Capital Budgeting Notes

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 + r_0,1) + F₂/( 1+ r_1,2) + … + F_T/(1 + r_T-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?

    • NPV_mine = -200 + 0 + 100/1.08² + 300/1.08³ - 100/1.08⁴ = 50.38

    • NPV_note = -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.

    • NPV_mine = 11.28

    • NPV_note = -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 + … + F_T/(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(F_t)].

  • Rate of return becomes “expected risk adjusted rate” [E(r_t)]. 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:

What is the expected future value of the promised cash flow?

• E(F) = p₁F₁ + p₂F₂ + p₃F₃ + … p_nF_n = ⁿ_j=1p_jF_j

• 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.

• 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

• Empirical Mean:

  • Expected value requires knowledge of the “true” probability distribution of the future cash flows.

  • This, in practice, means reliance on historical observations.

    • The realisation of cash flows (share prices, dividends, etc.);

    • For K periods;
      This means that, the longer the period, the more accurate the approximation will be.

    • With empirical mean as an approximation of the expected value.

• The NPV formula with credit risk accounted for:
  NPV = F₀ + E(F₁)/(1 + r_0,1) + E(F₂)/( 1+ r_1,2) + … + E(F_T)/(1 + r_T-1,T)

  • But what is the appropriate r (discount rate or cost of capital)?

• Covariance and Linear Regression

  • Covariance: Measures the extent to which two random variables (x and y) move in conjunction with one another.

    • Say, two securities or a security and the market (FTSE 100).

    • Expected value of the product of the difference between each variable’s possible outcome and its expected value.

    • cov(y,x) = (ⁿ_j=1 (y_outcome - y_expected)(x_outcome - x_expected))/n

  • Linear Regression: the line that “best” predicts the realisation of y given the realisation of x.

    • Y = a + Bx

  • Example:
    The following table depicts the rate of return on a particular security and the market portfolio (say FTSE 100).

• Linear Regression:
  The rate of return for a particular security and the market. The straight line is the closest prediction of the rate of return on our security (y) given...

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