#16811 - Capital Budgeting - Banking Law 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 the rate of return on the market (x).
  

• (market) B: measure of asset’s risk contribution in relation to the market portfolio.

  • If B > 0, the security tends to move together with the market.

  • If B < 0, the security and market move in opposite directions.

  • B = cov (r_market, r_security)/var(r_market)

• a: the higher a, the better the overall performance of the investment given any particular rate of return on the market.

  • a = y - Bx

The Capital Asset Pricing Model

• Goals:

  • To estimate an appropriate cost of capital for a particular project; and

  • To plug into the denominator of the NPV formula.

• Assumptions:

  • Investors are risk averse and are interest only in high expected returns and low standard deviations;

  • Investors are diversified and hold various assets that closely mirror the market portfolio; and

  • There is a risk-free asset (such as government bonds).

• CAPM:
  E(r_i) = r_F + [E(r_M) - r_F]*B_i

  • Three inputs:

    • r_F: the risk-free rate of return.

    • E(r_M): the expected rate of return on the overall market.

    • B_i: a project’s market beta.

  • [E(r_M) - r_F]: the equity premium or the market risk premium.

  • Expected rate of return = time premium + risk premium.

    • NOT the default risk (which enters through the NPV numerator).

• The market beta (B_i) of a project:

  • The higher the B_i, the more the asset moves in conjunction with the market (i.e. goes up or down when the market goes up or down).

  • A negative B_i means that the asset and the market move in opposite directions.

  • Adding an asset with a negative B_i to a portfolio decreases the overall portfolio risk and should be cheaper. Conversely, the higher the B_i, the higher the cost of capital.

  • This follow from the assumption that investors are interested in reducing their risk through greater diversification. Thus, the more an asset’s value moves in accordance with the market — the higher the B — the less investors like this asset, and the higher the expected return the asset must offer for investors to want them. Conversely, the more an asset’s value moves against he market — the lower its (negative) B — the better diversification other asset offers and the more desirable it is from the investor’s perspective, and the asset can thus offer a lower expected rate of return.

  • Market betas (B_i) and other inputs are estimated on the basis of historical data.

• CAPM and NPV:
  You have the opportunity to invest in the following project:

  • A restaurant franchise that requires an upfront investment of 100,000 (including 50,000 for equipment);

  • Similarly situated restaurants generate on average annual net profits of 14.5% of the initial investment over a 10-year period;

  • Salvage value of equipment after 10 years = half of its upfront costs;

  • Market Beta (B_i) = 1.8;

  • Market return for the next 10 years = 10%; and

  • Expected rate of return on 10-year gilts (e.g. bonds issued by the US government) = 5%.

• Should you invest? And what if B_i = 0.5?

  • Required rate of return (through CAPM):

    • E(r_i) = 5% + [10% - 5%]*1.8 = 14%

  • NPV:

    • NPV
      = -C_o + (E(F_t)/E(r_i))*(1 - 1/(1 +E(r_i))¹⁰) + S_T/(1 + E(r_i))¹⁰
      = -100,000 + (14,500/0.14)*(1 - 1/(1 + 0.14)¹⁰) + 25,000/(1 + 0.14)¹⁰
      = -17,622.72

  • But what if B_i = 0.5?

    • E(r_i) = 5% + [10% - 5%]*0.5 = 7.5%

    • NPV
      = -100,000 + (14,500/0.075)*(1 - 1/(1 + 0.075)¹⁰) + 25,000/( 1+ 0.075)¹⁰
      = 11,659.02

  • Notes:

    • The risk of default is handled in the NPV numerator.

    • The 14,500 annual average net profits takes account of the fact that some restaurants make no profit at all and fail.

    • The respective probabilities are factored into the average.

• Evaluating CAPM:

  • Intuitively powerful;

    • By...