Financial Markets
• Laurent Calvet calvet@hec.fr
• John Lewis
john.lewis04@imperial.ac.uk
Topic 4:
HEC – MBA Financial Markets
The Internal Rate of Return
4-1
IRR Definition
• IRR = The discounting rate that makes the Net
Present Value (NPV) equal to zero
• IRR is also called the Yield to Maturity
• YTM used for securities
• IRR used for capital expenditures, Venture Capital
and Private Equity.
HEC – MBA Financial Markets
4-2
IRR
• IRR is calculated using the same formula as the NPV
except that the rate r is unknown:
N
Fn
=V 0
∑
n
n =1 (1 + r )
Where V0 is the initial investment in the project
• The IRR is the rate r where the market value is equal
to the present value of the future cash flows from the
investment.
HEC – MBA Financial Markets
4-3
Computing IRR
• The solution to the IRR formula is found through trial
and error
• Use a calculator or spreadsheet
• Timing of cash flows is very important to calculation
• Accurate models very carefully model the cash flows
• Implementation shortfall arrives in an optimistic
model of cash flows and the reality of actual
magnitude and timing
HEC – MBA Financial Markets
4-4
IRR as an Investment Rule
• Many companies have a target return on
investment – either through policy or historical
results.
• In the NPV example, a project was started if the
NPV was positive.
• In the IRR case, the project is not started unless
the return (r) of the project is above a certain
threshold.
HEC – MBA Financial Markets
4-5
Limitations of IRR and NPV
• NPV and IRR can give opposing views of a project
• Competition among investors and mechanisms of arbitrage
theory move NVP towards zero
• Therefore exceptional rates of return move towards the
required rate of return
• Can another investment with IRR = 15.89 be found to replace
investment A in year 4?
Year
Investment A
Investment B
Market Value Investment
10
-10
12.25
-12.25
Required Rate
HEC – MBA Financial Markets
1
9
5
2
3
5
3
0.5
0
4
5
6
7
0
2.1
0
5.1
NPV
IRR
$1.29 15.89%
$2.32 11.05%
0.05
4-6
Which Investment A or B
Year
Investment A
Investment B
Market Value Investment
10
-10
12.25
-12.25
Required Rate
1
9
5
2
3
5
3
0.5
0
4
5
6
7
0
2.1
0
5.1
NPV
IRR
$1.29 15.89%
$2.32 11.05%
0.05
• Investment A has the lower NPV and higher IRR
ƒ Capital is returned at the end of 3 years
ƒ Years 4+ most likely to be reinvested at Required Rate of 5%
• IRR assumes that the return in year 4+ will be the
same as from years 1 to 3. Will it?
HEC – MBA Financial Markets
4-7
Modified IRR - MIRR
• Assumes cash flows are reinvested at the required rate of
return
• Calculate the terminal value of project by compounding all
cash flows at required rate
• Find the IRR with initial cash outlay at beginning of the
project and the terminal value of project
Year
Investment A
Investment B
Market Value Investment
10
-10
12.25
-12.25
Discount Rate
Investment A
Investment B
HEC – MBA Financial Markets
1
9
5
2
3
5
3
0.5
0
4
5
6
7
0
2.1
0
5.1
NPV
IRR
$1.29 15.89%
$2.32 11.05%
0 16.5
0 20.5
M-NPV MIRR
16.4975
7.41%
20.4971
7.63%
0.05
-10
-12.25
0
0
0
0
0
0
0
0
0
0
4-8
Modified IRR
• Reconciles NPV and IRR models
• Realistic view that follow-on investments will not
have abnormally high returns
• The value computed by MIRR matches real
outcomes
• If the IRR returns are not possible to replicate –
might it be better to return the cash flows to the
shareholders? Question for Corporate Finance
Class.
HEC – MBA Financial Markets
4-9
Special Cases for IRR
• No IRR - NPV remains positive for all discount rates.
• Multiple IRRs
ƒ Due to initial cash inflows and later high costs
ƒ Unsure which IRR is the correct IRR
• Investments that may have initial inward cash flows before
expenses must be paid.
ƒ This is effectively a loan
ƒ Subscriptions before materials are produced
• IRR does not have a time varying rate of return
• You know you have one of these when Excel returns #NUM
HEC – MBA Financial Markets
4 - 10
Mutually Exclusive Projects
• Must select from a list of promising projects
• Cannot select Project A and B – they are mutually
exclusive
• Inability to select both is often from Capital
Rationing
ƒ Hard rationing – cannot obtain funds at market rate
• Market imperfections, transaction costs, agency costs…
ƒ Soft Rationing – Internally imposed financial constraints
• Limits on borrowing and unable to raise equity cash
• Steady growth strategy
• Annual capital limits within divisions
HEC – MBA Financial Markets
4 - 11
Mutually Excl. - Scale
• Another form of the NPV and IRR disagreement
problem.
• A large scale project has a high NPV and lower
IRR while a small scale project has a low NPV
with a higher IRR.
• The large scale, high NPV, project creates more
wealth for shareholders
• From a straight IRR view, the small scale project
is a better use of funds.
HEC – MBA Financial Markets
4 - 12
Incremental IRR
• To select between a large scale high NPV and a small
scale high IRR project, create an incremental project that
has the cash flows equal to the difference of the cash
flows for the two projects.
• Decision rule: If incremental IRR is higher than the
required rate then select the large scale project. If
incremental IRR is lower than required rate then select
small scale project.
• The incremental project is a combination of the two
projects. Both projects have a positive NPV and IRR
above the threshold. If we accept the large scale project it
is the same as accepting the cash flows from the
incremental project and the small scale project. The
decision rule is on the incremental project.
HEC – MBA Financial Markets
4 - 13
Incremental IRR - example
• Required rate = 10%, IRR threshold = 15%
• Order of creating incremental project is important
ƒ subtract cash flow so initial investment remains negative
ƒ Apply decision rule – select larger project if incremental IRR is
above the threshold (or use NPV Rule)
Project
Small scale
Large scale
Incremental
HEC – MBA Financial Markets
F0
F1
-10
-100
-90
IRR
NPV(10%)
15.00
50%
$3.64
120.00
20%
$9.09
105.00
17%
$5.45
4 - 14
Timing Problem
• Projects with the same initial outflows but different
inflow characteristics.
• In straight IRR terms, the project with near term
inflows will have a better IRR than a project with
later, and larger, inflows
• Must use the NPV to decide between projects.
Decision depends on the discount rate.
• Reliance on just IRR leads to ‘short-term’ optimal
results. A problem in public companies that are
very conscious of quarterly numbers.
HEC – MBA Financial Markets
4 - 15
Timing Problem - example
• Two projects with identical outflows and different
inflows.
• Decision between projects depends on discount rate
used to calculate NPV.
ƒ Notice change in NPV at different discount rate
ƒ Which project would you choose and why?
ƒ At high discount rate – money is more valuable today
Project A
Project B
HEC – MBA Financial Markets
0
-10000
-10000
1
8000
0
2
3 IRR NPV (5%)
3000 1000 14% $1,203.97
2000 11500 11% $1,748.19
NPV(12%)
$503.38
$293.01
4 - 16