Capital Budgeting and Risk (Ch. 9)

COMING UP WITH THE CAPM R_f – SHORT-TERM R_f OR LONG-TERM R_f?

·        Which R_f should be used?

o       No consensus

o       Seems to make sense to say it should depend on how you’re going to use the E(R) or required return that you come up

o       If you want to know the E(R) from investing in a portfolio for 2 months, use a short-term R_f

o       If you’re trying to come up with the required return for computing the price of a stock, use a long-term R_f (why?)

·        Short-term R_f today, by convention, is the yield today on 3-month T-bills

·        Long-term R_f today, however, is not the yield today on long-term T-bonds

·        Table 7.1 (p 155): Avg returns for 75 year period 1926-2000

o       T-bills      3.9%

o       T-bonds    5.7%

o       Stocks      13.0%

·        Why is return on t-bonds > return on t-bills?

o       Let’s start with: what makes yld for l-t treasuries diff from yld for s-t treasuries?  There are 2 factors:

§        T-bonds are riskier.  Return on T-bonds included a risk premium for interest rate risk

§        L-t rates also reflect whether s-t rates are expected to increase or decrease in the future (expectations hypothesis)

=>       L-t yield = s-t yield + risk premium for interest rate risk + expectations factor

o       When short-term rates are expected to increase in the future, the expectations factor is positive.  When short-term rates are expected to decrease in the future, the expectations factor is negative.

o       Half the time we would expect s-t rates to increase, and half the time we would expect them to decrease

o       So when you take average returns over 75 years, the expectations factor will average out to zero

o       Hence, the diff in returns between t-bills and t-bonds that we see in Table 7.1 is just the avg. risk premium for intt rate risk:

Avg historical l-t yield = average historical s-t yield + avg risk premium

o       Avg risk premium for l-t bonds = avg t-bond return of 5.7% -  avg t-bill return of 3.9% = 1.8%

o       We can’t use the yield on long-term t-bills today as the long-term R_f because it contains a risk premium as well.  So we compute the long-term R_f today as follows:

l-t R_f today = yld on 20-year T-bonds today – avg. historical risk premium of 1.8%

o       What’s the avg l-t riskfree rate for 1926-2000? 

§        Would have to be the avg t-bond yield of 5.7% - the avg risk premium of 1.8% = 3.9%. 

§        And this is just  the average s-t riskfree rate (avg t-bill return of 3.9%)

§        This is always going to be true. 

§        avg l-t R_f = avg t-bond return – avg risk premium

      = avg t-bond return – (avg t-bond return – avg t-bill return)

      = avg t-bill return 

      = avg s-t R_f

·        To summarize, we have 2 alternative ways to compute required return

·        USING SHORT-TERM R_f

yield on 3-month t-bills today + b_i * (avg hist. return on stocks – avg hist. s-t R_f) = .011 + b_i* .085

·        USING LONG-TERM R_f

l-t risk free rate today + b_i*  (avg hist. return on stocks – avg hist. l-t risk free rate) = (yield on 20-year t-bonds today - .018) + b_i* .091 = .(0475 - .018) + b_i* .091 = .0295 + b_i* .091

·        Based on today’s values for 3-month t-bill yield and 20-year t-bond yield, the long-term E(R) would be higher by 1.85%.

(Note: in case you missed this in the book the first time around, it’s in footnote 8 on p. 226)

USING THE CAPM IN CAPITAL BUDGETING

·        The basic implication of the CAPM is: a project’s net cashflows must be discounted at a required return which reflects the systematic risk of the project

·        Project involves acquiring assets to generate cashflows.  So “systematic risk of the project” means systematic risk of the net cashflows generated by the project

·        In theory, applying CAPM to capital budgeting is simple:

o       Estimate the project’s beta

o       Use the SML equation to compute required return (using either a s-t riskfree rate or a l-t riskfree rate)  

o       Discount the project’s cashflows at this required return

·        Graphical illustration: 

      Positive NPV projects plot above the line, negative NPV projects plot below the line

·        Using the SML equation is straightforward.  So the basic question for us is: how do we estimate project’s beta?

ASSET BETA, DEBT BETA, EQUITY BETA  (Note: Frontpage has gone and changed all the beta symbols to b; I'm having trouble changing them back, so I'm just going to leave them this way.)

o       Assume we have a firm that is financed just by ordinary debt and equity (no convertible debt or preferred stock, etc).      =>  V = value of the firm = D + E

o       Also, for the moment, we’ll ignore taxes.  (Will take taxes into account later when we discuss capital structure decisions; Ch. 17, 18)

o       Let’s say the firm has three divisions:

o       The firm can be viewed as a portfolio of these three divisions. Like any portfolio, the beta of the firm is just the weighted average of the betas of the three divisions:

b_FIRM = x_Bb_B  + x_Pb_P + x_SDb_SD = (25/46)*.15 + (8/46)*.8 + (13/46)*.5 = .362

o       This beta represents the systematic risk of all the firm’s assets together.  We call this the asset beta (b_a).

o       From the asset side we can think of the firm as a portfolio of its different divisions, or a portfolio of its different projects.  In the same way, from the liabilities side we can think of the firm as a portfolio of debt and equity.  This means the firm’s beta is also a weighted average of the debt beta (b_d) and the equity beta (b_e).

