Investment Decision Rules (capital Budgeting Techniques) - Example 31491l

Investment Decision Rules (Capital Budgeting Techniques) The initial investment (I0), the net operating cash inflow (CFi, i = year 1, 2, …, n) and the terminal cash flow (CFn) must be carefully analysed in order to establish whether an investment is to be accepted or not. The capital budgeting techniques represent an important criteria used by a company to identify the most efficient investment projects among all those available. By applying such techniques and their decision rules the firm will be able to choose the most relevant investments that increase its market value. Some of the techniques used are simple or unsophisticated because they do not use the time value procedures, while others are more complex or sophisticated as the CFs projected for the future are to be brought at the present moment. A. The Unsophisticated Capital Budgeting Techniques 1. The Average Rate of Return (ARR) – establishes the most likely rate of return of the investment taking into the average profits after taxes or the average net CF and the average investment. The decision rule: ARR as highest as possible 2. The Payback Period (PP) – represents the number of years, months and days required to recover the initial investment from the cash inflows. 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑙𝑒𝑓𝑡 𝑡𝑜 𝑏𝑒 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 𝑎𝑓𝑡𝑒𝑟 𝑦𝑒𝑎𝑟"𝑡" 𝑥 12, 𝐶𝐹𝑡+1 where: t – number of full (whole) years needed to recover the initial investment from the annuity 𝑃𝑃 = 𝑡 +

(equal CFs) or from the mixed stream (different yearly CFs). The decision rule: a shorter PP is preferred B. The Sophisticated Capital Budgeting Techniques These techniques give explicit consideration to the time value of money because the CFs expected to be received every year need to be brought at the present moment (discounted) using a specified rate. The rate used to discount CF is also referred to as the firm’s cost of capital or opportunity cost. 1. The Net Present Value (NPV) – is established as the difference between the present value of n CFi the investment’s benefits (  ) and the present value of its costs (I0). i i 1 1  r  n CFi NPV =   I0 i i 1 1  r  The decision rule: if NPV > 0, accept the project; otherwise (NPV < 0), reject the project. When making an investment decision for several projects available, take the alternative with the highest NPV.

Example: In the table below you can see the relevant cash-flows for a 4 year investment: Year 1

Year 2

Year 3

Year 4

Initial investment = $70,000 CFi 25,000 29,000 34,000 27,000 Terminal CF 3,000 Calculate the NPV for this investment, if the firm’s cost of capital, applicable to the proposed investment is 18%. a. $8180.91 4

𝑁𝑃𝑉 = ∑ 𝑖=1

b. $7340.87

c. $8260.33

d. $7955.78

𝐶𝐹𝑖 25000 29000 34000 27000 3000 − 𝐼 = + + + + − 0 (1 + 𝑟)𝑖 1 + 0.18 (1 + 0.18)2 (1 + 0.18)3 (1 + 0.18)4 (1 + 0.18)4

−70000 = 21186.44 + 20827.35 + 20693.45 + 13926.30 + 1547.37 − 70000 = = 78180.91 − 70000 = $8180.91 2. The Payback Period using the present value of CFs (PPPV) - represents the number of years, months and days required to recover the initial investment from the cash inflows brought at the present moment. 𝐼𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑙𝑒𝑓𝑡 𝑡𝑜 𝑏𝑒 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 𝑎𝑓𝑡𝑒𝑟 𝑦𝑒𝑎𝑟"𝑡" 𝑥 12, 𝐶𝐹𝑃𝑉𝑡+1 where: t – number of full (whole) years needed to recover the initial investment from the annuity (equal 𝑃𝑃𝑃𝑉 = 𝑡 +

CFs) or from the mixed stream (different yearly CFs). The decision rule: a shorter PPPV is preferred Example: For the investment presented before calculate the replacement period I (recovery period or payback period) and the replacement period using the present value of CFs II I.

𝑃𝑃 = 2 +

a. 2 years 4 months 3 days

b. 2 years 7 months 14 days

c. 2 years 2 months 23 days

d. 2 years 5 months 19 days

70000 − (25000 + 29000) 𝑥 12 = 2 𝑦𝑒𝑎𝑟𝑠 5 𝑚𝑜𝑛𝑡ℎ𝑠 19 𝑑𝑎𝑦𝑠 34000 II.

