Intro to Finance: week 13

Expected market risk premium:

(ErM) - rf

Expected risk premium from stock i:

(Eri) - rf

Capital Asset Pricing Model (CAPM)

-E(ri): Expected return of stock i -rf: The risk-free rate -𝛃i: Beta of stock i -E(rM): Expected market return -E(rM) - rf: Expected market risk premium (EMRP) -𝛃i[E(rM) - rf]: Expected risk premium of stock i

Implications of EMRP: According to the CAPM (Equation 6.13),

-If EMRP increases, so will the required returns of investors from risky investments. If required returns increase, prices will have to decrease to generate those higher required returns, holding expected future cash flows the same. -If EMRP declines, so will the required returns of investors from risky investments. This will cause the value of those investments to increase.

Using the market value approach:

-Market value of equity: E = market capitalization = share price x number of shares outstanding. -Market value of debt: D = the present value of the company's future interest payments on its bonds and loans, discounted at the company's pre-tax cost of debt.

CAPM

-you dont get rewarded for bearing diversifiable risk -reward depends on systematic risk -risk premium exists to compensate investors for bearing systematic risk and occasionally taking a loss on that investment at bad times

There are three ways of estimating the EMRP:

1. historical premiums 2. surveys 3. implied EMRP

from 1926-2020, the average return for the U.S. stock market was basically

10%

[E(rm)-rf]= EMRP

5-6%

A high beta (above one) means relatively high systematic risk (above the beta of the market and the average stock).

A low beta (below one) means low systematic risk (below the market's).

A positive alpha indicates the stock/fund has performed better than its beta would predict.

A negative alpha indicates the stock/fund has underperformed. CAPM alpha, Sharpe Ratio, and Treynor Ratio are three alternative ways of measuring risk-adjusted performance.

Recall that we learned how to calculate portfolio beta -- it is the weighted average of the individual betas.

Above we also learned how to compute an asset's expected return using the Capital Asset Pricing Model. How about the expected return of a portfolio?

Equation 6.18: Alpha =

Actual return - CAPM expected return

Important question 2: OK, so beta matters for diversified investors. But what if I am not diversified? Won't my required return from a stock be related to volatility instead of beta?

Answer: Yes, sure, but that's too bad for you. What we are trying to find is the market's required return from a stock, not individual by individual. And since most investors are diversified, market prices will be set based on required returns of diversified investors. So, it's beta that we need to care about. And since diversified investors will face lower risk and will have lower required returns, this will put undiversified investors at a big disadvantage. They will in general value the same risky assets at a lower value compared to diversified investors. But, hey, diversification is easy and free. No one's stopping them from diversifying and getting rid of unnecessary risk.

The risk-free rate comes from the current yields on Treasury securities (maturity depends on the specific application. In stock valuation and capital budgeting, it's common to use the annual yield on 10-year Treasuries).

Beta can be estimated from historical returns using the slope of the characteristic line or the formula method as we covered earlier. What about EMRP? What is it and where does it come from?

Consider a mutual fund that is down 7.1% over the past year. S&P500 was down 7.3%. The beta of the mutual fund is 1.2. Risk free rate is 4%

CAPM expected return= 4%+1.2[-7.3%-4%]=-9.56% Alpha=-7.1%-(-9.56%)=2.46%>0

Expected market return:

E(rM)

Expected return from stock i:

E(ri)

Suppose you are considering investing in a mutual fund -- say SC Growth Fund -- and you are trying to evaluate whether the fund has performed well in the recent past.

For example, suppose SC Growth Fund generated an average annual return of 13.58% over the last 10 years (compounded average of course! Remember arithmetic vs. compounded average discussion from earlier this chapter).

Until now, we used risk and standard deviation (the volatility of a stock's returns) interchangeably.

Now that we know there are two types of risk, we need to be more careful about what's exactly measured by standard deviation versus beta.

Standard deviation measures the total variation of the returns of a stock or portfolio over time, including both systematic AND unsystematic risk.

So, standard deviation measures an investment's total risk.

Example 6.36: A company has a stock beta of 1.4. What is its cost of equity? Suppose the risk-free rate is 2% and the expected market risk premium is 5%.

