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Capital Budgeting Notes

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This is an extract of our Capital Budgeting document, which we sell as part of our Banking Law Notes collection written by the top tier of King's College London students.

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Capital Budgeting
● Methods to decide which projects to invest in and which to reject:
○ The Net Present Value Rules (which is the topic for today);
○ The Internal Rate of Return; and
○ Others (See I Welch, Corporate Finance, Chapter 4).
Net Present Value
● The present value of all future cash flows of a project minus the present value of its costs.
○ I.e. The sum of the present value of all future positive and negative cash flows.

○ NPV = F0 + F1/(1 + r0,1) + F2/( 1+ r1,2) + … + FT/(1 + rT-1,T)
● Example:
You can buy a project today for 100, next year it will generate a return of 20, the following year of 50, and in year three, when the project comes to an end, of 75. The constant interest rate is 10%. What is the NPV?
○ NPV = -100 + 20/(1 + 0.1) + 50/(1 + 0.1)2 + 75/(1 + 0.1)3
● The NPV Rule:
○ Accept only projects with NPV > 0.
○ Accepting projects with a positive NPV increases firm value.
○ Rejecting projecting with a negative NPV that will decrease firm value.
○ Present (market) value of the future cash flow - cost = the profit (or loss)
form the project.
○ Positive NPV means "free" money.
● Application:
As the CFO of your company you are faced with the following investment options
(with the cost of capital being constant at 8%):
○ Exploitation of a mine:
■ Cost today = 200;
■ Return in year 1 = 0;
■ Return in year 2 = 100
■ Return in year 3 = 300
■ Clean up costs in year 4 = 100.
○ Investment in a 4-year note with annual coupons:
■ Price today = 200;
■ Coupon in year 1 = 20
■ Coupon in year 2 = 20
■ Coupon in year 3 = 20
■ Coupon + Principal in year 4 = 220
○ Which project should we invest in?
■ NPVmine = -200 + 0 + 100/1.082 + 300/1.083 - 100/1.084 = 50.38 ■ NPVnote = -200 + 20/1.08 + 20/1.082 + 20/1.083 + 220/1.084 = 13.25
■ Conclusion = Invest in both if possible, but, if not enough cash, then only in the former, because it has a greater NPV.
○ But what if the interest rate is 16% instead of 8%? The bond is no longer feasible.
■ NPVmine = 11.28
■ NPVnote = -33.56
The Internal Rate of Return
● IRR = the rate-of-return-like number of NPV = 0.

○ The Internal Rate of Return:
0 = F0 + F1/(1 + r) + F2/(1 + r)2 + … + FT/(1 + r)T
○ Solve for r. But this is not possible to do by hand.
○ Invest if IRR > Required rate of return (that is the discount rate or the opportunity cost).
■ E.g. In the previous example, the required rate of return is the cost of capital of 8%.
● Advantages:
○ Single number that is easy to understand; and
○ All you need to know is the cash flows emanating from the project.
● Disadvantages:
○ There can sometimes be multiple IRRs;
■ When this is the case, the IRR method is then no longer accurate.
○ The IRR is not defined sometimes;
○ Comparison problems as it does not adjust for project scale.
● Overall, NPV is the most reliable.
Valuing Risk
● The NPV formula is easy.
○ But, in the presence of uncertainty, the inputs become difficult.
○ Future cash flows become "expected" cash flows, [E(Ft)].
○ Rate of return becomes "expected risk adjusted rate" [E(rt)]. It is this concept of credit risk or risk of default that is being reflected in the expected cash flows and expected risk adjusted rate.
● Expected value:
The probability-weighted average outcome of a random variable.
● Example:
A bond for 200 promises to pay 210 the next year. The issuer's business is risky. The bondholder may receive only a fraction of the promised cash flow at the end of the year. The probability distribution of payoffs (our random variable) is as follows: Payoff

Probability

210 50%

150 10%

90 10%

50 10%

20 10%

0 10%

What is the expected future value of the promised cash flow?
○ E(F) = p1F1 + p2F2 + p3F3 + … pnFn = n∑j=1pjFj
○ E(F) = (0.5 x 210) + (0.1 x 150) + (0.1 x 90) + (0.1 x 50) + (0.1 x 20) + (0.1 x 0) =

136 ● Variance and Standard Deviation:

○ Variance = The expected value of the squared deviations from the mean.
Payoff

Probability

Deviation (from the expected future value of 136)

210 50%

74 150

10%

14 90

10%

-46

50 10%

-86

20 10%

-116

0 10%

-136

○ Var = (0.5 x 742) + (0.1 x 142) + (0.1 x (-46)2) + (0.1 x (-86)2) + (0.1 x (-116)2) +
(0.1 x )-136)2) = 6,904

○ Standard Deviation = Square root of variance; that is, the expected squared deviation from the mean.
■ SD = √var = √6,904 = 83.01
○ We can expect to receive 136 plus/minus 83.01 on this bond. This is very risky. The bond promises a rate of return of 5%. What is the expected rate of return?
■ E(r) = (E(F)-C)/C = (136 - 200)/200 = -0.32

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