Finance Theory: Risk and Return (Part 1)
Welcome to the first article of our finance theory series, focused on investment. My goal is to give an overview of financial research over the past few decades and demonstrate how it has shaped our way of understanding risk, return, and asset allocation decisions. While this series doesn’t replace a comprehensive finance course, I do hope that you get the key insights and principles, which will help you make better investment decisions.
Risk and Return
Let’s begin by delving into the concepts of risk and return. To grasp these ideas fully, it’s important to recognize a fundamental characteristic of investments: you put money down today and get repaid at a later date, ideally with a premium.
In a free market economy, money can flow freely through the economy and be put to use for its most needed. For example, I could delay the benefit of using my own money now and lend it to an entrepreneur who needs to buy machinery for his factory. On a later date, when this entrepreneur generates enough revenues, he can pay me back and reinvest in his business.
So, how do we measure how well an investment is doing? One way is to look at the profit or loss, which is just the difference between what you originally invested and what your investment is worth now.
While this profit and loss approach can be effective in certain situations, it is challenging to compare different investments, since it presents results in absolute dollar terms. An alternative way to measure return is by determining the proportion of future portfolio value relative to the initial investment in period 0, known as the gross simple return.
However, when most people talk about return, saying for example: “My portfolio yielded a 20% return over the past year.”, they are referring to the net simple return. The net simple return is a measurement of profit and loss as a percentage of the portfolio’s initial value.
These are all measures of return, but what about risk? To think about the risk component of an investment, we need to look at the payoff, which has two important characteristics. First is that it happens over a time period. And second is that it’s uncertain. This uncertainty over a time period defines the investment’s risk. In this article, we’ll discuss how we can quantify this risk.
Let’s illustrate this concept with an example. Consider a stock (S), with an initial value of $100 in period 0 (present). To simplify, let’s envision two states of the economy: an “upstate” where the stock’s value climbs to $125, and a “downstate” where it drops to $80. Since we cannot predict with certainty which state the economy will enter at time 1 (future), we classify this investment as risky.
On the other hand, let’s consider a bond (B), which starts at $1 in period 0 and, regardless of the economic state, rises to $1.5 in period 1. This investment is considered risk-free, because the outcome is certain in either state of the economy.
Now, let’s introduce the concept of arbitrage, a fundamental principle in understanding financial markets. Arbitrage entails generating profit seemingly out of thin air. How does this work? Suppose, at time 0, I borrow stock S from a friend and promptly sell it on the market, getting $100 in cash. This action, known as short selling, allows me to invest the $100 by purchasing 100 bonds of B. What do you think occurs in the upstate and downstate of the economy?
In the upstate of the economy, my bond investment is now worth $150, but I must repay my friend $125 for the borrowed stock. Consequently, I realize a $25 profit. In the downstate, my bond investment is also worth $150, while I only owe my friend $80, resulting in a $70 profit. What’s impressive with this strategy, is that it incurs no initial investment, yet generates a profit. We can express the portfolio created with the following formula:
The minus sign for stock S on the formula means we shorted stock S. The plus 100 times bonds B part, means we bought 100 bonds B. This portfolio, initiated at time 0, carries no initial cost and yields a profit in both economic states:
Arbitrage opportunities are rare in the market, but it’s a fundamental concept to ensure assets are fairly priced. When investors identify overvalued assets, they tend to sell them, thereby reducing their prices. Conversely, undervalued assets attract buyers, causing prices to rise. The dynamics of supply and demand is what helps maintain fair asset pricing in the market.
The simplified example we explored considered only two economic states. In reality, the financial world is more complex. For instance, consider the Standard and Poor’s (S&P) 500 price index from 1998 to 2018.
The graph looks like a roller coaster with lots of ups and downs. How can we make sense of this uncertainty? How can we quantify risk for this index? Instead of focusing on prices, we can examine the month-to-month returns on the price index and plot them over time.
The monthly returns graph is a better way to see the ups and downs in the market. Additionally, we can generate a histogram displaying the frequency of returns falling within specific ranges, as shown here:
If you’ve ever taken statistics class, you’ll recognize this graph’s shape as a bell curve, known as the normal distribution. And this distribution has an important characteristic: its probability can be determined by two factors, the mean and the standard deviation. The mean is the average of the data points. The standard deviation indicates the extent the data points are spread apart. Using the roller coaster analogy, the higher the standard deviation, the more thrilling the ride.
It’s important to note that the distribution’s shape may vary depending on the data and sample timeline. It could exhibit a slight leftward skew, potentially influenced by how people react in bear markets versus bull markets, as fear often triggers stronger responses than greed. Moreover, the distribution may have fatter tails, indicating a higher likelihood of rare, extreme events, such as substantial market declines, compared to what a normal distribution predicts. For our discussion, we’ll assume normal distributions with probabilities determined by mean and standard deviation.
What’s Next
In the second article of this series, we’ll apply our newfound understanding of risk and return to compare various assets. We’ll also explore the implications of these metrics on a portfolio. If you have any comments or feedback about this article, please reach out to us at feedback@bearnbull.com. Make sure to also visit BearNBull’s website, where you can find more resources to make you a better investor.
