### @antoineantoine

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## Should we choose a model that gives a flawed but mostly usable coefficient, or should we choose a model that may give a very good or at times a very bad estimate depending on the nature of the data?

This is the question when choosing between the CAPM, or the Black Scholes for your discount rate in discounted cash flow valuations.

Let’s compare both models and see which is better, or is it a matter of philosophy in finance.

Risk is a combination of downside (danger) and upside (opportunity). Every risk & return model in finance is about measuring the danger in investment and asking the question: how many opportunities would I need in order to compensate me for taking that danger.

## What Does a Good Risk & Return Model Allow Us:

It should come up with a measure of risk that is universal – It has to apply for all assets (not just for stocks). Because in comparing opportunities for investment we need to make comparisons based on consistent measurements.

It should specify what kinds of risks you should get rewarded for – and those you shouldn’t get rewarded for (Market-specific risk vs company-specific risk).

It should give you a standardized measure of risk. This allows us to see if we are dealing with above average or below average risk – and by how much.

It should be able to convert the risk measure into the desired return (%) which we can use as a hurdle rate – a threshold that shows us if an investment is profitable.

It should work. The risk of the past should correlate with future returns.

“The alternative models (which are richer) do a much better job than theCAPMin explaining past return, but their effectiveness drops off when it comes to estimating expected future returns.”A. Damodaran

## Option Pricing Models

When we think of options, we are dealing with derivatives – they derive their value based on something else. Options have limited lives. What makes options limited is that they have a contingency that determines the payoff.

In real options terms, we are not paying for what the cash flows are today, but rather what the cash flows could be tomorrow.

Real options may not always be appropriate. We can check by answering these three questions:

*When is there a real option embedded in a decision or asset? *

Because if there is no actual option, we shouldn’t be bringing out the option pricing model.

*When does the option have significant economic value? – Should we even bother?*

Can we estimate that value using an option-pricing model? – Most options do not have economic value, and even less can be assessed using an option pricing model.

Keep in mind that a core feature of real options is exclusivity – We are only dealing with options if we retain the exclusive option to do something. The more exclusivity we have, the greater the economic value.

The only macro-economic variable that plays a role in options is the risk-free rate. The higher the risk-free rate, the lower the value for put options and the higher the value for call options. Because, if interest rates are high, the present value of what you can buy at a fixed price in the future becomes lower.

Introduction – Black Scholes

The Black-Scholes-Merton (BSM) model, is a mathematical model simulating the dynamics of a financial market containing derivative financial instruments such as options, futures, forwards and swaps. Black and Scholes showed that “it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock”

It assumes these instruments (such as stocks or futures) will have a lognormal distribution of prices.

The model estimates the variation over time of financial instruments.

BSM is a differential equation used to solve for options prices. Standard BSM is used to price EU (type – not geography) options and does not take into account that US options can be exercised before their expiration date.

**The Model Incorporates:**

– The constant price variation of the stock.

– The time value of money.

– The strike price.

– The time to the contract’s expiry.

**It is Used to Calculate the Theoretical Value of Options Using:**

– Current stock prices.

– Expected dividends.

– The option’s strike price.

– Expected interest rates.

– Time to expiration.

– Expected volatility.

**BSM Makes the Following Assumptions:**

– EU option, that can be exercised at expiration.

– No dividends are paid during the life of the option.

– Markets are efficient (market movements cannot be predicted).

– No transaction costs in buying the option.

– The risk-free rate and volatility of the underlying asset are known and constant.

– The returns on the underlying asset are normally distributed.

Already we are seeing that the BSM, has certain limitations and may only be appropriate for valuing a certain subtype of assets that follow a string of assumptions and behave like European type options.

That is why the type of assets that come closest to being appropriate for using the BSM are those that have payoffs that are contingent on something else happening. This could be a discrete risk event. A discrete event example is whether a biotech company gets a drug approved by a health regulatory agency. If this happens the price of the stock behaves like an option, and suddenly has a lot of potential value to be exercised – if not, then at best it will stay around the same.

When considering the BSM, it is good to keep things into perspective, and start with an initial question, “* is there a simpler way to value risks – such as a binomial model or a simple decision tree*”.

The BSM tends to attract purists and is often used as an intimidation tactic to sell a valuation as being of superior quality. But if in assets that do not behave like options we can use a model that is simpler and works just as well, then we should ask ourselves, is the complexity-quality tradeoff positive?

**Introduction – Capital Asset Pricing Model**

The CAPM is the first risk-return model to enter the stage, and we will evaluate its performance and compare it to the newer models.

**What it Does:**

Measures risk using the variance of actual returns against expected returns. It is a mean-variance based framework.

The CAPM measures all non-diversifiable risk. It specifies that only the portion of variance that cannot be diversified away will be rewarded.

It uses a Beta measurement, which is standardized around 1. This measurement is relative to the general market. However, Betas work only if you (the marginal investor) are diversified.

In banks, consulting firms and valuations, the CAPM remains the core model that people go back to even though it has drawbacks that repulse a lot of analysts.

It takes the beta and produces an expected return by using the formula:

*Expected Return = Risk-free rate + Beta * Risk Premium*

**Breaking Down the CAPM:**

A risk-free rate. The guaranteed return rate of investment – has no default, and no reinvestment risk.

A beta for the company. Risk volatility relative to the comparative market.

