# Models and methods to estimate the appropriate r

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Models and methods to estimate the appropriate r
Moeller-Finance Cost of Capital Models and methods to estimate the appropriate r Remember the guiding principle: The r should reflect the riskiness of the cash flows The guiding principle for choosing the appropriate r is to use the r that reflects the riskiness of the cash flows. Building upon portfolio theory and asset pricing models, we will investigate several methods to estimate the appropriate r. We will begin the analysis by reviewing models and methods for calculating the risk of various components, i.e. equity, debt, other sources of financing. Cost of equity (RE): The return equity investors require on investments in the firm. REVIEW: A firm with extra cash can either invest in a project or pay a dividend so the stockholder can reinvest the money. When would shareholders want a firm to invest in a project? The firm should decide to invest in the project when the project return is as least as great as the return on other investments of comparable risk (NPV>0). We can look to the market place to find investments of comparable risk and estimate their r using portfolio theory and various asset pricing models. CAPM/Security Market Line (SML) Approach, use the SML To be consistent with the dividend growth model we will drop the E’s denoting expectations and write the CAPM/SML as: Remember: SML and CAPM are similar models with one basic assumption that is different. For the SML, we do not need to assume the market is in equilibrium while with the CAPM it is. Notes -Cost of Capital

Dividend Growth Model Approach
Moeller-Finance Dividend Growth Model Approach Re=(D1/P0) +g Typically used for equity Future dividends? Future growth rates? Analyst forecasts Analyst are optimists! (Realized growth 40-60% lower) Historical growth rates Other models Dividend Growth Model Approach, use the dividend growth model and rearrange for RE (shown above in slide). (Note any pricing model can be solved for r.) Implementing the Dividend Growth Model Approach Need the current stock price and the expected annual dividend. The current stock price can easily be obtained from the various financial sources. Depending on the firm, the expected dividend may be very easy to estimate. Need an estimate of the future growth rate in dividends 1. Analyst forecasts: Empirically we know analysts are typically optimistic so we may need to temper their estimates. One study suggests realized long term growth estimates are 40-60% of the long term estimate. 2. Historical growth rates 3. Other models Example: Estimate Caterpillar’s return on equity. Their current dividend is \$1.40 per share, the current share price is \$50 and the mean yearly long term growth estimates from analysts are 15%. Since we know analysts historically overestimate long term growth, we will estimate the growth rate to be 50% of estimate or 7.5%. Notes -Cost of Capital

Capital Asset Pricing Model (CAPM)
Moeller-Finance Capital Asset Pricing Model (CAPM) Perfect Competition All investors hold the universe of publicly traded assets and have unlimited access to borrowing/lending at the risk-free rate No taxes or transactions costs All investors plan for one identical holding period All investors are mean-variance optimizers All investors have homogeneous expectations Notes -Cost of Capital

Implementing the CAPM Approach
Moeller-Finance Implementing the CAPM Approach Theoretically can be used on any asset (equity, debt, assets, etc.) – typically used on equity Ri=Rf+Bi(Rm-Rf) Only systematic risk (beta) is priced in equilibrium Computing the components Risk-free rate: Treasury rates Market risk premium: Expected return on broad based index such as the S&P 500 or Wilshire 5000 Beta Many services estimate equity betas: READ THEIR METHODOLOGIES!!! Estimate with historical equity data Implementing the CAPM/SML Approach The beta only prices the systematic risk. Need estimates of the risk free rate, expected market risk premium and the expected beta. 1. Typically use the forward or historical treasury rate to estimate the risk free rate 2. Expected market risk premium: If we assume yesterday is a good predictor of tomorrow we can use historical premium. 3. Expected beta: If we assume yesterday is a good predictor of tomorrow we can use historical beta measure a. Use Investment services, like Value Line or Morningstar b. Estimate with historical data: run a regression of Ri on RM. Beta is the regression coefficient on RM. There are many difficulties in estimating beta including betas vary over time, betas are influenced by changing financial leverage and business risk and choosing the estimation period. T Notes -Cost of Capital

How do we estimate CAPM? Expected Return Model (CAPM)
Moeller-Finance How do we estimate CAPM? Expected Return Model (CAPM) Realized Return Model (Index Model) Notes -Cost of Capital

