrow from) bank at 10% pa and pay it back in one year. | 109.09 Step 3 |
-120 Step 2 |
| Total | 9.09 Step 4 |
0 Step 1 |
The steps used to calculate the table's values are given here.
Step 1: All future cash flows need to total zero, that way only the initial (t=0) cash flow will be non-zero.
Step 2: The bank loan cash flow at time 1 must equal -120 so that total cash flows are zero. Since we're paying this $120 at the end, we must be borrowing using this bank loan.
Step 3: Since we're paying back 120 in one year, we must be borrowing the present value of that which is 109.09, calculated as follows: ###V_0 = -\dfrac{C_1}{(1+r)^1} = -\dfrac{-120}{(1+0.1)^1} = 109.09###
Step 4: Adding up the total cash flows at time zero, -100+109.09 = 9.09 which is the NPV of the arbitrage.
Arbitrage tables are great since they show how to a create a positive arbitrage cash flow of the NPV right now ($9.09) with no risk and no capital required.
An investor owns an empty block of land that has local government approval to be developed into a petrol station, car wash or car park. The council will only allow a single development so the projects are mutually exclusive.
All of the development projects have the same risk and the required return of each is 10% pa. Each project has an immediate cost and once construction is finished in one year the land and development will be sold. The table below shows the estimated costs payable now, expected sale prices in one year and the internal rates of returns (IRR's).
| Mutually Exclusive Projects | |||
| Project | Cost now ($) |
Sale price in one year ($) |
IRR (% pa) |
| Petrol station | 9,000,000 | 11,000,000 | 22.22 |
| Car wash | 800,000 | 1,100,000 | 37.50 |
| Car park | 70,000 | 110,000 | 57.14 |
Which project should the investor accept?
The best project appears to be the car park because it has the highest IRR. But the other important decision criteria is NPV which isn't calculated. The NPV's are:
###V_\text{0, Petrol station} = -9,000,000 + \dfrac{11,000,000}{(1+0.1)^1} = 1,000,000 ### ###V_\text{0, Car wash} = -800,000 + \dfrac{1,100,000}{(1+0.1)^1} = 200,000 ### ###V_\text{0, Car park} = -70,000 + \dfrac{110,000}{(1+0.1)^1} = 30,000 ###So while the car park has the highest IRR, it has the lowest NPV. The petrol station has the highest NPV, but lowest IRR.
Because the projects are mutually exclusive, only one project can be chosen. Rationally it's best to make the most money and choose the project with the highest NPV. After all, would you prefer to make $30,000 on the car park, or 1 millions of dollars on the petrol station?
An investor owns a whole level of an old office building which is currently worth $1 million. There are three mutually exclusive projects that can be started by the investor. The office building level can be:
- Rented out to a tenant for one year at $0.1m paid immediately, and then sold for $0.99m in one year.
- Refurbished into more modern commercial office rooms at a cost of $1m now, and then sold for $2.4m when the refurbishment is finished in one year.
- Converted into residential apartments at a cost of $2m now, and then sold for $3.4m when the conversion is finished in one year.
All of the development projects have the same risk so the required return of each is 10% pa. The table below shows the estimated cash flows and internal rates of returns (IRR's).
| Mutually Exclusive Projects | |||
| Project | Cash flow now ($) |
Cash flow in one year ($) |
IRR (% pa) |
| Rent then sell as is | -900,000 | 990,000 | 10 |
| Refurbishment into modern offices | -2,000,000 | 2,400,000 | 20 |
| Conversion into residential apartments | -3,000,000 | 3,400,000 | 13.33 |
Which project should the investor accept?
Since the projects are mutually exclusive, the one with the highest NPV ##(V_0)## should be chosen.
###V_\text{0, Rent as is} = -900k + \dfrac{990k}{(1+0.1)^1} = 0### ###V_\text{0, Refurbish into offices} = -2m + \dfrac{2.4m}{(1+0.1)^1} = 0.1818m### ###V_\text{0, Convert to residential} = -3m + \dfrac{3.4m}{(1+0.1)^1} = 0.0909m###The refurbishment into modern offices has the highest NPV, so that's the best option. In this case it also has the highest IRR but that is a co-incidence. The best project is the one which makes the most wealth and that is best decided according to NPV.