ECON1202 Lecture 2
Evaluating Time-Money Choices
UNSW Economics
c©UNSW Economics
Agenda
1 Equations of value;
2 A campus investment conundrum;
3 The Net Present Value (NPV);
4 The Internal Rate of Return (IRR);
5 Conclusions.
c©UNSW Economics
Equations of value
Scenario
You owe your parents some money. You have agreed to pay them in
three installments: $100 today, $500 in 6 months, and $350 in 9 months.
You’d like to restructure the payments so that the value of the debt
remains unchanged. They have agreed to use compounded (quarterly)
interest to value the debt.
You may either pay everything now, or pay everything in 12 months.
How much do would you pay, in each of the options?
c©UNSW Economics
Solution technique:
1 Note the timing for each cashflow item;
2 Calculate the value of each cashflow item at the FOCAL DATE;
3 Add the values up to obtain the net value.
3 6 9 12
Debts: $500 $350$100
t (months)
↑
focal date
c©UNSW Economics
Example
Using a focal date of now, and compound interest at the annual value of 7% and
compounded quarterly, how much would you owe at the focal date?
c©UNSW Economics
Example
Using a focal date of now, and compound interest at the annual value of 7% and
compounded quarterly, how much would you owe at the focal date?
Let x be the payment now. Then,
value of repayment = value of debts
x = 100
(
1+
0.07
4
)0
+ 500
(
1+
0.07
4
)−2
+ 350
(
1+
0.07
4
)−3
x = 100+ 482.95+ 332.25
x = $915.20
c©UNSW Economics
Example
Using a focal date of 12 months later, and compound interest at the annual value of
7% and compounded quarterly, how much would you owe at the focal date?
c©UNSW Economics
Example
Using a focal date of 12 months later, and compound interest at the annual value of
7% and compounded quarterly, how much would you owe at the focal date?
Let y be the payment in 12 months. Then,
value of repayment = value of debts
y = 100
(
1+
0.07
4
)4
+ 500
(
1+
0.07
4
)2
+ 350
(
1+
0.07
4
)1
y = 107.19+ 517.65+ 356.13
y = $980.96
c©UNSW Economics
Question:
Do you get a better deal if you pay today, or in 12 months?
Checking the two payments (now = $915.20, 12 months = $980.96), the
present value of the 12 month payment is,
P = 980.96
(
1+
0.07
4
)−4
= $915.20
That is, the present value of the future payment is the same as the
current payment.
Important!
When we use compound interest, as we did here, the focal date
doesn’t matter.
c©UNSW Economics
Scenario: A Coffeeshop Business Plan
Your friend has come to you with a business plan to open a
delivery-only coffeeshop called “The Bean Runner”. Her business
plan includes details about projected cashflows. She has asked you to
run the numbers. Will it work out?
c©UNSW Economics
The numbers ...
The numbers (according to your friend) look like this:
Year ending Costs Income Net Note
0 50,000 0 -50,000 Set-up costs
1 34,000 25,000 -9,000 Two-wages @ $17,000
2 34,000 45,000 11,000
3 44,000 60,000 16,000 Third wage @ $10,000
4 44,000 70,000 26,000
5 44,000 75,000 31,000
c©UNSW Economics
Steps to a decision...
1 Converting the NET cash-flows into PRESENT VALUES (based on
the going alternative rate of return);
2 Sum the cash-flow present values to get the NET PRESENT VALUE;
3 Make a decision:
NPV
{
> 0 : profitable
< 0 : unprofitable
c©UNSW Economics
Problem:
Suppose your friend has access to a bank which offers an annual rate
of 12% (compounded annually). Should The Bean Runner get off the
ground?
PV = S(1+ r)−t = (I − C)(1+ 0.12)−t
Year Costs Income I - C (1+ 0.12)−t PV
end
0 50,000 0 -50,000 1.000 −50, 000
1 34,000 25,000 -9,000 0.893 −8, 036
2 34,000 45,000 11,000 0.797 8, 769
3 44,000 60,000 16,000 0.712 11, 388
4 44,000 70,000 26,000 0.636 16, 523
5 44,000 75,000 31,000 0.567 17, 590
NPV −3, 764
c©UNSW Economics
Definition: Net Present Value (NPV)
If Ft is the estimated cash flow for a T period project at the end of
period t and the yearly compounded rate of interest, the COST OF
CAPITAL is r, then the NET PRESENT VALUE is the sum of present
values, assuming cash flows arrive at the end of the year,
NPV = F0 + F1(1+ r)
−1 + · · ·+ FT(1+ r)
−T . (1)
c©UNSW Economics
Interpreting the NPV rule
• Remember: if the NPV > 0 then the project is worthwhile. Why?