Solution: Cost of equity is the same as the CAPM expected return from Equation 6.17: Cost of equity = CAPM expected return = E(r) = 2% + 1.4 x 5% = 9%

Example 6.29: What is the beta of a portfolio where you invest 70% of your money in T-bills (assume risk-free) and the rest in a stock with a beta of 1.2?

Solution: The portfolio weights are 70% for the risk-free security with a beta of zero and the rest (30%) for the stock with a beta of 1.2. Using Equation 6.15: ßp = (0.7 x 0) + (0.3 x 1.2) = 0.36

Alpha:

The difference between a stock or fund's actual return and its expected return based on CAPM (or an alternative expected return model).

EMRP:

The premium that investors demand for investing in an investment with average risk (beta=1) or for investing in the entire stock market. Also known as "equity risk premium".

CAPM gives us the expected return of shareholders from a stock based on its systematic risk. We think of this as investors' required return from the stock as well as the company's cost of equity.

This is because a high required return means a lower stock price, which makes it costlier for the company to raise money by issuing stock.

Market equilibrium:

Treynor Ratio of Stock A = Treynor Ratio of Stock B = Treynor Ratio of all stocks

Expected return estimates provided by CAPM are used in three main applications:

Valuation, Capital budgeting, Performance evaluation

Capital budgeting:

WACC is also a critical component of capital budgeting and project choice as we learned in Chapter 5.

Method 2)

We already know how to calculate the portfolio beta as the weighted average of individual betas. We can plug the portfolio beta in CAPM to find the portfolio expected return:

Recall that it is quite difficult to calculate the volatility of portfolios based on volatility of the individual securities in the portfolio.

We had trouble with even two securities and we didn't even attempt to tackle volatility of larger portfolios.

CAUTION: It is common to confuse expected risk premium and expected return.

When we say "expected risk premium", we are referring to the difference between expected return and the risk-free rate.

Alpha=

actual return-CAPM expected return

In any case, all that's to say, below we will assume that we are looking at a well-functioning market with reasonable prices

so that Expected Return = Required Return.

It is a relative model, in the sense that it gives us the expected premium from an investment based on

some multiple (beta) of the expected market premium.

There is a very subtle difference between the two. As we will discuss briefly under the market efficiency topic, if a financial market is functioning well (financial securities are priced reasonably, reflecting their true values)

the two are the same thing: market's required return = expected return from the investment.

We can compute portfolio expected returns in

two ways.

Portfolio beta is simply the weighted average of individual security betas

where the weights are portfolio weights.

portfolio beta

ßp: Portfolio beta wi: Portfolio weight of stock i ßi: Beta of stock i

Why is CAPM called the capital asset pricing model?

-Model: In science, a model is a simplified representation of the real world in the form of an equation. It may not be perfect, but it is still useful. -Capital asset: Any asset used to make money, as opposed to assets used for personal enjoyment or consumption. Includes stocks and bonds, but also real estate, and even collectibles and art. Therefore, in theory, this model can be applied much more generally than just stocks. But in practice, its primary use is with stocks. -Pricing model: The reason it is called a pricing model is that its output, expected returns, is taken as investors' required return and used as the discount rate in DCF stock valuation (pricing stocks) and capital budgeting (pricing projects).

Equation 6.19: WACC

-rE: Cost of equity, typically the required return from CAPM -rD: Pre-tax cost of debt, typically the average interest rate the company pays on all its debt -rD(1-T): After-tax cost of debt, equal to pre-tax cost of debt times (1 - tax rate) -E/(D+E): The weight of equity in the company's capital structure. -D/(D+E): The weight of debt in the company's capital structure.

There are three inputs in the CAPM:

1. The risk-free rate 2. Beta 3. Expected market risk premium (EMRP).

Option 2 (The better approach):

Compare performance against CAPM expected return

Recall that when we covered the risk-return tradeoff earlier, we came up with the Sharpe Ratio. Back then we were still working with standard deviation as our risk measure.

Now that we are focusing on systematic risk rather than total volatility, let's revisit the risk - return trade-off subject.

