LECTURE 11
Financial Mathematics: Investment appraisal
1.) Concept: Net Present Value (NPV)
Present value and future values concepts can be used to assess investment opportunities. Remember that due to interest rates, a dollar in future is worth more than a dollar today.
In business analysis, you might consider the initial cash outlay and the following cash inflows, and decide if a project is worth pursuing. Example 1 illustrates this idea.
Example 1a: Net Present Value (NPV)
A business project requires a cash outlay of $24,000 for buying special equipment. After this equipment is purchased, it will be possible to make certain products and sell them.
a.) The project is estimated to generate net positive cash flows of RM1000, RM8,000 and RM17,000 in the following three years. Find the NPV, given an interest rate of 6%. Comment.
Answer:
The cash flows can be summarized as follows: note negative numbers for cash outflows and positive numbers for cash inflows. Note that the numbers for Years 1 to 3 are all FUTURE values
|
Year |
Cash Flow |
|
0 |
-24,000 |
|
1 |
1,000 |
|
2 |
8,000 |
|
3 |
17,000 |
Convert all of the cash flows to present values (i.e todays dollars think of Ex. 16 in last tutorial)
Year 0: PV = -24,000
Year 1: P = FV/(1+ i) n = 1,000/ (1+ 0.06) 1 = **943.40
Year 2: PV = 8,000 /(1+ 0.06) 2 = **7119.97
Year 3: PV = 17,000 /(1+ 0.06) 3 = **14273.53
NPV = -24,000 + 943.40 + 7119.97+ 14273.53= -1,663.10
Negative NPV project gives relatively poor returns, reject
NOTE:
THE
FOLLOWING FIN CALC. NPV METHOD WILL NOT BE ACCEPTED BUT YOU CAN USE IT
TO JUST CHECK YOUR COMPUTATIONS
2ndF, CA,
6, i, -24,000, CFi, 1000, CFi, 8000, CFi, 17,000, CFi, NPV
Fin Calc: ** only for computing this value give the working (i.e. formula & substitution shown above)
Year 1: 2ndF, CA, 1000, FV,6,i,1,n,COMP PV Ans: 943.40 *
Year 2: 2ndF, CA, 8000, FV,6,i,2,n,COMP PV Ans: 7119.97 *
Year 3: 2ndF, CA, 17000, FV,6,i,3,n,COMP PV Ans: 14273.53 *
* Ignore negative sign because you receive this money
NPV = -24,000 + 943.40 + 7119.97+ 14273.53= -1,663.10
b.) The project is estimated to generate net positive cash flows of RM1000,
RM8,000 and RM17,000 in the following three years. Find the NPV, given an interest rate of 2%. Comment.
Year 0: PV = -24,000
Year 1: P = FV/(1+ i) n = 1,000/ (1+ 0.02) 1 = ** 980.39
Year 2: PV = 8,000 /(1+ 0.02) 2 = **7,689.35
Year 3: PV = 17,000 /(1+ 0.02) 3 = **16,019.48
NPV = -24,000 +980.39 +7,689.35+ 16,019.48= 689.22
Positive NPV project gives relatively attractive returns, accept
Fin Calc: ** only for computing this value give the working (i.e. formula & substitution shown above)
Year 1: 2ndF, CA, 1000, FV,2,i,1,n,COMP PV Ans: 980.39 *
Year 2: 2ndF, CA, 8,000, FV,2,i,2,n,COMP PV Ans: 7,689.35*
Year 3: 2ndF, CA, 17,000, FV,2,i,3,n,COMP PV Ans: 16,019.48*
* Ignore negative sign because you receive this money
NPV = -24,000 +980.39 +7,689.35+ 16,019.48= 689.22
Positive NPV project gives relatively attractive returns, accept
Interpreting NPV:
NPV > 0; project earns more that discount rate
NPV = 0; project earns same as discount rate
NPV < 0, project earns less that discount rate
Exercise 1a:
A project has the following net cash flows over 4 years. Find the NPV, given an interest rate of 5%. Comment.
Year 1: - 10,000
Year 2: 2,000
Year 3: 3,000
Year 4: 8,000
Exercise 1b:
A food caterer is considering a 3-year contract to supply food for a cafeteria in a certain college. His revenues and costs are summarized below:
|
Year |
Revenues |
Costs |
|
0 |
- |
10,000 * |
|
1 |
20,000 |
15,000 |
|
2 |
21,000 |
15,000 |
|
3 |
30,000 |
26,000 |
* Costs of bidding for the contract.
Given an interest rate of 10%, is this project worthwhile?
2.) Concept: Internal Rate of Return (IRR)
The internal rate of return calculates the rate of the total cash flows and compares it with the prevailing interest rates. The IRR is the interest rate at which the NPV = 0
To compute IRR, find a discount rate at which NPV is positive and another rate at which NPV is negative. Then apply the formula:
Formula:
IRR = (N1I2 N2I1) / (N1 N2)
Where,
Discount rate I1 gives NPV of N1
And discount rate I2 gives NPV of N2
Example 2:
Using the NPV at 2% and 6% for the cash flows given in Example 1, and find the IRR
Cash flows from Eg. 1:
|
Year |
Cash Flow |
|
0 |
-24,000 |
|
1 |
1,000 |
|
2 |
8,000 |
|
3 |
17,000 |
Answer:
At 2% discount rate, NPV = 689.22
At 6% discount rate, NPV = -1,663.10 (answer for Eg. 1)
I1 = 0.02, N1 = 689.22
I2 = 0.06, N2 = -1,663.10
IRR = [689.22(0.06) (-1,663.10)(0.02)]/ [689.22 (-1633.10) ]
= 0.0319
Ans: IRR = 3.19 %
Fin Calc:
2ndF, CA, 24000, +/-,CFi, 1000, CFi, 8000, CFi, 17000, CFi, IRR
Ans: IRR = 3.11 % (wait for a while for the calculator to finish calculations)
Exercise 2:
For a certain project, a discount rate of 15% yields an NPV of RM14000 while a discount rate of 17% yields an NPV of RM17,000
Find the IRR. Note: You might need to set your decimals to 4 places in the financial calculator for this question.
Interpreting the IRR:
IRR > i, IRR exceed prevailing interest rates
IRR < I, IRR is less than prevailing interest rates
IRR = I, IRR is equal to prevailing interest rates
3.) Comparing two or more projects
When you compare several projects, those with the highest NPV or IRR are preferred. This concept will be covered in further detail in the tutorial questions.
Independent
FRA Chapter 22
