Approaches to Calculating the Cost of Capital

The capital asset pricing model helps investors assess the required rate of return on a given asset by measuring sensitivity to risk.

When considering assets for diversification in an investment portfolio, investors and financiers use various tools to project the required rate of return and risk of a given investment. This means that investors are on the lookout for ways to minimize risk, maximize returns, and invest intelligently in well-priced assets. The key assumption is that the market will self-correct to adjust each investment option's expected return to the relative risk of investing.

The Capital Asset Pricing Model

The capital asset pricing model (CAPM) can be a useful tool for measuring the ratio between risk and return on a given investment. This model measures a given asset's sensitivity to systematic risk (or market risk) about the expected return compared to a theoretical risk-free asset.

How CAPM Works

This sounds complicated, but it's simpler than it seems. All this means is that the CAPM tries to measure the risk the market will offer the asset compared to the risk-free rate and make sure the expected return will offset that risk. To understand this concept, there are a few variables that are useful to identify upfront:

• E(Ri) is an expected return on security.
  
  
• E(RM) is an expected return on market portfolio M.
  
  
• β is a non-diversifiable or systematic risk.
  
  
• RM is a market rate of return.
  
  
• Rf is a risk-free rate.

There are many ways to rearrange the relationship between these variables to derive meaningful information. Like all equations, depending on what you know, you can solve for what you do not know.

For basic CAPM calculations, you want to solve for the expected return on a security, which looks like this:

\(E(R_i)=R_f+β_i(E(R_m)−R_f)\)

Security Market Line

By rearranging these variables, you can also look at the concept of the security market line (SML), which underlines a security's relationship with systematic risk and respected return in a graphical format. This is written as:

\(SML:E(R_i)=R_f+β_i[E(R_M)−R_f]\)

Security Market Line This graph illustrates the security market line, which visualizes an asset's expected return.

Risk and Return

A final application of the CAPM variables in relation to one another is in deriving a ratio that illustrates the relationship between risk and return. On the left side of the equation below, you have an assessment of the overall risk relative to a risk-free asset. On the right side, you have the overall return (similarly relative to a risk-free asset). This can be written as follows:

\(\dfrac{E(R_i)−R_f}{βi}=E(R_m)−Rf​\)

Investors can utilize the CAPM equation and its various implications to assess various market investment opportunities to diversify a portfolio and identify undervalued assets.

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The SML is the graphical representation of CAPM used to determine if an asset is priced to offer a reasonable expected return for the risk.

The SML Approach

The Security Market Line (SML) is the graphical representation of the capital asset pricing model (CAPM), with the x-axis representing the risk (beta) and the y-axis representing the expected return. It graphs the relationship between beta (β) and expected return, i.e., it shows expected return as a function of β. The y-intercept of the SML is equal to the risk-free interest rate, while the slope is equal to the market risk premium (the market's rate of return minus the risk-free rate). The slope also represents the risk-return tradeoff at a given time. The SML applies to any asset.

\(SML : E(R_i) = R_f + \beta_i(E(R_M) - R_f)\).

SML Equation The SML is the graphical representation of CAPM and thus is found using the same equation.

Applications of the SML

Individual assets that are correctly priced are plotted on the SML. In the ideal world of CAPM, all assets are correctly priced and thus lie on the SML. In real market scenarios, we can use the SML graph to determine if an asset being considered for a portfolio offers a reasonable expected return for the risk. If an asset is priced at a point above the SML, it is undervalued since, for a given amount of risk, it yields a higher return. Conversely, an asset priced below the SML is overvalued since, for a given amount of risk, it yields a lower return.

CAPM-SML The Security Market Line for the Dow Jones Industrial Average over 3 years, with the x-axis representing beta and the y-axis representing expected return.

Another way to think about the SML's real market applications is in terms of buying and selling securities. If an asset is priced above the SML and thus undervalued, it should be bought. If an asset is priced below the SML and thus overvalued, it should be sold.

A discounted cash flow analysis is a highly useful tool for calculating the net present value of a given product, process, asset, or organization.

The discounted cash flow approach is a valuation method investors and organizations can use to assess the net present value (NPV) of an asset, process, product, or the overall organization. NPV analyses using the discounted cash flow approach are widely used across various industries to decide which projects to invest in. They are also used as valuation devices for investors when looking at company performance, using past performance to project future performance.