The Treynor Ratio is the "systematic risk" version of the Sharpe Ratio, because it scales excess returns by beta instead of standard deviation.

So, it is an alternative measure of risk-adjusted returns, where risk is only systematic risk and is measured by beta.

Example 6.30: Consider a stock that generated a 10% annual return on average when the average risk-free rate was 2.5%. The stock has a beta of 1.25. What was its Treynor Ratio?

Solution: Using Equation 6.16, Treynor Ratio = (0.10 - 0.025) / 1.25 = 0.075 / 1.25 = 0.06.

Finance is not just about analyzing numbers.

Sometimes we are called to use our judgment in order to make wise financial decisions.

Generalizing this equilibrium to all stocks:

Treynor Ratio of all risky securities should be identical and equal to the Treynor Ratio of the market.

Beta measures systematic risk,

and it is therefore the correct risk measure for diversified investors.

E(r)-rf

expected risk premium

Now, we are going to take a leap of faith and claim the following:

in equilibrium all stocks should offer the same risk-reward tradeoff (i.e., the same Treynor Ratio).

So, when you need to find a company's cost of equity,

use CAPM to find shareholder's expected return. The two are the same.

-Initially, Treynor Ratio of Stock A > Treynor Ratio of Stock B. -Step 1: Investors buy the better stock (Stock A with higher Treynor Ratio) and sell (or don't buy) the worse stock (Stock B with lower Treynor Ratio). -Step 2: Investor demand increases the price of Stock A and reduces the price of Stock B.

-Step 3: The price increase reduces the expected return and therefore the Treynor Ratio of Stock A. The price decline increases the expected return and therefore the Treynor Ratio of Stock B. -Step 4: Market reaches an equilibrium when the two stocks offer an identical risk-reward tradeoff. The two stocks are now equally attractive from a risk-reward perspective.

Key Implications of CAPM: According to the CAPM:

1. Investors require a risk premium from investments, equalling the beta of the investment times the expected market risk premium. 2. Investors' required return equals the risk-free rate plus their expected risk premium from the investment 3. Investors will require higher risk premiums (and therefore returns) from stocks with high beta, and vice versa 4. Required risk premiums (and therefore returns) will be greater than those of the market's for stocks with a beta greater than one, and vice versa. 5. If the market is efficient (and CAPM valid), all investments will be priced such that expected returns from the investment will equal investors' required return from that investment given by the CAPM.

Determinants of EMRP:

1. The riskiness of the entire market/economy: If investors begin to worry that aggregate risk is increasing, the required premium for holding risky investments and thus EMRP will increase. 2. The risk-tolerance of investors in the market: If investors become more risk-averse, they will demand a a greater compensation to hold risky investments. Thus, EMRP will increase. This happens during crises and economic recessions, as people become increasingly fearful of further financial losses and their risk-tolerance declines.

Surveys:

A logical forward-looking approach is to ask people (investors, managers, academics, etc.) what their expected market return or risk premium is. For example, Duke University has been conducting a Business Outlook Survey of Chief Financial Officers on a quarterly basis, which includes a question about the CFOs' expected long-term stock market performance. According to the March 2023 survey, U.S. CFOs expect S&P 500 to earn an average annual return of 8.4% per year over the next 10 years. As of the end of March 2023, the 10-year Treasury yield was 3.48%. These figures imply an EMRP of 8.4% - 3.5% = 4.9% based on the survey responses.

Important question 1: Why are we going to relate required returns to betas (measure of systematic risk), and not to volatility (measure of total risk)?

Answer: Because, total risk is irrelevant for diversified investors!

As we learned, diversified investors should only care about their investments' systematic risk.

Beta measures only systematic risk and completely ignores the investment's unsystematic risk since it wouldn't matter within a diversified portfolio.

Ex: you have 30% of your investment in a stock with a beta of 1.6, 25% on an index fund that tracks S&P500 and the rest in Tbills. Expected market return is 10% and the risk free rate is 4%. What are the beta and expected return of your portfolio

Bp=0.3x1.6+0.25x1+ 0.45x0=0.73 E(rp)= 4%+0.73[10%-4%]= 8.83%

Note: Based on the EMRP estimates from the three methods outlined above, we will be using an EMRP of 5 - 6% for most examples going forward.

