Beta in the Context of the Capital Asset Pricing Model

Before we start this article, it is important to understand the foundations of the capital asset pricing model. So let’s talk about beta in the context of the CAPM for a bit.

The risk-free rate is what you get from a treasury bill or similar instrument; it is known and guaranteed so there’s no risk, but it doesn’t earn much either. The risky asset is typically some sort of equity, as issued by a company on a stock market like that in New York or Tokyo. The lower the beta, the less risky that is on a daily basis. The higher the beta, the more risky it is. When you buy a stock, someone else has already bought it and might sell it back to you with no risk, but at a higher price. If you can borrow against other people’s stocks to buy them and they don’t default on their promises, that’s known as "asset leveraging." This can happen with bonds too but historically this was more common with stocks. Asset-backed securities are one of the ways that has been used to leverage investments other than simply buying a stock outright.

The capital asset pricing model assumes that different companies have the same beta for some reason, so the returns from one are directly proportional to those from another. The beta is known to be almost constant over time, so it’s assumed to be constant as well.

Beta is also known as alpha relative risk or as market-to-book risk, and has a couple of other names. It’s called something different in other models like the value at risk model, but whatever you call it, this is what it means. At any given moment in time there’s a single beta or something close to it. The CAPM assumes it’s constant over time, and it’s not really important in how the model works.

If you want to measure this risk, you can compare the returns to those from a benchmark and let that tell you something about how risky that investment is - but that’s not perfect either and it only tells you part of the story.

The capital asset pricing model has been refined over time. Originally, it was a simplified way of giving some sense as to whether or not an investment was worth while based on its past performance alone. However, as we will see below, this is bad practice in today’s environment and we should look at other factors as well.

Using the capital asset pricing model to estimate the value of assets is actually one of the more acceptable uses for it. This is because:

The risk-free rate is known and guaranteed, so there is no risk The asset in question has a beta, which can be calculated from historical returns or other measures It also has a beta as part of another model that can be estimated from historic data That other model assumes that the risk-free rate is constant over time All assets have similar ability to earn market returns, based on their beta What that looks like over time depends on many factors - such as inflation, growth rates in certain industries and parts of the economy, GDP levels, etc.

I’m not going to try and prove it or anything; you can read papers and books on the topic if you want to know more. However, I will give a few examples using Excel. This is not really necessary, but it gives you an idea of how this works in practice.

We’re going to start with a few assumptions here:

You want to invest in risk-free T-Bills at 3% per annum The stocks you’re considering all have similar betas, so their performance is highly correlated to each other You want to hold all your shares for a year and then sell them, not thinking about them again for one year.

Let’s use this data:

I’m going to use the past 10 years, because it’s more likely to be accurate. I’ll also assume that all these stocks are going to stay at their current market prices for the whole time. If you wanted to buy them and hold them, you could create a portfolio with each of the three investments and then buy and sell at different times so that you don’t have to get into the math here.

So here are the results from this Excel spreadsheet:

This is good information for comparing different investments, but it doesn’t tell us very much about estimating value in all situations.

So let’s try another example. This time we have:

You have to assume we’re going to pay the same 3% interest on these bonds as we do on treasury bills. That’s fine, but the difference is that they’re 30 years in duration and they don’t pay much interest at all right now, so you have to assume something else as well. I think it makes sense to assume their current yield is the same as 3% and that you will be able to buy them for $100 at any point in that time - basically giving yourself an option for $3 of downside protection.

Here’s what happens if you create a portfolio that represents $10,000 of each:

So what can we learn from this? We know now that if we invest in some stocks and some bonds, the expected return is 14.3% based on everything above. If you want to get a higher return, there are two things you can do:

Average out the risk by adding different investments with different risk levels Change your time horizon by reducing your interest rate or increasing the time period

There’s also another factor here: volatility. We can reduce volatility by changing our time horizon; this keeps us from diving in and out too much.

Conclusion:

The capital asset pricing model is a useful tool for comparing different investments, but it’s not perfect. It can indeed help you decide between two investments with different betas and revenue streams. However, when you try to assume that the beta of an investment is constant over time, it’s going to give you a value above the true market value when in reality it might be going down in some cases. This can be particularly confusing because the 3% on treasury bills declines with inflation and interest rates - you have to look at all these factors. We need to look at every investment in every time period for what it really is and what is expected if we invest into that investment at that point in time.