Estimation Issues Beta is non-stationary Data Frequency
Moeller-Finance Estimation Issues Beta is non-stationary What estimation period? How often do you revise? Beta moves toward one Data Frequency Daily, Weekly, Monthly? Market portfolio S&P 500? Other equity indices? Other real assets? There are many empirical things we know about estimating Beta’s and the goal is to get the most accurate measure possible. Two rules of thumb for accuracy: Statistically correct: use proper estimation techniques Adhere to finance theory For instance, the r should reflect the riskiness of the cash flows. How long is the project? Does your estimation period reflect the expectation for the life of the project? Should you adjust the beta? Move toward one, since empirically we know it trends toward one? If appropriate, should you add a liquidity, small capitalization, or other type of premium? Be careful, theory doesn’t allow for this so you need a preponderance of empirical proof in order to correctly add a premium. Adding a premium should be a very rare. See what Delaware Law says about these methods: As with all statistical estimations, there are advantages and disadvantages of the data frequency. The more frequent the data, the greater the noise while the less frequent you loose some of the relationship by smoothing the data. The CAMP theory calls for the market portfolio to be the entire market (including all types of investments) so you want to proxy for the market with the broadest based index you can reasonably find. Notes -Cost of Capital

Return on Debt Opportunity Cost of Debt Financing
Moeller-Finance Return on Debt Opportunity Cost of Debt Financing Use the YTM of outstanding debt which reflects opportunity cost Historical borrowing costs are irrelevant Coupon rate is irrelevant Use credit ratings to estimate cost of debt Find firms with similar debt risk (probability of bankruptcy) The Cost of Debt (RD): The interest rate that the firm's creditors demand on new borrowing. Cost of debt is observable 1. Yield-to-Maturity on currently outstanding debt 2. Yields-to-Maturity on newly-issued bonds with similar credit ratings Historical debt cost is irrelevant. The coupon rate on the firm's outstanding debt that was issued at par is inconsequential. Notes -Cost of Capital

Other Asset Pricing Models
Moeller-Finance Other Asset Pricing Models Many other models both proprietary and scholarly APT: Arbitrage Pricing Theory Fama/French Model 3 Factor: Market return, small stock versus big stocks and high versus low book/market (value versus growth stocks) 4 Factor: Additional momentum factor discovered by Carhart Other Asset Pricing Models There are many asset pricing models that have some scholarly evidence and many more that are proprietary. Some other well known models are several iterations based on Fama and French’s work, including the 4 factor model which includes a size, market/book, market risk premium and momentum factor. Fama and French update the benchmark returns approximately two weeks after the end of each month. The benchmark factors summarize (1) the overall market return (Rm), (2) the performance of small stocks relative to big stocks (SMB, Small Minus Big), and (3) the performance of value stocks relative to growth stocks (HML, High Minus Low). The Fama-French benchmark portfolios are rebalanced quarterly using independent sorts on size (market equity) and the ratio of book equity to market equity. The book-to-market ratio is high for value stocks and low for growth stocks. (description and table on the next page are from Ken French’s website, ) Notes -Cost of Capital

Risk is Difficult to Empirically Measure
Moeller-Finance Risk is Difficult to Empirically Measure Data is necessary for empirical observations Usually estimate equity betas because of data availability (asset beta is difficult to observe) Equity risk comparables are difficult to find Need to have the same capital structure Adjust for different capital structure by levering and unlevering beta Need to have the same business (asset) risk Industry Estimation: May use industry mean/median Other companies, other projects, divisions, etc. Usually these measures are a combination of asset or business risk and other types of risk (i.e., capital structure) Adjust by levering and unlevering beta Computing beta (focus on the intuition from the CAPM beta): Remember the beta is computed by figuring the riskiness of a series of cash flows compared with the market. However, in the past, we usually used the equity returns of the individual stock to estimate that risk. Equity risk is a combination of two types of risk of the firm: asset risk and financing risk. Because the equity holder is the residual claimant they receive the cash flows from assets minus the costs of financing. (Note: these cash flows can also have other unusual deductions including one time accounting charges that alter the flows but we will ignore those for simplicity.) The beta (return) of a portfolio of assets is merely the weighted average of the beta (return) of the individual assets. Remember this relationship does not work for volatility because when calculating the variance of a portfolio, there is an additional covariance/correlation term (see third term of the below equation). Notes -Cost of Capital

How Do We Manage This Problem?
Moeller-Finance How Do We Manage This Problem? We use the relationship between the total firm market value (V), asset (A), equity (E), debt (D) and the NPV of the capital structure/financing (D) Think of the firm value expressed as V=A+D= E+D The NPV of the capital structure/financing is the value created by the capital structure choice of the firm In our simple world with no bankruptcy costs, this is basically the tax shield of debt In perfect capital markets NPV of financing is zero (no taxes) The firm (assets) can be viewed as a portfolio of its financing (assume equity, debt and NPV of capital structure/financing) Notes -Cost of Capital

