PPT OUTLINE
1. What is Rate of Return?
 Rate of Return, also referred to as the Return of investment, is the net gain or loss of an
investment over specified time period, expressed as a percentage of the investment’s
initial cost.
 When calculating rate of return, you are determining the percentage change from the
beginning of the period until the end.
2. What is Net Present Value (NPV)?
 Net Present Value (NPV) is a method that calculates the value of the future cash flows
which the project will produce during the particular period of time by taking into account
different factors.
3. What is Internal Rate of Return (IRR)?
 Internal rate of return, also referred to as “breakeven interest rate”, is a methods that
solves for the interest rate that equates the equivalent worth of the cash inflows (receipts
or saving) to the equivalent worth of cash outflows (expenditures, including investment
costs) .
 It can also be defined as, the interest rate at which the benefits are equivalent to the
costs, or the present worth (PW) is 0.
 It measures the rate of return on a project or investment
 Also defined as the annual rate of growth that an investment is expected to generate
 A discounted cash flow technique which considers the time value of money
 Can be solved using the IRR formulas.
4. Comparison of concepts:
1. Rate of Return VS Internal Rate of Return
a. Example for Rate of Return: You invest an amount of $100 today. With that
investment, you get back $3 every year, FOREVER. In this problem, we have 3% rate
of return per year.
b. Example for Internal Rate of Return: You invest $100 today. Then you get back $60
after 1 year. After 2 years, you get back another $60. And then it stops. (So…It is not
FOREVER.) In this case, what is the rate of return? We know that the rate of return in
this scenario is HIDDEN. This Hidden Rate of Return is what we call now as the
INTERNAL RATE OF RETURN.
From the examples given above,
2. Net Present Value (NPV) VS Internal Rate of Return (IRR)
𝑅
𝑡
The net present value is calculated using the equation 𝑁𝑃𝑉 = (1+ⅈ)
𝑡 ; where, NPV is the net
present value, 𝑅𝑡 is the net cash flow at time t, ⅈ is the discount rate, and the 𝑡 is the time of
the cash flow. It is also important to note that in finding the NPV, the rate is present. On the
other hand, the Internal Rate of Return is calculated using the formulas and methods
presented in the next following slides: the r or the rate is missing and the net present value is
equal to zero.
5. How to solve for IRR? (Formulas that can be used to solve for IRR)
 IRR Formula
𝑁
𝑁𝑃𝑉 = ∑
𝑐𝑛
(1 + 𝑟)𝑛
𝑛=0
Where,
NPV = Net Present Value
N
= total number of periods
n
= non-negative integer
𝑐𝑛
= cash flow
r
= internal rate of return
 Trial and Error Method
Process:
i.
Choose two IRR guesses wherein one of the two will yield a positive net present
value while the other will yield a negative one. Note, it is better if you chose two
slightly closer values, like 10% and 15% or 10% and 20%.
ii.
After guessing two IRRs, substitute to the formula
𝑁𝑃𝑉𝑎
𝐼𝑅𝑅 = 𝑅𝑎 + (𝑅𝑏 − 𝑅𝑎 ) (
)
𝑁𝑃𝑉𝑎 − 𝑁𝑃𝑉𝑏
where,
Ra = lower rate
Rb = higher rate
NPVa = net present value at lower rate
NPVb = net present value at higher rate
 EW (PW, FW, or Aw) methods
i.
IRR using Future worth (FW) formulation
Equation:
FW of cash flows (receipts) – FW of cash flows (disbursement) = 0
[R (F/A, i*%, N) + S] – [I (F/P, i*%, N) + E (F/A, i*%, N)] = 0
ii.
IRR using Present worth (PW) formulation
Equation:
PW of cash inflows – PW of cash outflows = 0
[R (P/A, i*%, N) + S (P/F, i*%, N)] – [I + E (P/A, i*%, N)] = 0
iii.
IRR using Annual worth (AW) formulation
Equation:
AW of receipts – AW of disbursement = 0
[R + S (A/F, i*%, N)] – [I (A/P, i*%, N) + E] = 0
Note, using the IRR formula might not help in solving the IRR especially when problems have cash flows
involving larger amount partnered with higher number of years. In this case, it would be better to use the
Trial and Error Method, though it might take time to solve for the IRR. The third method mentioned on
the other hand, might be confusing when you do not have knowledge about the equations involving of
Present Worth, Future Worth and Annual Worth. Hence, it would be safe and fast to use the IRR formula
6. Decision Guidelines:
 Accept When
 Reject When
7. Terms to remember in solving for the IRR using different formulas:
a) Net Present Value (NPV)
b) Initial Investment (I)
c) Salvage Value (S)
d) Annual Revenue (R)
e) Annual Expenses (E)
f) Study Period (N)
g) MARR (i)
8. Sample problems
9. What is Incremental IRR?
10. Sample Problems
11. Cases where IRR might not work
 Case 1 (Unconventional Cash Flows)
 Case 2 (Mutually Exclusive Projects)