But keep in mind that the estimate varies over time and this value may not be valid in the future.

If, on the other hand, you are looking at a market that is not functioning well (think for example of the stock market in an emerging country, or a wacky cryptocurrency), there may be a discrepancy between what return you'd require in order to invest vs. what return you'd expect to earn.

For example, you may analyze a stock and think to yourself you'd require a 15% annual return in order to consider investing in it, but based on its current valuation and future prospects you consider it overvalued and estimate an expected return of -3%. In this case, you'd not invest in it since you expect it to earn less than what you require.

Performance evaluation:

How do we decide whether a professional money manager has performed well or not? We've learned that riskier investments have higher required returns, so we must take this into account when evaluating returns from risky investments. CAPM offers a practical way to do this.

Method 1)

If we know the expected returns of the individual securities, we can calculate portfolio expected return as the weighted average of individual expected returns. Recall Equation 6.11 for portfolio return: So, method 1 requires us to estimate the portfolio expected return as the weighted average of individual expected returns, which are estimated first using CAPM.

EMRP is a key input for the CAPM, but it is not a number that we can observe.

It is a number that needs to be estimated based on one's beliefs and expectations about the future. What EMRP should we use when applying the CAPM?

Implied EMRP:

It is possible to back out the EMRP from observed prices (e.g., the level of the S&P 500) and analysts' market-level cash flow forecasts using a discounted cash flow valuation model. As of March 2023, Damodaran's EMRP estimate was around 4.9%, ranging between 4.6% and 5.7% depending on the estimation methodology used.

As we saw above in Equation 6.15, portfolio beta is the weighted average of the individual betas. Therefore, adding a relatively safe investment (beta less than portfolio beta) in a portfolio reduces portfolio risk and adding a risky investment (beta greater than portfolio beta) adds portfolio risk

Later, we will learn that this is why risky investments have higher required returns by investors; adding a high beta asset increases portfolio risk, which requires extra compensation in the form of a greater required return.

Important question: Was 13.58% per year over the last 10 years good enough? It depends. We cannot evaluate investment performance in isolation. We need to consider how comparable options performed during the same time period. We also need to consider the amount of risk the fund exposed its investors to. 13.58% may be great performance if the fund makes fairly safe investments, but not so good if the fund tends to invest in very risky securities.

Option 1 (Not great, but better than nothing): Compare performance against a reasonable benchmark. For example, if the S&P 500 was up only 8% per year during that time period, 13.58% looks great. In contrast, if the S&P 500 was up 16% per year, 13.58% looks relatively bad. You can improve on Option 1 by choosing the most appropriate benchmark possible. For example, a better benchmark for our SC Growth Fund may be the S&P 500 Growth index, which invests in a subset of S&P 500 companies with growth characteristics (e.g., a high P/E ratio), instead of using the entire S&P 500 as the benchmark. Nevertheless, Option 1 is not great because it ignores possible differences in the riskiness of the SC Growth Fund vs. the benchmark you are using for performance comparison.

Yahoo Finance estimates betas based on monthly returns going back 5 years.

Other platforms may use different frequencies (e.g., daily or weekly returns) or different time windows (e.g., past year or last three years). This may lead to discrepancies in beta estimates for the same company.

Equation 6.16: Treynor Ratio

Recall that Sharpe Ratio= r-rf/o The Sharpe Ratio gives you excess return (over the risk-free rate) per unit of total risk (systematic and unsystematic). The Treynor Ratio gives you excess return per unit of systematic risk, which is what matters for diversified investors.

If the market is in equilibrium and CAPM is valid, all stocks must have the same Treynor Ratio, which equals the market's Treynor Ratio.

Since the beta of the market equals one by construction, this universal Treynor ratio equals the expected market risk premium (EMRP).

Furthermore, note that these historical estimates are based on past performance. But recall what we are trying to do here. We are trying to estimate a stock's expected return in the future.

So what really matters for us is the stock's future beta. Naturally, future beta is not observable, which is why we are relying on historical estimates. But keep in mind that such historical analyses can produce strange estimates that should not be extrapolated to the future. So, if a beta estimate doesn't make sense, it may be necessary to use your judgement rather than rely solely on the historical estimate.

