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Net Present Value (NPV),
Internal Rate of Return (IRR)
and Modified IRR (MIRR)
Lonnie Chrisman, Ph.D.
Lumina Decision Systems
Analytica User Group
Part 1 : 20 Nov 2008
Part 2: 4 Dec 2008
Copyright © 2008 Lumina Decision Systems, Inc.
Uses NPV and IRR
• Capital budget planning, e.g.
Choosing between investments
Deciding whether to fund new projects
• NPV and IRR are metrics used to
compare the valuation of alternative
cash flows over time.
Copyright © 2008 Lumina Decision Systems, Inc.
Present Value
Question: Which is worth more? $10,000
to be received with certainty one year
from today, or:
• $10,000 received right now?
• $9,500 received now?
• $9,000 received now?
• $8,000 received now?
Estimated present value: __________
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Factors influencing
time-value of money
•
•
•
•
•
•
•
Inflation.
Risk-free rate of return.
Opportunities to put money to work.
Alternative investments.
Your remaining lifespan.
Financing rate.
Uncertainty (?)
Copyright © 2008 Lumina Decision Systems, Inc.
Discount Rate
• Quantifies your time-value of money, as a % per
time-period (often %/year).
• Exercise 1: Using your present value of $10,000
received in one year, compute your implied
discount rate.
pv  (1  r )  $10,000
$10,000
r
1
pv
If you felt $10,000 in one year
is worth $9,300 now, your
discount rate would be:
r = 10000/9300 – 1 = 7.5%
Copyright © 2008 Lumina Decision Systems, Inc.
Present Value Exercises
Exercise 2: Using a discount rate of
10%/yr, what is the present value of
$10,000 received in:
• 2 years?
• 10 years?
• 6 months?
• 3 months?
Copyright © 2008 Lumina Decision Systems, Inc.
Cash Flow & NPV
Exercise 3: A $10,000 investment (a possible
project) will return $2,000 annually for 10
years.
• Represent this cash flow in Analytica.
(There are 11 cash flow “events”)
Using a 12% discount rate:
• Compute the present value (in Analytica) of
each cache flow event.
• Compute the net present value for this
investment.
• Should you make the investment?
Copyright © 2008 Lumina Decision Systems, Inc.
Interpreting NPV
Assuming perfect cash-flow knowledge:
• NPV > 0 Investment adds value. Proceed with
project.
• NPV < 0 Reject project
• Projects (investments) are often mututally
exclusive. Pursue option with max(NPV) as long as
its NPV>0.
• NPV = maximum you are willing to pay today to
guarantee the future cash flow.
• But… NPV is only one consideration. Organizations
also have non-monetary objectives to consider.
Copyright © 2008 Lumina Decision Systems, Inc.
NPV function’s offset
• The NPV (in both Analytica and Excel)
assumes the first point is 1 time period
in the future.
Present value is:
• NPV(r,cashFlow,T) * (1+r)
or
• cashFlow0 + NPV(r,cashFlow,futureT)
Copyright © 2008 Lumina Decision Systems, Inc.
NPV Exercises
Exercise 4: A real estate investment of $200K
will result in monthly rental earnings of $1K
for 8 years, and will then be sold (with
certainty) for $250K.
• Compute the NPV for this investment.
Assume annual discount rate = 8%
• Graph the NPV-curve for all discount rates
from 0% to 20%.
• Compare NPV-curve if you increase rental
earnings 2% each year.
Copyright © 2008 Lumina Decision Systems, Inc.
More NPV exercises
Exercise 5: A $1.5B nuclear power plant
with a 30 year lifetime will generate
$200M per year for 30 years (realized
at the end of each year), and then cost
$5B to decommission on the 31st year.
• Compute the NPV using an 8% discount
rate.
• Graph the NPV for discount rates
ranging from 0% to 20%.
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XNPV – unequal time periods
Exercise 6:
• A $100K investment on 4 Dec 2008 pays
a $2K dividend on 1 Jan 2009, another
on 1 June 2009, and then returns your
$100K on 1 Aug 2009.
• Using discount rate = 10%, use XNPV to
compute the net present value.
Copyright © 2008 Lumina Decision Systems, Inc.
Investment Decisions using NPV
Exercise 7: A $10,000 treasury note with
3.5% coupon rate, matures on 15 Dec
2009. If you require a 2% return, how
much are you willing to pay on 4 Dec
2008?