𝑃𝑃𝑃𝑉 = 3 +

a. 3 years 6 months 9 days

b. 2 years 2 months 23 days

c. 3 years 5 months 20 days

d. 2 years 5 months 19 days

70000 − (21186.44 + 20827.35 + 20693.45) 𝑥12 = 3 𝑦𝑒𝑎𝑟𝑠 5 𝑚𝑜𝑛𝑡ℎ𝑠 20 𝑑𝑎𝑦𝑠 13926.30 + 1547.37

3. The Internal Rate of Return (IRR) – represents the discount rate that equals the present value n CFi of cash inflows with the initial investment associated with the project (   I 0 ) or the i i 1 1  r  n CFi discount rate that equals NPV of an investment opportunity with zero (   I 0  0) . i i 1 1  r  Mathematically, IRR is found by solving the following equation: NPV = 0 

n

CFi

 1  IRR  i 1

i

 I0 = 0

The decision rule: if IRR > cost of capital (r), accept the project; otherwise (IRR < r), reject the project. For a project to be acceptable, the IRR must exceed the firm’s cost of capital or opportunity cost. This guarantees that the firm is earning at least its required return and assures that the market value of the firm will increase. The figure below plots the NPV as a function of the discount rate. The curve crosses the horizontal axes at the IRR, because this is where NPV equals zero. NPV (USD, EUR, RON)

0

Discount rate (%) IRR

In most cases, the final decision is easy to be taken according to NPV and IRR (which are the most important criteria of selection) because they offer the same conclusion. Still, there are cases that are described by the following situations: a) nonexistent IRR: no real solution b) multiple IRR: several solutions. In such cases, we cannot apply the IRR rule of decision and the company will rely only on the NPV rule. Example: A company has two alternative investment projects available which are mutually exclusive. The initial investment and the annual CFs are presented in the table below: Year Investment A Investment B

0 - $80,000 - $130,000

1 $25,000 $40,000

2 $29,000 $47,000

3 $34,000 $52,000

4 $35,000 $50,000

The company uses for both investments the following financial resources: -

in proportion of 20% the shareholders’ equity (own capital) with a cost (dividend rate) of 13.75%;

-

in proportion of 55% a bank loan with an interest rate of 11.5%;

-

the rest up to the total amount of money necessary to implement project A or B is represented by a leasing with a cost of 15.24%. The company pays tax on profit and the interest expenditures offer a tax facility (16%). Which one is

the most efficient for the company? Use NPV, PPPV and IRR. r = cost of the capital used to finance the investments = 0.20 x 0.1375 + 0.55 x 0.115 x (1- 0.16) + + (1 – 0.20 – 0.55) x 0.1524 x (1 – 0.16) = 0.1126 => r = 11.26% 𝑁𝑃𝑉𝐴 =

25000 29000 34000 35000 + + + − 80000 = 22469.89 + 23427.17 + 24686.63 + 2 3 1.1126 1.1126 1.1126 1.11264

+ 22840.83 − 80000 = $13424.52 > 0 => project A is efficient for the company and increases the market value by ≈ $13425. 𝑁𝑃𝑉𝐵 =

40000 47000 52000 50000 + + + − 130000 = 35951.82 + 37968.18 + 37756.02 + 1.1126 1.11262 1.11263 1.11264

+ 32629.75 − 130000 = $14305.77 > 0 => project B is efficient for the company and increases the market value by ≈ $14306. According to the NPV technique (rule), project B is preferable to project A (NPVB > NPVA). 𝑃𝑃𝑃𝑉 𝐴 = 3 + 𝑃𝑃𝑃𝑉 𝐵 = 3 +

80000 − (22469.89 + 23427.17 + 24686.63) 𝑥12 = 3 𝑦𝑒𝑎𝑟𝑠 4 𝑚𝑜𝑛𝑡ℎ𝑠 28 𝑑𝑎𝑦𝑠 22840.83

130000 − (35951.82 + 37968.18 + 37756.02) 𝑥12 = 3 𝑦𝑒𝑎𝑟𝑠 6 𝑚𝑜𝑛𝑡ℎ𝑠 22 𝑑𝑎𝑦𝑠 32629.75

According to the PPPV technique (rule), project A is preferable to project B (PPPV A < PPPV B). 25000