Example 6.39: Suppose you are analyzing a company with the following characteristics. If the risk-free rate is 2% and the expected market risk premium is 5%, what is this company's WACC? Market value of debt: $160 million Market price of each share: $32 Number of shares outstanding: 20 million Tax rate: 20% The average yield on current debt (pre-tax cost of debt): 3.5% Beta: 0.8

Solution: Let's first calculate the cost of equity. Using CAPM, rE = rf + β [E(rM) - rf] = 2% + 0.8 x 5% = 6% Market value of equity = Share price x # of shares outstanding = $32 x 20 million = $640 million The capital structure weights: Weight of equity = E/(D+E) = $640 million / ($640 million + $160 million) = 80% Weight of debt = D/(D+E) = $160 million / ($640 million + $160 million) = 20% Let's plug everything in the WACC formula: Using Eq. 6.18: WACC = 6% x 80% + 3.5% x (1-0.2) x 20% = 4.8% + 0.56% = 5.36%

Example 6.38: A company has a beta of 1.2, pre-tax cost of debt of 6% and an effective corporate tax rate of 25%. 30% of its capital structure is debt and the rest is equity. The current risk-free rate is 2% and the expected market return is 7%. What is this company's weighted average cost of capital?

Solution: Let's first calculate the cost of equity. Using CAPM, rE = rf + β [E(rM) - rf] = 2% + 1.2[7% - 2%] = 8% The capital structure weight of debt is 30% (meaning that 30% of new financing is expected to come from new debt). The rest (100% - 30% = 70%) is equity. Using Eq. 6.19: WACC = 8% x 70% + 6% x (1-0.25) x 30% = 5.6% + 1.35% = 6.95%

Example 6.34: Consider a stock with an expected return of 9.2%. What must be this stock's beta if the risk-free rate is 1.2% and the expected market return is 6.4%? Assume CAPM is valid.

Solution: Here, the unknown is the stock's beta. Let's plug the parameters in the CAPM model and then solve for the unknown: Using Equation 6.17: 9.2% = 1.2% + beta x [6.4% - 1.2%] = 1.2% + beta x 5.2% Let's rearrange to leave beta alone: beta = (9.2% - 1.2%) / 5.2% = 1.54

Example 6.37: SC Growth Fund earned annual average returns of 13.58% over the last 10 years. During that time, the average risk-free rate was 2% and the average market return was 14%. If SC Growth Index had a portfolio beta of 1.1, what was its annual alpha?

Solution: Let's first compute the expected annual return of SC Growth Fund during the last 10 years based on CAPM. Using Equation 6.17: E(r) = 2% + 1.1[14% - 2%] = 2% + 13.2% = 15.2% Using Equation 6.18: Alpha = 13.6% - 15.2% = -1.6% SC Growth Fund had a negative alpha of 1.6% per year, meaning that it underperformed its CAPM expected return by that amount each year.

Example 6.32: A stock has a beta of 1.6. Suppose the expected market risk premium is 5.5% and the risk-free rate is 2%. What is this stock's expected return according to the CAPM?

Solution: Notice that the question provides the expected market risk premium, which is expected market return minus the risk-free rate. Using Equation 6.17, E(r) = 2% + 1.6[5.5%] = 10.8%.

Example 6.31: A stock has a beta of 1.4. Suppose the expected market return is 8% and the risk-free rate is 2%. What is this stock's expected return according to the CAPM?

Solution: Using Equation 6.17, E(r) = 2% + 1.4[8% - 2%] = 10.4%. Interpretation: Expected market return is 8% and expected return from this stock is 10.4%. Why is the stock's expected return higher? Because of its high beta of 1.4. A beta higher than one means expected return greater than the market expected return.

This is a more sensible proposition than may appear at first glance.

Suppose there are two stocks in the market with different Treynor Ratios, such that the Treynor Ratio of Stock A is higher than the Treynor Ratio of Stock B. What would rational investors do?

Volatility reflects the total risk of investments.

That's fine as a risk measure if you are considering an investment in isolation, but not within a diversified portfolio.

