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## Capital Budgeting Notes

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.

The following is a more accessble plain text extract of the PDF sample above, taken from our Banking Law Notes. Due to the challenges of extracting text from PDFs, it will have odd formatting:

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%.
○ Single number that is easy to understand; and
○ All you need to know is the cash flows emanating from the project.
○ 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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