Actual quote (4 Dec 2008): $10,459.06
Should you buy it?
Copyright © 2008 Lumina Decision Systems, Inc.
Uncertain Cash Flows (ENPV)
A proposed product:
• Requires Poisson(9) months to develop,
at LogNormal(µ:10K,σ:3K) per month.
• Will launch if successful, P(success)=60%
• After launch, monthly earnings for Poisson(24)
months:
Earningsm~Normal(10K,8K)
Exercise 8: (use monthly discount_rate=1%)
• Build a model of this cash flow.
Graph: Bands(cash_flow)
• Compute NPV.
Compute: E[NPV], Graph: PDF(NPV)
Copyright © 2008 Lumina Decision Systems, Inc.
Interpreting NPV with Uncertainty
• What is the criteria for pursuing a
project/investment?
ENPV>0?
getFract(NPV,10%)>0?
ENPV>0 and getFract(NPV,1%)>-100K?
• Expected return / risk tolerance
tradeoff…
Copyright © 2008 Lumina Decision Systems, Inc.
Net Present Utility
Motivations:
• Incorporating risk-tolerance or risk-adversity.
• Non-monetary considerations, e.g.:
Community good will
Economic development
Strategic fit
Staff skill development
• U(earnings,cgw,ed,sf,skd) – utility is a non-linear
function of earnings and other factors.
• NPV(r,U,T) – can be used to compute and base
decisions on NPU.
• When U captures risk-tolerance, ENPV>0 serves as a
go/no-go decision criteria.
Copyright © 2008 Lumina Decision Systems, Inc.
Obtaining Corporate
Discount Rate from Stock Price
• From Capital Asset Pricing Model (CAPM)
Parameters:
• d = corporate discount rate (rate shareholders expect
from investments).
• β = stock price “beta”=Cov(stock,market)/Var(market)
• rf = Risk-free rate of interest
• E[Rm] = Expected return from the stock market.
d  rf   ( E[ Rm ]  rf )
Copyright © 2008 Lumina Decision Systems, Inc.
CAPM Exercise
Exercise 9: Find the shareholder-implied
discount rate for AAPL, using β=1.93, E[Rm]=8%
(ave. return from NASDAQ 1980-2008),
rf=1.5%.
d

 ( E[ Rm ]  rf )  rf
 1.93(8%  1.5%)  1.5%

14%
Copyright © 2008 Lumina Decision Systems, Inc.
Part II
Internal Rate of Return
(IRR)
Copyright © 2008 Lumina Decision Systems, Inc.
Rate of Return
• Exercise 10: A $10,000 investment today
pays $15,000 in 3 years. What is the
annual rate of return?
$10,000  (1  r )  $15,000
3
Copyright © 2008 Lumina Decision Systems, Inc.
Internal Rate of Return (IRR)
Exercise 11: A $1M investment returns
$200K per year for 10 years. What is
the rate of return?
• Plot the NPV of this investment as a
function of discount rate (from 0-20%)
• Use IRR function to find rate or return.
• Compare to an annual return of $180K
or $220K per year.
• Graph IRR as annual return varies from
$100K to $300K.
Copyright © 2008 Lumina Decision Systems, Inc.
IRR in perspective
• IRR measures the quality or efficiency
of an investment, but not the
magnitude.
• Has intuitive appeal, extremely widely
used in practice.
• Does not require a discount rate.
• Many downsides – can be extremely
misleading. Highly non-robust.
• Shunned by textbooks and academics.
Prevailing wisdom: Use NPV, avoid IRR!
Copyright © 2008 Lumina Decision Systems, Inc.
IRR has many downsides
• Exercise 12: Return to NPV graph for
our nuclear power plant in Exercise 5
($1.5B initial cost, $5B decommission
cost at 30y, $200M earnings per year).
• What is the rate of return for this
project?
Copyright © 2008 Lumina Decision Systems, Inc.
IRR is not uniquely determined
• IRR may have multiple solutions:
Up to as many solutions as there are sign
changes in the cash flow.
• IRR may also have zero solutions:
E.g., Zero sign changes.
Common example: Project never reaches
profitability.
Copyright © 2008 Lumina Decision Systems, Inc.
IRR can mislead
Contradictory results when costs come after
revenue. Lower IRR can be better.
