60 36 56.31 55.75 68.37 69.26 68.46 67.78 67.2 103.81 39.01 55.75 55.37 69.26 68.46 67.78 67.7 103.81 72.646 71.358 70.25 70.412 69.612 68.932 68.352 200.96
The unrecovered balance at any time t during the life of the project is given by the expression
1
0 1
( ) (1 ) (1 ) ...
Cash flow at time ; to the project.
t t
t t
t
F r CF r CF r C
CF t
r IRR
= + + + − + +
=
=
F
On the basis of the form of the unrecovered stream, we can define the following type of investments:
(i) Pure investment A pure investment is one for which the unrecovered investment balance is either zero or negative throughout the life of the project, i.e.,
1
0 1
1
0 1
( ) (1 ) (1 ) ... 0; 1, 2, 2..., 1 and ( ) (1 ) (1 ) ... = 0;
is the life of the project.
t t
t t
n n
n n
F r CF r CF r CF t n
F r CF r CF r CF
n
−
−
= + + + + + ≤ = −
= + + + + +
A pure investment is one from which the firm does not borrow at any time during the life of the investment but recovers fully its investment at the end of the project. Hence the internal rate of return in case of a pure investment is the return earned on the funds invested in the project by the firm.
(ii) Mixed investment A mixed investment is the investment for which F rt( ) is greater than zero for some t and less than or equal to zero for some other t. Thus a mixed investment contains unrecovered investment balance ( ) and over-recovered investment balance ( ). In other words, there are periods when the firm lends to the project and there are periods when the firm borrows from the project.
( ) < 0
F rt F rt( ) > 0
Two-fold classification On the basis of the above types of investments, we can classify investments as follows.
(i) Simple investments (Simple investments are always pure investments);
(ii) Non-simple, pure investments; and (iii) Non-simple, mixed investments.
Now, we shall study the relevance of IRR for such investments.
Consider the following information relating to the four projects viz. A, B, C, and D:
Table 10.28
Cash flows (Rs.) Year
A B C D
0 -5,000 -2,000 -50 -1,600
1 1,000 600 200 10,000
2 1,000 500 -200 -10,000
3 1,000 40 - -
4 1,833 -378 - -
5 5,000 1,800 - -
IRR (%) 20 12 100 25, 400
We, now, calculate the unrecovered investment balance for each of these IRR
Table 10.29
Unrecovered investment balance (Rs.)
D Year
A B C
r = 25% r = 400%
0 -5000 -2000 -50 -1600 -1600
1 -5000 -1640 100 8000 2000
2 -5000 -1337 0 0 0
3 -5000 -1097 - - -
4 -4167 -1607 - - -
5 0 0 - - -
Type of investment Simple Non-simple, pure Mixed Mixed Mixed
If we interpret the above facts, we see that throughput its life, project A is a net borrower although the investment is fully recovered at the end. The IRR in this case is internal to the project. Again, project B is a non-simple, pure investment and IRR is internal to the project.
For project C, the unrecovered balance is negative in the beginning, positive at the end of first year and zero at the end of the project. This implies that the firm has been a lender to the project from period 0
to 1 and a borrower from the project from period 1 to 2. The investment is a mixed investment and it is both a user of and a source to the funds of the firm. In such cases, there is a problem of interpretation of the IRR. If the rate of return imputed to the funds borrowed from the project is same as the rate of interest on the funds lent to the project, then the rate of return can be viewed as internal to the project.
This is not so in general and in general, the rate of interest on the funds lent to the project is different form the rate of return imputed to the funds borrowed from the project. Thus in case of mixed investment the rate of return cannot be considered as internal to the project.
For project D, besides interpretation there is problem of multiple rates of return. Then IRR is not very meaningful for such projects.
To sum up, IRR is meaningful for pure investments but not for mixed investments.
Decision-making in case of mixed investments Mixed investments are to be analyzed differently as the usual computation of IRR in these cases is not very meaningful. Then we need to have some criterion for decision making in case of mixed investments. First of all, we see how to identify a mixed investment.
Test for pure and mixed investments To test whether an investment is a pure or mixed, we compute imin, the minimum interest rate that makes investment balance F rt( ) ≤ 0; t = 0,1, 2...n−1. Such a value always exists since F r0( ) < 0.
Since F it
( )
min ≤ 0 ∀ =t 0,1, 2...n−1( )
minF it 0 i i and t 0,1, 2... 1
⇒ ≤ ∀ > = n− (As i increases, Ft decreases)
Then the test procedure is as follows
(i) Find the value of iminfor the investment.
(ii) Find i* which satisfy F in( *) = 0
(iii) Compare i* with imin. If i* >imin, the investment is pure, otherwise it is a mixed investment.
Criterion for mixed investment- Return of invested capital In case of a mixed investment;
the firm commits funds to the project part of the time and borrows funds from the project rest of the
time. Define r* as the return on the invested capital and k as the cost of the funds borrowed from the project.
When the unrecovered balance from the project is negative (i.e. the firm has funds committed to the project) it is compounded at r*; When the unrecovered balance from the project is positive (i.e. the firm has overdrawn funds from the project) it is compounded at k. Thus the unrecovered balance is a function of both r* and k. Let we denote this by F r kt
( )
*, . As the life of the project is n years, so( )
*,F r kn =0 is the condition for realizing a return r*on invested capital. To obtain r*, we proceed as follows.
(i) Determine whether the investment is a pure investment or a mixed investment. For a pure investment r* = k.
(ii) For a mixed investment, calculate F r kt
(
*,)
as follows:( )
( ) ( )
( )
( ) ( )
( )
( ) ( )
*
0 0
* *
1 0 1 0
0 1 0
* *
1 1
1 1
* *
1 1
,
, 1 if 0
1 if 0
, 1 if 0
1 if 0
, 1 if 0
t t t t
t t t
n n n n
F r k CF
F r k F r CF F
F k CF F
F r k F r CF F
F k CF F
F r k F r CF F
− −
− −
− −
=
= + + <
= + + >
= + + <
= + + >
= + + <
M
M
( )
1 1
= Fn− 1+k +CFn if Fn− > 0
(iii) Obtain r* by solving F r kn
( )
*, =0 subject to 0 ≤ r* ≤ imin= For project D
(i) r = 25%, 400% and imin 525%. So the project is a mixed project.
(ii)
( )
( ) ( )
*
0 0
* *
1 0
*
, -1600
, 1600 1 10, 000 ( 0) 8400 1600
F r k CF
F r k r F
r
= =
= − + + <
= − −
Since r* = 5.25 ( imin) ⇒1600r* < 8400
i.e., F r k1
( )
*, ≥ 0and it is compounded at k in the next step.( ) (
* *) ( )
2 , 8400 1600 1 10000
F r k = − r +k −
(iii)
( )
( ) ( )
* 2
*
*
, 0
8400 1600 1 10000 0 5.25 6.25
1 F r k
r k
r k
=
⇒ − + − =
⇒ = −
+
r* can be obtained for different values of k. Following points can be observed:
(i) r*, the return on the invested capital increases with k, the imputed cost of surplus funds;
(ii) r* →iminasymptotically; and (iii) If k = r*, then k = r* = i*.
5.25 6.25 1
(1 ) 5.25(1 ) 6.25 0.25, 4.00
k k
k k k
k
= −
+
⇒ + = + −
⇔ =