Finance Theory: Tangency Portfolio (Part 4)
Welcome to the fourth article of our finance theory series, focused on investment. In the previous article, we talked about diversification. We broke down the volatility of an asset or portfolio into two parts. One part, called specific or idiosyncratic risk, which is easily eliminated (diversified) by adding more assets to a portfolio. Another part, called market or systematic risk, which is not diversifiable. Investors should be paid a premium for bearing market risk only. In this article, we’ll dive deeper into the meaning of the Tangency portfolio.
Tangency Portfolio
We’ve discussed the significance of the Tangency portfolio and why it’s the preferred choice for rational investors, but some important questions remain: What is the Tangency portfolio in practice? Can we determine what composition of assets it contains, so we know what to invest in? To answer these questions, we’ll examine market dynamics through a simplified scenario.
Let’s assume the market only has two companies, company A and company B. We are going to use specific notation and a few formulas, please don’t get caught up on the details. The most important thing is to understand the insights from the analyzes.
First, we define the share prices for Company A and Company B as SA and SB respectively. The number of outstanding shares for each company is denoted as NA and NB. Now we can calculate the market value (market capitalization) of each company by multiplying the share price by the number of outstanding shares.
Now we have VA as the market value for company A and VB as the market value for company B. The value of the market portfolio can be defined as the total market capitalization of all stocks. Since we only have two companies in this economy, we can say that the value of the market portfolio (Vmkt) can be expressed as the sum of VA and VB.
We can also compute the return of the market portfolio over one holding period, from t0 to t1.
The total market value at t1 is the sum of the market cap of company A at t1 and the market cap of company B at t1. Market cap is calculated by multiplying the number of outstanding shares by the share price at that specific period.
We can also write the total market value t1 as the product of the total market value at t0 times one plus the rate of return on the market over this holding period. The same is true for calculating what the share price of each company is at t1. We can expand it in terms of share price of the company at t0 times one plus the rate of return of each specific company during the holding period.
With a bit of algebra, we can isolate the return on the market, revealing that it’s a weighted average of the returns of all stocks in the market. Each stock’s weight is determined by dividing its market capitalization by the total market value. This insight is fundamental: the market portfolio is a value-weighted portfolio of all stocks in the market.
Now, let’s introduce investors into this simplified market scenario. Suppose there are only three investors, each holding a portion of both Company A and Company B in their portfolios, resulting in a total portfolio value as shown in the table below:
We know that the value of the investment for each investor is the sum of their investment in company A and the investment in company B. These numbers can be calculated by multiplying the number of shares they have in each specific company by the stock price of that company.
If we assume market equilibrium, meaning investors have transacted as they wished and supply equals demand, there is nothing left over to buy or sell in this market. We can say that the total number of shares for company A is equal to the sum of shares that each individual investor has for company A. The same is true for company B. And we can also say that the total value of the market equals to the sum of each individual investment in this market.
Now suppose that investors are rational and want to invest in the Tangency portfolio, which is the most efficient portfolio of risky assets, having the highest Sharpe ratio and giving the highest return per unit of volatility. We can write three equations to represent each individual investment as a percentage of their investment in the overall tangency portfolio. In the following formulas, lambda (λ) represents the percentage:
We can also add these three equations. The result shows that the sum of all the individual investments equals to the sum of the percentage of their investments in the Tangency portfolio multiplied by the overall value of the Tangency portfolio.
We know that all the lambdas must add up to one, because investors are rational and they invest in the Tangency portfolio. This brings us to our final result, where we see that the sum of each individual investment is equal to the Tangency portfolio. Since we know that each individual investment is also equal to the market portfolio, we conclude that in economic equilibrium the Tangency portfolio should be equal to a value weighted portfolio of all assets in the market.
Our simplified two-company, three-investor scenario has helped us grasp the essence of the Tangency portfolio. This explains why proponents of passive investing advocate for market index investments, arguing that low-cost index Exchange-Traded Funds (ETFs) often outperform actively managed funds. Even renowned investor Warren Buffett made a $1 million bet in 2007, predicting that the S&P 500 would outperform a basket of hedge funds selected by Protégé Partners over a decade. He won the bet convincingly, with the S&P 500 returning 7.1% compounded annually compared to the basket’s 2.2%.
There is a strong case for passive investment and it’s one of the reasons why we’re seeing a massive migration of money from actively manage funds to low-cost ETFs. However, not all indexes are value weighted indexes. Some indexes split the weights equally by stock which makes for a different portfolio composition than the one for the Tangency portfolio. Make sure you understand how the indexes are calculated before you invest in them.
Another important reality check: the whole market is unobservable! It should include every asset in the economy, such as stocks, bonds, real estate, options, human capital, etc. We don’t have a perfect proxy for the overall market. Most people use approximations of the whole market. Some of the most widely used proxies are the S&P 500 index or a total US equity market index or even a total world equity index.
What’s Next
In the fifth article of this series, we’ll take a closer look at asset pricing models, including the Capital Asset Pricing Model (CAPM) and other multi-risk factor models. 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.