Moeller-Finance The Relationship The beta of a portfolio is the weighted average of the components therefore The return of a portfolio is also the weighted average of the components Substitute return (r) for beta (B) in the relationship Note: The use of this relationship is typically called levering and unlevering Portfolio betas: Weighted average of individual assets’ beta We then make one modification to allow us to fully specify the relationship. Assume that the value of the firm is the value from the business/economic/asset risk (A), i.e. what goes on inside the company walls, and the value derived from the financing/capital structure (D). So For simplicity, we are going to assume the firm only finances with equity and/or debt. However, if the firm had other sources of financing it would simply be expressed as an additional term. Below is the expression with the additional term included where bo is the beta of other financing and O is the market value of the other financing. Also bA is the asset (unlevered) beta, bE is the equity (levered) beta, bD is the debt beta and bD is the NPV of financing beta. Notes -Cost of Capital

What is the risk of financing NPV(D)?
Moeller-Finance What is the risk of financing NPV(D)? Assume BA=BD (Case 1) Assume BD=BD (Case 2) Assume BD=BD and D=tD (Case 3) If possible, we can make various assumptions about the risk of the value of financing/capital structure to simplify the relationship. So the question that needs to be answered is, what is the riskiness of the cash flows derived from financing? In the perfect world, capital structure has no value but market imperfections may lead to value creation through capital structure. For instance, capital structure may create value (positive or negative) from the tax debt shield and/or the probability of bankruptcy. To help gauge the riskiness of these types of flows we need to understand the expected capital structure over the life of the project and the expected tax rate over the life of the project. Notes -Cost of Capital

Assumptions about NPV of financing (D)
Moeller-Finance Assumptions about NPV of financing (D) BA=BD Asset (business) risk related to the financing risk? More likely if leverage is constant proportion of market value BD=BD Debt risk related to the financing risk? More likely if leverage is constant dollar amount Assume BD=BD and D=tD Most restrictive assumptions rd is the appropriate rate, debt is constant dollar amount and a tax deductible perpetuity (tD= tDrd/rd) How about floating rate debt? Reasonable assumptions? Think of the NPV(financing/capital structure) as primarily coming from the debt tax shield (we will ignore the cost of bankruptcy for this example). Remember the value of the debt tax shield is the sum of the expected discounted cash flows of the debt tax shield (D*Rd*tax rate). Think of the riskiness of each of these three elements, D, Rd and tax rate, as a function of the riskiness of assets and debt. If there is an overall positive relationship of the debt tax shield with assets (debt), Case 1 (Case 2 or 3) is the best. When is the risk of financing similar to the asset risk? Restated, when is the riskiness of the debt tax shield cash flows similar to the asset cash flows? Restated again, when is the debt tax shield positively related to assets? One answer, when leverage is a constant proportion of market value. This means that riskiness of debt (D) (therefore the debt tax shield especially when holding Rd and the tax rate constant) is positively related to the riskiness of the asset value. When is the risk of financing similar to the debt risk? One answer, when leverage is a constant dollar amount which means that D is constant, so the riskiness of Rd is positively related to the financing flows riskiness (especially while holding the tax rate constant). Notes -Cost of Capital

Moeller-Finance Additional Assumptions What is the beta of debt? Can we assume it is zero? If the firm has fixed rate debt and a low probability of bankruptcy, its very close to zero Rule of thumb: Keep the assumptions to a minimum, in other words, lever and unlever only when necessary! What is the beta of debt? Can we assume it is zero? If the firm has fixed rate debt and a low probability of bankruptcy, its very close to zero Notes -Cost of Capital

Computing An Asset Beta
Moeller-Finance Computing An Asset Beta Asset beta is usually difficult to observe How do we estimate an asset beta? Strip out the asset risk by unlevering the beta Or find an all equity (pure play) firm Find the Beta for a new hotel project. The industry Be is 1.5%, average industry debt level is 20% and Bd is 0.2%. (assume Case 1). Assume Rf is 3% and Rm is 13%. Does this relationship hold for R also? Finding r when the project has a different risk than the firm If you can find an estimate of the business risk (asset risk) for a similar risk project then you can just use that estimate. However, usually these estimates are difficult to find and instead you use equity returns to proxy for the risk. When equity risk is used, the measure implicitly includes the average asset risk and financial risk of the industry. However, your firm may not have the same capital structure as the industry. In this case you will want to take the industry equity beta, unlever it by the industry capital structure (what we did in the example on this slide) and then re-lever it to your firm’s capital structure (example shown on the next slide). NOTE: If Case 1 is not the correct assumption about the NPV of capital structure/finance then simply replace the formula with the appropriate relationship. Notes -Cost of Capital

