Investment Decision Rules (Chapter 5)

ALTERNATIVE DECISION RULES

·        NPV RULE (correct, maximizes s/h wealth)

·        BOOK RATE OF RETURN (arbitrary)

·        PAYBACK PERIOD (arbitrary)

·        IRR (correct, maximizes s/h wealth)

·        PROFITABILITY INDEX (used only in one situation; correct, maximizes s/h wealth)

We’ll return to the NPV rule in Chapter 6.  For now we focus on the other two correct criteria: IRR and PROFITABILITY INDEX

IRR

·        Definition: project’s expected return (i.e. average E(R) per year over life of project)

·        Computationally, IRR is the discount rate which makes PV of inflows equals the PV of outflows (or the discount rate at which NPV is 0)

·        Graphically:

·        IRR rule for independent projects: accept if IRR > required return

o       Clearly this rule maximizes s/h wealth if NPV curve is downward-sloping

o       When will NPV curve have this shape?  When we have CONVENTIONAL CASHFLOWS:

o      First non-zero cashflow is negative

o      Sign of cashflows changes only once

o      Examples:

  C₀      C₁      C₂      C₃           

                        -100     30      30      30        => conventional

                        -100    -50      90      90        => conventional

                             0    -50      40      40        => conventional

                        -100     80      80     -30        => non-conventional

                           20    -50      30      30        => non-conventional

                         100    -30     -30     -30        => non-conventional

A project will rarely have non-conventional cashflows.  But we also have to work with the difference between the cashflows of two  projects.  We can easily get non-conventional cashflows then.

o      As long as you have conventional cashflows, the NPV curve is downward sloping.  If IRR > required return, then the NPV at the required return is positive.  So the IRR rule guarantees that we pick positive NPV projects.

·        IRR rule for mutually exclusive projects

o       If we have two mutually exclusive projects, we have to pick the better of the two.  The project with the higher IRR is not necessarily better (does not necessarily have higher NPV).  We can see that this will happen if the NPV curves for the two projects intersect:

Which project is better depends on the required return (whether it is higher or lower than the intersection point).

o      Even though we can’t just pick the project with the higher IRR, we can still use IRR to figure out which project to pick. 

·       Take the incremental CFs (e.g. CF of project B – CFs of project A), and compute their IRR

·       Assume for now the incremental CFs are conventional CFs. 

·       Compare the IRR of the incremental CF to the required return

·       If the IRR > required return, this means the NPV of the incremental CFs is positive. 

·       [NPV of (B – A)] > 0 =>  (NPV of B) – (NPV of A) > 0 => (NPV of B) > (NPV of A)

·       Hence, if you start with B – A, then pick B over A whenever IRR of the incremental CFs > required return.

·       If IRR of the incremental CFs < required return, pick A over B

·       Picking B over A means we get the CFs of A plus we get the incremental CFs.  “Should we pick B over A?” boils down to “are the incremental CFs worthwhile, do they have positive NPV?”  This is what we figure out by comparing the IRR of the incremental CF to the required return.

Example: We need to choose between these two mutually exclusive projects using IRR alone.

            0               1               2               3

Project A            -1,000         505           505           505

Project B          -11,000      5,000        5,000        5,000

Required return 10%

IRR_A = 24%, IRR_B = 17%

·       A has the higher IRR but it is not necessarily the better project. 

·       If we cheat by computing the NPVs at a 10% discount rate, we see that B indeed has the higher NPV.  (NPV_A = $256, NPV_B = 1,435) 

·       Why does this happen?  A gives you a higher E(R) on a much smaller investment, so the total dollar NPV you get is small; B gives you a smaller percentage return on a much bigger dollar investment, so the total dollar NPV you get is large. 

·       So we see that with mutually exclusive projects, you cannot just pick the project with higher IRR

We compute IRR of the incremental cashflows B - A

        0                  1                   2                   3

B - A      -10,000           4,495            4,495            4,495

IRR turns out to be 16.6% => IRR > required return, so we choose B

Graphical interpretation:

·       IRR of the incremental cashflows = discount rate at which NPV of the incremental cashflows is zero. 

·       In other words, it’s the discount rate at which (NPV of B) = (NPV of A), i.e. the rate at which the NPV curves for the two projects intersect.

·        Other issues:

o       Borrowing or lending

Suppose we had computed incremental cashflows for A – B. 