The Discounted Cash Flow Formula

The discounted cash flow formula focuses on determining the relative time value of money of each projected cash flow (i.e., monthly, quarterly, annually, etc.), bringing each forecast of future value into present value terms. Just as in an NPV analysis, the ultimate end product will be the value of future profits (or losses) in today's terms. It's calculated as follows:

\({\displaystyle DCF={\frac {CF_{1}}{(1+r)^{1}}}+{\frac {CF_{2}}{(1+r)^{2}}}+\dotsb +{\frac {CF_{n}}{(1+r)^{n}}}}\)

In situations of multiple cash flows over multiple periods of time, it is necessary to create a summation that can incorporate variance in both variables:

\( DPV=∑_{t=0}^N\dfrac{FV_t}{(1+r)^t} \)

Sometimes, cash flows are considered continuous. In such scenarios, the appropriate adjustment to the equation is:

\( DPV = ∫_0^TFV(t)e^{−λt}dt \),

Inputs

When considering these formulas, you'll need to understand your inputs. The inputs for a discounted cash flow analysis are:

• DPV – The discounted present value of the future cash flow (FV), or FV, adjusted to compensate for the units of time in the future it will be received.
  
  
• FV – The nominal value of a cash flow amount in a future period (i.e., the amount of the cash flow before taking the time value of money into account).
  
  
• r – The interest rate or discount rate reflects two important information: the opportunity cost of foregoing other investments and the intrinsic risk of not receiving the projected cash flow.
  
  
• n – The time (in a given unit) before the future cash flow occurs. This is usually done annually but can be done monthly or quarterly. Keep in mind that r must be adjusted according to the time period!

Strategic Use

Like any projection, the most important thing to keep in mind as either investor or strategist is uncertainty. The primary purpose of a future cash flow analysis is to balance expectations to consider existing and future resources to make the optimal decision (from a profit perspective). As a result, determining 'r' (required rate of return due to opportunity cost and risk) is absolutely critical to the success of these calculations.

A Discounted Cash Flow Example This is a good example of a what a discounted cash flow analysis would look like on paper, particularly as a prospective investor. There's actually quite a bit more information here than you may strictly need to understand the calculation, but it's a great way to see how each piece of information fits together.

The bond yield plus risk premium (BYPRP) approach is another method for determining the value of an asset, specifically a company's publicly traded equity. BYPRP allows us to estimate the required return on equity by adding the equity's risk premium to the yield to maturity on the company's long-term debt.

\(R_r = Y_{ltd} + E_{rp}\)

Bond Yield Plus Risk Premium Equation States that the required return on equity equals the yield of the company's long-term debt plus the equity's risk premium.

Bond Yield vs. Risk Premium

Simply put, the yield on a bond is the rate of return received from the investment. In the BYPRP approach, we use a bond's yield to maturity, the discount rate at which the sum of all future cash flows from the bond (coupon payments and principal payments) equals the bond price. This is also called the internal rate of return (IRR).

Yield To Maturity Graph: A hypothetical graph showing yield to maturities (or internal rates of return) for corresponding present values.

The equity risk premium is essentially the return that stocks are expected to receive in excess of the risk-free interest rate. The normal historical equity risk premium for all equities has been just over 6%.

Generally, an equity's risk premium will be between 5% and 7%. Common methods for estimating the equity risk premium include:

• The Fed Model (forward operating earnings yield [earnings per share divided by share price] minus the 10-year U.S. Treasury Bond yield)
  
  
• The dividend yield plus projected earnings growth minus the 10-year Treasury yield
  
  
• The historical stock returns minus the 10-year Treasury yield

Estimating the value of an equity using the bond yield plus risk premium approach has its drawbacks. We can only utilize the BYPRP approach if the entity has publicly traded debt and it does not produce as accurate an estimate as the capital asset pricing model or discounted cash flow analysis.

Moreover, equity risk premium estimates can be highly inaccurate and vary wildly depending on which model is used. It can be very difficult to get an accurate estimate of the risk premium on equity with a duration of roughly 50 years using a risk-free rate of such short duration as a 10-year Treasury bond.

\(R_r = 0.06 + 0.04\)

Example Equation Required return = 6% + 4%