Estimate return on equity for a new capital structure
Moeller-Finance Estimate return on equity for a new capital structure Use when you can reasonably estimate New capital structure Changes in cost of debt From the previous example, the industry Ra is 15.4%. Your hotel project is going to have a debt level of 40% and the Rd is 7%. What is your Re? What if the project changes the capital structure? There are several ways to deal with estimating the NPV when the new project will alter the existing firms capital structure. We will focus on two basic methods: adjusting the WACC and the adjusted present value (APV). WACC adjustment Essentially we will adjust the WACC when the new project supports a different capital structure but we can reasonably estimate the new capital structure and potential changes in the return on equity and debt. Again we will use the process of levering and unlevering returns and betas to accomplish our adjustment. For simplicity, lets assume Case 1 is the best assumption. Then substitute in return for beta and solve for Re. Now take the new Re, new capital structure, new Rd and plug it into the WACC to get the new WACC. Adjusted Present Value On way to decompose APV is break the value of the project into the unlevered NPV and additional financing benefits or costs. This allows capital structure to vary but it only shows up in the financing benefits or costs. Notes -Cost of Capital

“Whole Firm” Risk Measures
Moeller-Finance “Whole Firm” Risk Measures From portfolio theory Portfolio risk is the weighted average of the components’ risk Works with beta and return (but not volatility) Think of the “whole firm” as components of... Financing: Debt, equity, other financing (i.e., WACC) Value: Business (unlevered) and financing flows Other logical breakouts? Divisions/business units Assets in place and growth opportunities Look to the available data and logical economic components Used for any complex asset (does not have to be “whole firm”) Notes -Cost of Capital

Weighted Average Cost of Capital
Moeller-Finance Weighted Average Cost of Capital When is WACC the appropriate discount rate? Proposed investment project is similar to the overall business activities of the firm Project is financed with same capital structure weights as the firm Target weights versus actual weights Represents cost of the next dollar a firm would raise Simplify the capital structure to debt and equity View the firm as a portfolio of securities Cost of Equity Cost of Debt WACC reflects average riskiness of firm's securities Another source of an estimate of r is the firm's cost of capital where it reflects the average riskiness of all its funding, i.e., the firm’s mixture of debt and equity. In other words, we estimate the appropriate r by estimating the cost of the next dollar a firm can raise assuming the firm does not alter its capital structure or riskiness. For simplicity we are going to assume that the firm only consists of simple debt and equity. If you have a firm with many types of financing the WACC is still the value weighted average of the after tax costs of financing. Think of the Firm as portfolio of debt and equity Let E = the market value of the firm's equity (share price * shares outstanding) Let D = the market value of the firm's debt (bond price * bonds outstanding) V = the market value of the firm (E+D) Note it appears we have ignored the NPV of financing from the previous specification. However, when using the WACC you assume the capital structure does not change so we can use the ex ante capital structure weights which implies the NPV financing is zero. The firm's capital structure weights are: Notes -Cost of Capital

Moeller-Finance What if the projects are not similar to company risk? WACC may lead to poor decisions! Incorrect Investment Decisions B A What if the WACC is not an appropriate estimation of r? Two violations: 1. The project risk is not similar to the risk of the existing firm’s assets 2. The capital structure of the firm will (significantly) change due to the project Violation #1: Different risk projects. The SML and the WACC: What is the effect of using the wrong r? Using WACC to evaluate investments with risks that are substantially different from the over all firm can lead to poor decisions. You may either accept projects whose true NPV<0 (projects whose return is less than the SML but greater than the WACC, area B) or reject projects whose true NPV>0 (projects whose return is greater than the SML but less than the WACC, area A). Firm’s overall cost of capital Rf Project’s security market line Notes -Cost of Capital

Weighted Average Cost of Capital (WACC)
Moeller-Finance Weighted Average Cost of Capital (WACC) Capital Structure weights (portfolio weights) Use market values WACC (not adjusted) = value-weighted average of cost of capital WACC = (E/V)Re + (D/V)Rd WACC (adjusted) = value-weighted average of after tax cost of capital Tax-Advantage of Debt Implies: Adjusted weighted average cost of capital (WACC) is the weighted average of the costs of after tax debt and equity Interest payments are tax deductible: The after-tax cost of debt is the pre-tax cost of debt multiplied by (1 - corporate tax rate). Let tC denote the corporate tax rate. Estimating the WACC: 1. The return on equity can be estimated using the dividend growth or SML approach that we previously discussed. 2. E, D and V can be estimated by multiplying the market value of equity (debt) of the firm by the number of shares (bonds) outstanding. (Since the assumption is that the capital structure doesn’t change, we can express V as V=D+E.) 3. The corporate tax rate is usually given or can be estimated from the expected level of taxable income for the firm. 4. The cost of debt is estimated as follows Unadjusted weighted average cost of capital is the weighted average of the costs of debt and equity ignoring taxes John Graham has used a sophisticated algorithm to estimate corporate taxes. You can access them through his web site. WACC = (E/V)Re + [(D/V)Rd*(1-tc)] Estimates of Corporate tax rate? Notes -Cost of Capital