• The NPV comparison tells us whether the project ‘beats the bank’
– that is, whether the project is more profitable than the
alternative of investing your money with the bank.
• In other words: if (and only if) NPV < 0, then you’d be better off
not doing the project and putting your money in the bank to earn
interest instead.
Challenge: do the same calculation as in the example above, but at 8%
cost of capital. Is the project now worthwhile?
c©UNSW Economics
Suppose we do the calculation of the NPV for a range of interest rates
...
0.05 0.10 0.15
0
−4
−8
−12
−16
−20
4
8
12
16
20
NPV($′000) r
b
b
b
b
b
b
b
b
IRR
c©UNSW Economics
• The INTERNAL RATE OF RETURN asks the question,
What cost of capital would mean an NPV = 0, the break-even
point?
• Which means that if,
r = IRR
then the return on the project (over its life) is exactly the same as if
we put our money into the bank! (the competition is a dead-heat!)
c©UNSW Economics
Definition: Internal Rate of Return (IRR)
The internal rate of return of a project is the solution to the following
question:
What interest rate, when used in the project’s NPV calculation,
would produce an NPV equal to zero?
NPV = F0 + F1(1+ r)
−1 + · · ·+ FT(1+ r)
−T = 0 . (2)
In words, the project’s IRR is the interest rate that – if offered by a bank
– would make it just as worthwhile for me to save my money with that
bank rather than invest it in the project.
Caution! Care with the IRR
When interpreting the IRR, notice that it is (by definition) independent
of the current cost of capital (what is actually offered by the banks). It is
tempting to think that IRR somehow depends on this value. It doesn’t!
(But we compare to it.)
c©UNSW Economics
Example (IRR)
Suppose a project requires an initial investment of $20,000 and returns $7,000
and $16,000 at the end of the first and second years respectively. Find the IRR
of the project assuming yearly compounding.
c©UNSW Economics
Example (IRR)
Suppose a project requires an initial investment of $20,000 and returns $7,000
and $16,000 at the end of the first and second years respectively. Find the IRR
of the project assuming yearly compounding.
NPV = −20, 000+ 7, 000(1+ r)−1 + 16, 000(1+ r)−2
Set NPV = 0 and solve for r. Multiply through by (1+ r)2:
20, 000(1+ r)2 = 7, 000(1+ r) + 16, 000
20, 000+ 40, 000r + 20, 000r2 = 7, 000+ 7, 000r + 16, 000
20, 000r2 + 33, 000r − 3, 000 = 0
Solve with the quadratic formula:
r =
−33, 000±
√
33, 0002 − (4)(20, 000)(−3, 000)
(2)(20, 000)
r = 0.0864,−1.736 so... r∗ = 8.64%
c©UNSW Economics
• There are simple cases (esp. when t ≤ 2) that can be solved by
hand using the quadratic equation;
• Obviously, it is difficult to solve for r in most cases (other than
trial-and-error approximation), so we often use a software
package (such as MICROSOFT EXCEL);
c©UNSW Economics
Example (optional)
Using a computer program of your choice, find the IRR and NPV, at 3%
interest, of the following stream of net profits: (-45, -25, -2, 12, 27, 30, 31).
c©UNSW Economics
Example (optional)
Using a computer program of your choice, find the IRR and NPV, at 3%
interest, of the following stream of net profits: (-45, -25, -2, 12, 27, 30, 31).
Going to my program, I enter the data into one column (cells A1:A7). Then, I use the
function IRR:
=IRR(A1:A7)
Giving: 7.81%.Now, for the NPV care is required, the syntax is:
=NPV(r,F1,F2,...,FT)
Note: no F0 – that’s right, to get the correct outcome, you would have to write (in our
case):
=NPV(0.03,A2:A7)+A1
Giving: $9. Try it with r = 0.0781.
c©UNSW Economics
Does The Bean Runner go ahead?
• Clearly, at cost of capital 12%, the The Bean Runner isn’t worth it –
we’d do better by putting our money in the bank at the offered
rate of 12%;
• However, by calculating the IRR, we could see that for any cost of
capital less than 10%, the The Bean Runner is a good idea! ... we’d
beat the best interest going at the bank!
• Finally, suppose that we had two different projects – the coffee
roaster being one, the other being a simple bakery, if they both
have a positive NPV, we still wouldn’t know how to pick between
them.
• However, if you can’t do both, choosing the higher NPV project
will be the highest returning project.
c©UNSW Economics