 0                 1               2                3

A - B      10,000         -4,495       -4,495        -4,495     

What IRR do we get now?  What project dod we choose?

o       Why do we now end up picking the wrong project?

o       These incremental CFs are not conventional CFs

o       Conventional CFs correspond to lending – first cash goes out and later cash comes back in.  The IRR is the E(R) you are earning; when we lend we want higher returns.  That’s why it’s good when IRR is greater than the required return.

o       The CFs we have here (A – B) are the exact mirror image of conventional CFs.  They correspond to borrowing rather than lending.  First cash comes in and later it goes out.  The IRR is now the E(R) you borrow at.  We want to borrow at as low a return as possible, so you’re better off when IRR is less than the required return.

o       In other words, when the CFs reverse from lending to borrowing, we also have to reverse our rule.  So we pick A over B if IRR < required return.  Since here IRR > required return, we will make the same choice as we did before: pick B over A.                 

o       Multiple IRRs: 

With non-conventional cashflows, you can sometimes get multiple IRRs.  It gets a little more complicated, but we can still make the investment decision using IRR alone.

Suppose we get these incremental cashflows when comparing two mutually exclusive projects (which have required return = 10%)

0         1           2           

C - D        50      -145      105         

Solving for IRR, we would get 40% and 50%

o       Both IRRs are much greater than required return.  Should we just “accept” C – D, i.e. pick C over D?

o       We need to ask: what does it mean for a project to have two IRRs?

o       Graphical interpretation: there are two possibilities for the NPV curve

o       We need to figure out which of these curves actually describes C-D.  All we need to know is whether the y-intercept for the NPV curve is positive or negative.  And the y-intercept = NPV at a zero discount rate =  the sum of the cashflows

o       Here, the sum of the cashflows is positive.  This means the NPV curve starts with a positive y-intercept. 

o       Now, from the graph we can see that C – D has a positive NPV at a discount rate of 10% => pick C over D.                             

o       IRR doesn’t exist: With non-conventional CFs, sometimes there is no IRR.  However, we can still make the investment decision using IRR

0              1               2           

E - F       1,000        3,000      -2,500     

·  Solving for IRR, we would get -32% and -468%

·  Negative discount rates are meaningless => no meaningful discount rate.

·  What does this mean?

·  Graphical interpretation:

=> If the y-intercept is positive, NPV is positive at all discount rates; if the y-intercept is negative, NPV is negative at all discount rates;

·  Here, the sum of the CFs is positive => positive y-intercept => pick E over F                                   

o       Non-flat term structure:

·  The yield curve is not flat or horizontal.  This means the 1-year rate is different from the 2-year rate which is different from the 3-year rate, etc

·  Each cashflow has its own required return; we don’t have one single required return for the project.

·  This is the only time the IRR rule totally breaks down.  We can compute IRR but we no longer have a number to compare it to.

·  Whenever we use IRR, we are implicitly assuming a flat yield curve, and therefore a constant required return.

E           5                 7.5             2.5           0.5

      ·        If you pick the projects with highest NPV till budget is used up, pick A and B for total NPV of $17

·        If you pick the projects with highest P.I. till budget is used up, pick C, E, B and D for total NPV of $21.5

·        Why do you fare better with P.I.?  It gives you the biggest bang for your buck.  Highest P.I. means the highest NPV per dollar invested.  Each dollar is invested at the highest available NPV per dollar.  Maximizing NPV per dollar invested maximizes total NPV for the $50 you invest.

·        There’s one caveat.  Picking projects in order of P.I. is guaranteed to give you the best combination, only if you exactly use up your budget.  Suppose the budget here was $65. 

o       Picking in order of P.I. you would still pick C, E, B and D.  You've spent $50 and picked up a total NPV of $21.5.  The only project left is A, which you can’t take because you have $15 left and A requires an investment of $30.

o       For the $50 you spent you got the maximum possible NPV; but you also wasted $15 (for which you got the minimum possible NPV, zero)

o       In such a situation, it is possible that some other combination, which uses up more of the budget, gives you a higher total NPV than $21.5.

o       By eyeballing different combinations, we can see that picking A, B, C and E, you exactly use up your $65 budget, and you pick up a total NPV of $25.5

·        Bottom line:

o       Try picking in order of P.I.

o       If you exactly use up your budget, that’s the best combination.

o       If not, check and see if some other combination gives a higher total NPV

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