Bringing it all together: Cash flow and r?
Moeller-Finance Bringing it all together: Cash flow and r? FCF using WACC Cash flows are the flows to the total firm WACC is based on the firm’s existing capital structure (RHS of the balance sheet) Adjusted Present Value (APV) Most common use: Break the flows into flows to assets and flows to financing NPV plus PV (other benefits or costs) Flow to Equity Approach (FTE) Only estimate the flows to the equity holders The appropriate r is the return on equity. From your first class is finance, you focused on calculating the relevant cash flows to the company and discounting them by a “whole firm” r (WACC). In this class we add two additional DCF valuation methods. One alternate method is to calculate the adjusted present value which is a method of dividing the project into two separate flows. First, the base case flows to the assets. Second, the flows to the financing of the project. Adjusted Present Value: If the financing and project are not independent, the adjusted present value (APV) can be used to decide whether to accept the project. APV is the addition of the base NPV as we have computed it in the past plus or minus additional financing benefits or costs. (Caution: Make sure that the financing and project are truly dependent. Many times they appear so but with some creative thinking the two decisions can be divorced.) This method can also be used to take the value of the unlevered firm and add the tax shield of debt. Another alternate method to capital budgeting is to calculate the expected dollar return to the shareholders, sometimes called the flow to equity approach. The expected equity income is the yearly cash flow minus the tax shield of debt. If you use this cash flow number then the appropriate r should only reflect the cost of equity. Notes -Cost of Capital

Moeller-Finance Comparison of 3 methods Assume the project is financed with \$50 of debt which costs 8% and \$50 of equity which costs 12%. The yearly perpetual project cash flow is \$8.8, the tax rate is 30% and assume you can perpetually take advantage of the tax shield of debt. What is the NPV? Notes -Cost of Capital

WACC FTE Equity income=cash flow minus after tax cost of debt
Moeller-Finance WACC FTE Equity income=cash flow minus after tax cost of debt The r should reflect the riskiness of the cash flow! In the WACC approach, we think of the firm as a portfolio of its financing. Because we exclude the tax advantage of debt in the cash flows, we need to adjust for it in r. Therefore we use the after tax WACC. Remember when determining relevant costs for capital budgeting projects and one exception was financing costs? The definition for a relevant cost is: Any future cash flows that are incurred due to accepting the project. Though financing costs (interest expenses, coupons, etc) may be incurred due to accepting the project we do not include them in operating cash flows because they will usually enter the NPV calculation via r. This difference can be seen in in the calculation of the WACC and the first approach of the APV (shown on the next slide). In the FTE approach, we are only calculating the cash flows that an equity holder receives. In this case, we need an r that represents that risk so we use the expected return on equity. Notes -Cost of Capital

APV: One approach (r based on who receives flow)
Moeller-Finance APV: One approach (r based on who receives flow) Second approach (r based on origin of flow) The r should still reflect the riskiness of the cash flow! In the APV approach, we need to find a logical way to break apart the cash flows. In this case, lets think of the firm as two different types of flows, first from the project and second from the tax advantage of debt. Now the tricky part, what are the appropriate r’s for these two flows? In the first approach, we look to who receives the cash flows to estimate the riskiness. The bond and stock holders have the claim on the cash flows of the firm, so we need can use the WACC to estimate the appropriate r. However, in contrast to the previous WACC example, we are going to include the cash flows due to the tax advantage of debt so we need to use the pre-tax WACC (otherwise we would double count the tax advantage of debt). In the second approach, we look to the origin of the cash flows to estimate the riskiness. The cash flows from assets come from the economic function of the firm so we can estimate the riskiness by using MM #2. However, the flows due to the tax advantage of debt have a different source of risk. When a firm is reasonably confident that it will be able to take advantage of the tax deduction (in other words, they will have positive taxable income), one reasonable proxy for r is the riskiness of debt. (We will cover this topic in more depth during our discussion of capital structure.) Notes -Cost of Capital

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