[1] "NVDA" "GSPC"
Workshop 2, Investment Theory
Abstract
This is an INDIVIDUAL workshop. In this workshop we learn the Single-Index Model, the Capital Asset Pricing Model and methods to estimate the systematic risk of a financial instrument (its beta)
1 The Single-Index Model
The Single index model, also called the Market Model, is a factor model to better understand how an asset return is linearly related with an index return over time. This index is usually the market index, so we can learn how much an asset return is following or is related with the return of the market.
The single-index model states that the expected return of a stock is given by its alpha coefficient (also named beta0 or b0) plus its market beta coefficient (beta1 or b1) multiplied times the market return. In mathematical terms:
E[R_{i,t}] = α + β(R_{M,t})
Where:
E[R_{i,t}] is the expected return of a stock i in the period t
α, also called b_0 is a coefficient that represents the return of the stock when the market return is equal to zero β is a coefficient that represents the sensitivity of the stock return to changes in the market return R_{M,t} is the Market return in the period t
The single-index model is usually calculated using historical monthly returns of a stock and the market of at least 36 months. It is common to use 60 months (5 years) to estimate the model.
It is strongly recommended to use continuously compounded returns instead of simple returns to estimate the model.
A more accurate mathematical expression of the Single-index model is adding a random shock also called random error to model the exact return of the stock (instead of its expected return):
r_{(i,t)} = b_0 + b_1*r_{(M,t)} + ε_t Where:
ε_t is the error at time t. Thanks to the Central Limit Theorem, this error behaves like a Normal distributed random variable ∼ N(0, σ_ε); the error term ε_t is expected to have mean=0 and a specific standard deviation σ_ε (also called volatility).
r_{(i,t)} is the return of the stock i at time t.
r_{(M,t)} is the Market return at time t.
b_0 is a coefficient that represents the return of the stock when the market return is equal to zero
and b_1 is a coefficient that represents the sensitivity of the stock return to changes in the market return.
We can use daily or monthly returns to estimate this model. It is recommended to use monthly historical returns of 4 or 5 years of monthly data (if possible) to better capture the relationship between the market return and the stock return.
1.1 Systematic vs non-systematic risk
Looking at the single-index equation for the stock return we can interpret that the stock return depends on the market return, the sensitivity of the stock return to changes in the market return, the excess return over the market (measured by b_0), and the random shock ε_t.
The market return (the single-index return) R_{M,t} is a factor that significantly influences the return of the stock according to the stock b_1 coefficient.
The b_1 coefficient is the sensitivity of how much the stock return -on average- changes when the market return changes in +1%.
The term b_1 * R_{M,t} represents the systematic risk of the stock since the stock return will be significantly affected by the movements of the market returns in a systematic way according to its sensitivity b_1.
The term ε_t is a random shock with expected mean equal to zero and with a standard deviation equal to the volatility of the stock. This random shock represents the unsystematic risk or idiosyncratic risk that is not related to the market. Actually, the value of this error or shock for each period represents the impact of all news announced to the market on the stock return. If the investors perceive (on average) that the news (internal about the firm or external that might affect the firm) announced to the market is good for the firm, then this error shock will be positive; if not, it will be negative.
The term b_0 is a measure of excess return over the market. According to the market efficient hypothesis b_0 should not be significantly different than zero since the market portfolio is the most efficient instrument in the market. In other words, there should not be a stock that systematically offers returns over the market most of the time.
1.2 Example
We can visualize the meaning of b_0 and b_1 coefficients by looking the scatter plot of the market return vs a stock return. Let’s show the scatter plot for the firm NVDA:
As you see, I indicated that the Market returns goes in the X axis and NVDA returns in the Y axis.
In the single-index model, the independent variable is the market returns, while the dependent variable is the stock return
Sometimes graphs can be deceiving. Always check the Y sale and the X scale. In this case, the X goes from -0.10 to 0.10, while the Y scale goes from -0.20 to 0.40. Then, the real slope of the line should be steeper.
We can change the X scale so that both Y and X axis have similar ranges:
Now we see that the the market and stock returns have a similar scale. With this we can better appreciate their linear relationship.
WHAT DOES THE PLOT TELL YOU? BRIEFLY EXPLAIN
1.3 Estimation of the single-index model with real data
We need historical data of 4-5 years for the stock and the market index. It is recommended to use monthly returns instead of daily returns in order to better capture the systematic risk.
Then we need 2 variables: monthly returns for the stock of interest and monthly returns for the market index.
b_1 estimated as the ratio of the covariance of the variables to the variance of the independent variable. In this case, the independent variable is the market return. Then:
b_1=\frac{cov(r_m,r_i)}{var(r_m)}
If you estimate b_1 for a stock and you find that b_1=1.5, this can be interpreted as follows: For each +1% movement in the Market return, it is expected that the stock return moves (on average) about +1.5%. However, if the market return goes down in -1%, then the stock return will have a higher loss of about -1.5%!.
