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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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