• Exercise 13: Graph the NPV-curve (0-10%) for
these two cash flows, and compute IRR:
Year 0
1
2
3
4
5
6
Proj 1
-200 200 200
200 200 200 -900
Proj 2
-200 195 195
195 195 195 -900
Copyright © 2008 Lumina Decision Systems, Inc.
IRR distorts apparent value of
intermediate returns
• Appears as if your positive intermediate earnings
can be re-invested at IRR rate.
Makes options with exceptionally high IRR look too good.
• Appears as if your negative intermediate expenses
are financed from the start at IRR rate.
Makes bad projects look too good, good projects with late
cash flow look bad.
In practice, these distortions are often huge, and
highly relevant – compensating for them drastically
changes relative merits.
Note: MIRR helps here (discussed later).
Copyright © 2008 Lumina Decision Systems, Inc.
IRR and Uncertainty
Expected Rate of Return
• E[IRR] is non-sense!
Is usually does not exist. (e.g., non-zero probability
that project never turns profit).
Will usually be NaN in a Monte-Carlo simulation
When it does exist, its results don’t make sense.
Median may exist when Mean doesn’t, but is also nonsense.
• Exercise 14: A $1000 investment returns $1100
with 50% or $900 with 50% after T years.
Compute the EIRR for T=[0.5, 1, 2, 4] years.
Note: The expected rate of return is “obviously” 0%
in all four cases, right? What is the EIRR?
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Expected Rate of Return
• “Expected rate of return” more
accurately describes the discount rate
where the ENPV-curve crosses zero.
This is not the same as E[Irr(v,T)]!
It is equal to Irr(E[v],T)
• Technically called “IRR of the Expected
Cash Flow”.
• Behaves more intuitively as expected.
Still suffers from same drawbacks as IRR
with certain cash flows, of course.
Copyright © 2008 Lumina Decision Systems, Inc.
Exercise: Expected Rate of Return
A proposed product (from Exercise 8):
• Requires Poisson(9) months to develop,
at LogNormal(µ:10K,σ:3K) per month.
• Will launch if successful, P(success)=70%
• After launch, monthly earnings:
Earningsm~Normal(10K,8K)
• Exercise 15:
Compute Irr of expected cash flow.
Compare to graph computed during NPV section.
Attempt to compute E[Irr]. Does it work?
Copyright © 2008 Lumina Decision Systems, Inc.
Modified IRR (MIRR)
• MIRR is a variation of NPV that
expressed as a rate-of-return.
Robustness of NPV
Intuitive appeal of IRR
Avoids IRR distortions
• Requires:
reinvestRate (rrate): Interest rate we can
expect to receive on positive flows.
financeRate (frate): Interest rate we
expect to pay to finance negative flows.
Copyright © 2008 Lumina Decision Systems, Inc.
MIRR details
Idea:
(index T := 0..n)
• All positive flows are re-invested at rrate
until project end (Time=Tn).
PV equivalent to all gains arriving on Tn.
gain := Npv(rrate,posFlows,T) * (1+rrate)n+1
• All negative flows financed at frate is
equivalent to all expenses occurring at T0
(index T:=0..n)
cost := Npv(frate,negFlows,T)*(1+frate)
• MIRR = (gain/-cost)1/n-1
Copyright © 2008 Lumina Decision Systems, Inc.
MIRR Exercises
Exercise 16:
• Create a User-Defined Function to compute MIRR.
Exercise 17:
• Compute the MIRR for these cash flows from the earlier “IRR can
mislead” exercise using rrate=frate=6%.
Year
0
1
2
3
4
5
6
Proj 1
-200
200
200
200
200
200
-900
Proj 2
-200
195
195
195
195
195
-900
Copyright © 2008 Lumina Decision Systems, Inc.
Summary
• It matters when a cash flow occurs. “Time value of money”.
• Discount_rate quantifies our own time-value of money.
Inflation, risk-free investment, other opportunities, remaining
lifespan, financing rate, etc.
• NPV captures the magnitude of a return.
• IRR and MIRR capture the efficiency (quality) of an investment.
• IRR is heavily used in practice, but often highly misleading. Its
upside is that it does not require any discount rate estimates.
• The Monte-Carlo simulation of IRR is nonsense. IRR is poorly
suited for uncertain cash flows.
• NPV and MIRR are robust and meaningful when used with
Monte-Carlo simulation.
Copyright © 2008 Lumina Decision Systems, Inc.