Then:
if b_1>1, then the stock will be riskier than the market.
if b_1<1, then the stock will be less risky than the market
if b_1=1, then the stock will have about the same risky than the market
That is the reason why b_1 is a measure of sensitivity and also a measure of market risk of a stock.
And b_0 can be estimated as the mean of the stock return minus b_1 times the mean of the market return:
b_0 = \bar{r_i} - b_1*\bar{r_m}
It is very common that b_0 will have a value very close to zero.
b_0 is a measure of excess return of the stock compared to the market return. It is very difficult to find a stock with positive high b_0 since it is supposed that there should not be a stock that offeres systematically higher returns compared to the market returns.
We can also estimate these beta coefficients by running a simple regression model. The regression model estimates the beta coefficients using the above formulas.
1.4 Challenge: estimate the Single-Index model
You have to estimate the Single-index model for the NVDA stock. You have to use monthly data of NVDA from Jan 2019 to March 2024.
Downdload monthly prices from Yahoo Finance to an Excel Sheet. You have to do the corresponding calculations before estimating the beta coefficients in Excel using the formulas explained above. Use the Excel formulas of covariance and variance for this estimation (instead of the manual calculation). Use continuously compounded returns to estimate the Single-Index model.
You have to interpret the calculation of both beta coefficients.
2 The CAPM Model
The Capital Asset Pricing Model states that the expected return of a stock is given by the risk-free rate plus its beta coefficient multiplied by the market premium return. In mathematical terms:
E[R_i] = R_f + β_1(R_M − R_f )
We can express the same equation as:
(E[R_i] − R_f ) = β_1(R_M − R_f )
Then, we are saying that the expected value of the premium return of a stock is equal to the premium market return multiplied by its market beta coefficient.
In this model there is NO b_0 coefficient since the CAPM assumes that b_0 should be zero since there should not be a stock that systematically offeres excess returns compared to the market returns. However, when we estimate the CAPM with historical data, it is very recommended to estimate the b_0 coefficient.
Then, the CAPM model used to estimate its beta coefficients is expressed as:
StockPremiumReturn_{i,t} = b_0 + b_1(MarketPremiumReturn_t) + ε_t Where:
StockPremiumReturn_{i,t}=(E[R_i] − R_f)
MarketPremiumReturn_{i,t}=(R_M − R_f)
ε_t ∼ N(0, σ_ε); the error is a random shock with an expected mean=0 and a specific standard deviation or volatility. This error represents the result of all factors that influence stock returns, and cannot be explained by the model (by the market).
As in the Single-Index model, you can estimate the b_1 coefficient of the CAPM as:
b_1=\frac{cov(MarketPremiumReturn,StockPremiumReturn_i)}{var(MarketPremiumReturn)}
And the b_0 coefficient can be calculated as:
b_0 = \overline{StockPremiumReturns_i} - b_1*\overline{MarketPremiumReturns} Remember that the bar over the variable means the arithmetic mean of the variable.
In the CAPM we use premium returns instead of returns for the calculation of the beta coefficients.
The interpretation of b_1 in the CAPM is the same than in the Single-Index Model, but we have to refer to premium returns instead of returns. Then, b_1 is a measure of market risk of the stock, while b_0 is a measure of excess return of the stock over the market.
If b_1=2.0, the means that the risk of the stock is the doble compared to the market. In other words, for each +1% of change in the market premium return, it is expected that the stock premium return will move in aboutg +2%.
3 CHALLENGE 2: Estimate the CAPM for NVDA
Download stock price of NVDA and S&P500 market index (^GSPC) from Yahoo Finance. Download the Treasury bill monthly rate from the FRED. Go to the FRED site and search for “3-month Treasury Bills” and download the dataset. Read the unit of measure and check whether you need to do any transformation to end up with monthly returns as in the case of NVDA and the market returns.
You have to estimate the CAPM b_0 and b_1 for NVDA in Excel using the premium monthly returns of the market and the stock, and using the formulas explained earlier.
You have to provide interpretation for b_0 and b_1
4 Quiz 2 and W2 submission
You have to submit your Excel file for Workshop 2 through Canvas.
Also, you have to respond Quiz 2 in Canvas. You will be able to try this quiz up to 3 times. Questions in this Quiz are related to concepts of the readings related to this Workshop. The grade of this Workshop will be the average between Quiz 1 grade and the grade of your Excel.
Remember that you have to submit your Excel file through Canvas BEFORE THE FIRST CLASS OF THE NEXT WEEK.