For example, if investors require an 11% return from a stock during the following year based on market conditions and the stock's risk, we think of that as also investors' expected return from that stock over the next year.

The actual return will differ from expected return, of course. It may be much higher or lower due to unforeseeable events and surprises.

Why after-tax cost of debt?

The after-tax cost of debt is what is included in WACC because interest paid by corporations is tax deductible in the U.S. This deduction lowers corporations' tax bill and increases their net income. As a result, the effective cost of debt of the company is lower than the interest rate it pays its lenders, and WACC reflects that by including the after-tax cost of debt.

Interpretation: In both cases, a higher ratio means more value per unit of risk.

The difference is in how they each define risk.

Valuation:

The expected return from CAPM is a proxy for investors' required return, which is used in weighted average cost of capital (WACC) calculations as the company's cost of equity. WACC is then used as the discount rate in DCF stock valuation.

Historical premiums:

The most common approach in estimating the EMRP is looking backward. The implied assumption behind this approach is that the future will resemble the past in terms of investors' required returns to make risky investments. The market return in the U.S. (S&P 500 including dividends) has averaged 11.5% per year between 1928 and 2022. During that time, the average annual inflation rate was 3.1% and the average return on 10-year T-bonds was 4.9%. So, the stock market investments generated a real return (return over inflation) of 11.5% - 3.1% = 8.4% and annual premium (return over the long-term risk-free rate) of 11.5% - 4.9% = 6.6% during the past 95 years [Data source: Damodaran's website]. So, the market risk premium has been 6.6% per year based on the long-term historical average since 1928

What are the capital structure weights?

The weight of debt and equity in the company's capital structure can be computed based on either book values from the company's balance sheet or market values. The market values approach is generally considered better as it is forward looking while the balance sheet approach is backward looking.

Note: Recall that Sharpe Ratio uses return volatility as the risk measure, which is inappropriate if you are evaluating an investment with considerable diversifiable risk.

Therefore, Sharpe Ratio is only used to evaluate risk-adjusted performance of highly diversified portfolios, such as the holdings of a mutual fund. CAPM alpha and Treynor Ratio do not suffer from this problem since those two use beta as the risk measure.

The market has beta of one (𝛃M = 1)

This is because the market moves one-to-one with the market, by definition (You can also see this mathematically by replacing the i's with M in Equation 6.11). Think of beta of 1 as your numerical anchor. A stock with a beta of one moves very closely with the market. The average stock beta is also one.

What this lower beta tells us is that, evidently (and not surprisingly) Walmart's business and therefore its stock returns are not nearly as sensitive to the US economic conditions (and therefore the aggregate stock market returns) as Citigroup's are.

This makes Walmart's stock a much safer investment within a highly diversified portfolio compared to Citigroup.

Beta = 0 means risk-free from a systematic risk perspective (𝛃f = 0)

This may be either because the standard deviation of the security's returns is zero, or its correlation with the market is zero.

E(rm)

expected market return

E(rm)-rf

expected market risk premium

E(r)=

expected return

The CAPM provides the expected return of an investment based on its beta,

expected return from the market, and the risk-free rate.

Instead of required returns, we will follow the lead of the academic literature and almost all finance textbooks and start talking about

expected returns from an investment.

Now, recall that the market beta equals one (𝛃M = 1) and

rearrange the equation to leave E(ri) alone on the left-hand side. This gives us the Capital Asset Pricing Model, CAPM.

Within a diversified portfolio (at least 30-40 stocks), diversifiable (unsystematic, idiosyncratic, etc.)

risk is eliminated, only non-diversifiable (systematic or market) risk remains.

The weighted average cost of capital (WACC) is the rate of return that a company is expected to earn for its investors

the average cost of all sources of funding for a company. A company's capital comes from debt or equity; this formula takes a weighted average of these sources of capital.

Voila! This is the discount rate ("i") for a company's future cash flow,

used in DCF stock valuation and capital budgeting.

As we'll soon learn, cost of equity from CAPM is used to calculate the company's weighted average cost of capital (WACC)

which is then used by investors in DCF stock valuation and by corporate managers in capital budgeting.