b_a = (D/V) b_d +  (E/V) b_e                                 (1)

o       Let’s ask ourselves what will happen to all three betas if we hold the firm’s assets constant, and change just its debt ratio.  (We call this a pure capital structure change.  The investment decision doesn’t change.  The assets don’t change.  All that changes is the capital structure.)

o       b_a must stay constant, since we are not changing the assets of the firm

o       As you increase the debt ratio, the debt will become riskier =>  b_d will increase

o       So the question becomes, how does the equity beta change? 

o       asset beta is a weighted average of the debt beta and the equity beta

o       if the asset beta stays constant, and the debt beta increases, we would intuitively expect the equity beta to decrease.

o       Strangely enough, that’s not what happens

o       Since the asset beta stays constant, (D/V) b_d +  (E/V) b_e has to stay constant.  We are changing D/V, b_d and E/V.  So b_e becomes a slack variable, taking whatever value is necessary to keep (D/V) b_d +  (E/V) b_e constant.  Here’s an example of how the numbers work out:

o       As we increase the debt ratio, b_d and b_e both increase but the weighted average stays constant.  How is this possible?

o      The key is that as both betas increase, we continuously shift some weight from the higher number (b_e) to the lower number (b_d).  The shift in the weights compensates for the increase in the betas, and the weighted average stays constant.

o       The equation for b_e is given by:

b_e= b_a  + (D/E) (b_a - b_d)                                     (2)

o       Since equity is riskier than debt, we have b_e > b_d

o       Since asset beta is a weighted average of b_d and b_e it must lie in the middle:      b_e > b_a > b_d

o       This means  (b_a - b_d) is positive => the second term as a whole is positive

o       As we increase the debt ratio, the second term in equation 2 increases

Thus, equity beta consists of two components: the constant term b_a, and the second term which increases with the level of debt.

WHICH BETA DO WE WANT: ASSET BETA OR EQUITY BETA?

·        Let’s return to the basic question, which was: how do we estimate a project’s beta?

·        Which beta do we want – the project’s asset beta or its equity beta?

·        How do these betas relate to the risk stockholders actually bear?

o       Equity beta is the systematic risk stockholders actually bear given that the project is partly debt financed

o       Asset beta represents the systematic risk stockholders would bear if there was no debt financing (from equation 2, when D = 0, b_e = b_a)

·        Surprisingly, we don’t want the equity beta (the true risk to s/h) but the asset beta (the risk s/h would bear in a hypothetical scenario)

·        Why?  This goes  back to our convention for computing the net cashflows of a project, and how we want to handle the impact of debt financing.

·        We compute the net cashflows assuming 100% equity financing.  The discount rate we apply to these cashflows must be consistent with that assumption. 

So just as the net cashflows are not the true cashflows but what they would be under 100% equity financing, the beta we use reflects not the true systematic risk but what it would be under 100% equity financing

The impact of debt financing is taken into account separately.

o       So the project’s cost of capital is the required return based on the asset beta:

R_a = R_f + b_a * RP_m

And just as b_a is a weighted average of b_d and b_e, this required return based on b_a is a weighted average of the required return on debt R_d and the required return on equity R_e:

R_a = (D/V) R_d + (E/V) R_e

We call it the weighted average cost of capital, WACC.  (There is a different formula for WACC once we take taxes into account.  We’ll talk about that stuff later.)

·        R_d, the return that has to be provided to b/h, is the cost of debt capital. 

R_e, the return that has to be provided to s/h, is the cost of equity capital. 

The weighted average of these two returns, WACC, is the overall return a project must earn to break even (i.e. have zero NPV).

ESTIMATING ASSET BETA FOR PROJECT

·        It’s relatively simple if:

o       this is a new project by an existing firm whose shares are publicly traded

o       the firm has only one line of business

o       the new project is the same business

·        Then the project’s asset beta = the firm’s existing asset beta.  So we can:

o       estimate the firm’s equity beta (by running a regression of R_i on R_m)

o       use equation 1 to compute the asset beta from the equity beta (this is called unlevering the beta)

o       (Aside: computing the equity beta from the asset beta is called levering the beta)

o       stick this asset beta into the SML equation

·        A different procedure is required if the firm’s shares are not publicly traded (can’t get the returns data for the regression), or if the firm has many different lines of business (the firm’s asset beta is a weighted average of the asset betas of different divisions; project’s asset beta does not equal the firm’s existing asset beta) or if the project is in a new business (again, project’s asset beta does not equal the firm’s existing asset beta).

·        The recommended procedure then is:

o       Pick a bunch of listed stocks in the same business as the new project

o       Estimate the equity beta for each stock

o       Unlever each firm’s equity beta to get its asset beta

o       Take the simple average of these asset betas

·        We pick a bunch of stocks because when you estimate the beta of one stock, the estimation error is usually large.  Averaging a bunch of independent estimates reduces estimation error.

·        Example: you need to estimate asset beta for a new project in the pharmaceutical industry.  You obtain the following information:

a)     What’s your estimate of the project’s asset beta?

b)     If the project will actually be financed with 25% debt, and the debt beta is estimated to be 0.11, what’s the project’s equity beta?

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