29000

34000

35000

NPVA = 0 <=> 1+𝐼𝑅𝑅 + (1+𝐼𝑅𝑅)2 + (1+𝐼𝑅𝑅)3 + (1+𝐼𝑅𝑅)4 − 80000 = 0 1

IRR > 0 => 1 + IRR > 1=> 1+𝐼𝑅𝑅 = 𝑥 < 1 Equation of NPV in x: 25000x + 29000x2 + 34000x3 + 35000x4 – 80000 = 0 / :1000  35x4 + 34x3 + 29x2 + 25x – 80 = 0 For the equation in “x” we should find two consecutive values (with two decimals) for which NPV equation changes sign (goes from “-“ to “+”; this means that somewhere in between we also have value 0). “x” usually takes the two consecutive values at around 0.75 – 0.95, which means that IRR will be around 5% to 35%. For x = 0.8, NPV = 35 x 0.84 + 34 x 0.83 + 29 x 0.82 + 25 x 0.8 – 80 = - $9.696 For x = 0.85, NPV = + $1.35 For x = 0.84, NPV = - $0.96 We found that for x1 = 0.84 and x2 = 0.85, NPV changes the sign from - $0.96 to + $1.35. But x = 1/(1+IRR).

1 1+𝐼𝑅𝑅1 1 1+𝐼𝑅𝑅2

= 0.84 => 𝐼𝑅𝑅1= 0.1905; IRR1 = 19.05%, NPV1 = - 0.96 = 0.85 => 𝐼𝑅𝑅2= 0.1765; IRR2 = 17.65%, NPV2 = + 1.35

The solution for IRR that we are looking for is placed somewhere between 17.65% and 19.05%, closer to 19.05% because NPV for this rate (- 0.96) is closer to 0 than NPV for 17.65% (+ 1.35). The exact value will be calculated from the mathematical equation of the line that goes through both points 1(19.05%, - 0.96) and 2(17.65%, 1.35). 𝐼𝑅𝑅− 𝐼𝑅𝑅2

The equation is: 𝐼𝑅𝑅

1 −𝐼𝑅𝑅2

𝑁𝑃𝑉− 𝑁𝑃𝑉

= 𝑁𝑃𝑉 −𝑁𝑃𝑉2  1

2

IRR−0.1765

0−1.35

= −0.96−1.35 =>𝐼𝑅𝑅 = 0.1905−0.1765

−1.35 𝑥 0.014 −2.31

+ 0.1765 =

= 0.1847; 𝐼𝑅𝑅𝐴 = 18.47%. 40000

47000

52000

50000

NPVB = 0 <=> 1+𝐼𝑅𝑅 + (1+𝐼𝑅𝑅)2 + (1+𝐼𝑅𝑅)3 + (1+𝐼𝑅𝑅)4 − 130000 = 0 1

IRR > 0 => 1 + IRR > 1=> 1+𝐼𝑅𝑅 = 𝑥 < 1 Equation of NPV in x: 40000x + 47000x2 + 52000x3 + 50000x4 – 130000 = 0 / :1000  50x4 + 52x3 + 47x2 + 40x – 130 = 0 For x = 0.8, NPV = 50 x 0.84 + 52 x 0.83 + 47 x 0.82 + 40 x 0.8 – 130 = - $20.816 For x = 0.85, NPV = - $4.01 For x = 0.86, NPV = - $0.41 For x = 0.87, NPV = + $3.26 We found that for x1 = 0.86 and x2 = 0.87, NPV changes the sign from - $0.41 to + $3.26. But x = 1/(1+IRR). 1 1+𝐼𝑅𝑅1 1 1+𝐼𝑅𝑅2

= 0.86 => 𝐼𝑅𝑅1= 0.1628; IRR1 = 16.28%, NPV1 = - 0.41 = 0.87 => 𝐼𝑅𝑅2= 0.1494; IRR2 = 14.94%, NPV2 = + 3.26

The solution for IRR that we are looking for is placed somewhere between 14.94% and 16.28%, closer to 16.28% because NPV for this rate (- 0.41) is closer to 0 than NPV for 14.94% (+ 3.26). The exact value will be calculated from the mathematical equation of the line that goes through both points 1(16.28%, - 0.41) and 2(14.94%, 3.26). 𝐼𝑅𝑅− 𝐼𝑅𝑅2