An equity risk premium. It is what you demand over and above the risk-free rate to invest in equity.

** Result:** The required return on an investment will be a linear function of its beta.

An interesting aspect of the CAPM is that it does not account for upside skewness, which is something investors seem to be quite fond of.

The CAPM assumes no transaction costs and no private information, which means you shouldn’t pick individual stocks to beat the market, and instead should focus on being a diversified investor. This is a false choice because in practice you can both pick stocks and be diversified.

In terms of diversification tactics, there is a transaction cost on the number of firms we include in our portfolio when attempting to beat the market. Since there are both transaction costs for stocks, and picking stocks is based on information, there should be a cutoff point in regards to the number of stocks an investor should have in a portfolio if an investor wants to retain the benefits of picking stocks.

**Limitations of the CAPM**

The model makes unrealistic assumptions.

This is important, but it does not kill the model. It is a realistic model that makes some unrealistic assumptions.

The parameters of the model cannot be estimated precisely.

The market index used can be wrong.

The firm’s beta may have changed during the estimation.

Having said that, there are multiple estimates other than the beta that we could have gotten wrong (Revenues, growth, margins etc).

The model does not work well.

**If the model is right, there should be:**

A linear relationship between returns and betas.

The only variable that should explain returns is the beta.

**In Reality:**

The relationship between betas and returns is weak (CCA 7%).

Other variables (Size, price/book value – EV/invested capital, etc…) seem to explain differences in returns better.

**So, let’s look at the practicality of the CAPM:**

You can value the equity of the business or the whole business.

When you value the equity of the business you are looking at the cash flows after debt and you discount them back at the rate of return that equity investors demand – Which is the cost of equity.

If you use cash flows to equity, use the cost of equity.

The other approach is to value the whole business. In this case, you take the cash flows before debt payments and you discount them back at a weighted average cost of capital. With this, you get the value of the business, and you subtract debt to get the value of equity.

If you have cash flows to capital use the cost of capital.

Discount rates reflect the risk of an investment. Higher risk investments have higher discount rates.

If we choose to do a valuation in real-terms, meaning you take inflation out when projecting numbers you don’t get the benefit of inflation. Then your discount rate has to be a real(terms) discount rate.

Then we get to the risk rate, which should reflect the risk perceived by the marginal investor. If done properly our discount rate will be risk-adjusted as seen by marginal investors.

This is precisely where the CAPM enters, and where analysts have the most reservations.

**Arguments against using betas – as part of the CAPM:**

Intrinsic valuation does not trust the market, while on the other hand beta is reflecting a regression against market prices. Thus, we are using a price-based measure of risk, within an intrinsic value model.

Beta assumes a diversified investor. They do not work well with concentrated portfolios.

**Alternatives to market price betas are:**

If you do not use beta as it is a price-based measure, you may want to substitute with other measures of risk within the CAPM:

Accounting betas. Where you use changes in earnings instead of changes in stock prices.

You can also look at the cost of debt and scale it up.

If you do not believe in diversified investors, you can focus this all on just the company you are analyzing – don’t use the regression.

We do not need a beta as our risk measure. We can tailor risk, specific to the company. We could use measures such as the number of clients, variability in revenue, etc.

**Alternatives such as the standard deviation instead of the beta work in the following manner:**

Take the SD of your stock.

Take the SD off the market index.

** Divide the target SD by the market SD**.

A noted characteristic of beta is that the more discretionary the product or service, the higher the beta. A discretionary product is something people can delay/defer buying, can live without. For example luxury stores vs. grocery stores.

Businesses with higher fixed costs have higher betas since fixed costs have to be paid in good time.

When you borrow the money you increase the beta for your equity because you are creating a fixed cost – new interest payments.

**Hamada Beta** **–** Reflects the proposition that the true value of beta is a function of the industry in which the company operates (% of revenues derived from different industries), and the additional fixed cost of interest payments on debt.

For developers you can get Apple Inc. beta from API calls using stocks API.

**We can compute this beta in the following manner:**

Take the average beta of the multiple companies in the same sector to get your unlevered beta. Check for debt and if there is debt compute the debt/equity ratio and add it on to the unlevered beta.

A quick History of Risk-Return models:

– CAPM 1964 – Capital Asset Pricing Model, as discussed above.

– APM – Arbitrage Pricing Model.

Why measure all of the risks in stock with one beta? The APM is still measuring risk that cannot be diversified away, but the risk can come from interest rates, inflation, real economic growth.

Uses a factor analysis to assess risk and gets multiple betas.

**Drawback** – We do not know the nature and name of the factors.

**MFM** – Multifactor model.

Same as the APM, but the factors **have been identified**.

Drawback – Is backward-looking, and the past factors are **not relevant for future changes**.

**Proxy models –** Use historical data to compare differences across the performance. The market performance is compared to the characteristics of individual indicators or firms. Low price to book ratio and a small(er) market capitalization company is shown to yield higher returns.

Proxy models allow something else to stand in for risk.

In Conclusion

There is no reason to give up on a valuation if we do not trust the current beta since we can use all kinds of substitutes.

When comparing Black Scholes vs. CAPM, we should use the Black Scholes when the firm has characteristics of an option – in the rest of the cases using the Black Scholes is an overshoot since the CAPM is simpler and works at least as well as the rest.

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