The equation is: 𝐼𝑅𝑅

1 −𝐼𝑅𝑅2

𝑁𝑃𝑉− 𝑁𝑃𝑉

= 𝑁𝑃𝑉 −𝑁𝑃𝑉2  1

2

IRR−0.1494 0.1628−0.1494

0−3.26

= −0.41−3.26=>𝐼𝑅𝑅 =

−3.26 𝑥 0.0134 −3.67

+ 0.1494 =

= 0.1613; 𝐼𝑅𝑅𝐵 = 16.13%. According to the IRR technique (rule), project A is preferable to project B (IRR A > IRRB). The two main techniques (NPV and IRR) offer different solutions regarding the best project. For such circumstances, we should use the incremental IRR rule. The incremental IRR investment rule applies the IRR rule to the difference between the cash flows of the two mutually exclusive alternatives A and B. Two or more projects are mutually exclusive if the

firm can, at most, accept only one of them. If you subtract the cash flows of opportunity B from the opportunity A, then you should take opportunity A if the incremental IRR exceeds the cost of capital. Otherwise, you should take opportunity B. We will calculate the incremental NPV and IRR. For this we will subtract the cash flows of A from the cash flows of B and then calculate the IRR for these cash flows. The incremental cash flows are: Year ΔCF = CFB - CFA 15000

0 - $50000

18000

18000

1 $15000

2 $18000

3 $18000

4 $15000

15000

𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝑁𝑃𝑉 = 1.1126 + 1.11262 + 1.11263 + 1.11264 − 50000 = 𝑁𝑃𝑉𝐵 − 𝑁𝑃𝑉𝐴 =14305.77 – -13424.52 = $881.25 Equation of incremental NPV in x: 15000x + 18000x2 + 18000x3 + 15000x4 – 50000 = 0 / :1000  15x4 + 18x3 + 18x2 + 15x – 50 = 0 For x = 0.85, NPV = - $5.36 For x = 0.87, NPV = - $2.88 For x = 0.89, NPV = - $0.29 For x = 0.9, NPV = + $1.04 We found that for x1 = 0.89 and x2 = 0.90, NPV changes the sign from - $0.29 to + $1.04. 1 1+𝐼𝑅𝑅1 1 1+𝐼𝑅𝑅2

= 0.89 => 𝐼𝑅𝑅1= 0.1236; IRR1 = 12.36%, NPV1 = - 0.29 = 0.9 => 𝐼𝑅𝑅2 = 0.1111; IRR2 = 11.11%, NPV2 = + 1.04

The solution for IRR that we are looking for is placed somewhere between 11.11% and 12.36%, closer to 12.36% because NPV for this rate (- 0.29) is closer to 0 than NPV for 14.94% (+ 1.04). The exact value will be calculated from the mathematical equation of the line that goes through both points 1(12.36%, - 0.29) and 2(11.11%, 1.04). 𝐼𝑅𝑅− 𝐼𝑅𝑅2

The equation is: 𝐼𝑅𝑅

1 −𝐼𝑅𝑅2

𝑁𝑃𝑉− 𝑁𝑃𝑉

= 𝑁𝑃𝑉 −𝑁𝑃𝑉2 

𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝐼𝑅𝑅 =

1

2

IRR−0.1111 0.1236−0.1111

0−1.04

= −0.29−1.04 =>

−1.04 𝑥 0.0125 + 0.1111 = 0.1209; 𝐼𝑛𝑐𝑟𝑒𝑚𝑒𝑛𝑡𝑎𝑙 𝐼𝑅𝑅 = 12.09%. −1.33

NPV NPVB > NPVA

NPVA > NPVB

B

A

18.47 IRR A

881.25

Discount rate (%)

0 16.13 12.09 Incremental IRR IRR B

A B

As this chart shows, the incremental IRR is 12.09%. In other words, the NPV on the incremental investment is zero when the discount rate is 12.09%. In addition, the NPV of the two projects are equal for this rate: NPVA = NPVB = $881.25. The incremental IRR is the rate that causes the incremental cash flows to have zero NPV. The incremental cash flows have zero NPV when the two original projects A and B have the same NPV. According to the figure presented above, the decision rule is taken as follows: -

if the discount rate (r) < 12.09% project B is more efficient and will be adopted (NPVB > NPVA)

-

if r = 12.09% the company may choose either project A or project B (NPVA = NPVB)

-

if 12.09% < r < 18.47% project A is more efficient and will be implemented (NPVA > NPVB)

-

if r > 18.47% reject both projects (NPVA < 0, NPVB < 0). In this example, r = 11.26% < incremental IRR (12.09%) => project B is eligible and the market

value of the corporation will increase by about $14306, while the internal rate of return brought by this investment alternative is 16.13%.