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Spring, 2000 -1-
Beta and Regression
Administrative Items
Getting help
- See me Monday 3-5:30 or tomorrow after 2:30.
- Send me an e-mail with your question. (stine@wharton)
- Visit the StatLab/TAs, particularly for help using the computer.
Problem set #
- Last problem set.
- Preparations for the project.
Makeup Exam Next week. Be there or keep the zero you now have!
Adjusting for Risk in Investments
What’s up with the stock market?
Dice assignment in Stat 101
- Returns on several “simulated” investments. Avg Annual Return SD of Return White 0% 5% “cash” Green 7.5% 20% “market” Red 71% 130% “internet”
- Which of these investments worked out for your group?
- What about the mixed investment, called “Pink” which is half in Red and half in White? Avg Annual Return SD of Return Pink 35.5% 65% “portfolio”
Effects of variation
- Variability in returns is expensive!
- Start with an initial wealth W 0 of $100.
- Assume return is up 10% on one day, and down by 10% the next.
- Where do you end up after a few days? S (1+.1)(1–.1) = S(1 – .01) = S (0.99) ‡ losing 1% / two days or losing 1/2 % every day!
Spring, 2000 -2-
Risk-adjusted return The risk-adjusted value of an investment with stochastic returns R (i.e., the returns change from day to day) is often calculated as (some will do this differently)
Value = E(R) – Var(R)/
The second term is often called the “volatility drag” on the returns.
Background (optional) For the interested ones, here’s an outline justifying the previous expression. Again, start with wealth W 0. If we denote the return on your investment on the i th^ day as R (^) i, then your wealth after n days is
W (^) n = W 0 (1+R1)(1+R 2 ) … (1+R (^) n)
Rather than work with products, convert this to sums by taking logs.
log W (^) n = log W0 + Σ log (1+Ri)
and use the Taylor approximation log(1+x) ≈ x – x 2 /2 to get
log W (^) n ≈ log W0 + Σ (R (^) i – R (^) i^2 /2)
which has long run average
E log W (^) n ≈ log W 0 + n (ERi – Var(R (^) i)/2)
So, if you want to maximize your expected wealth, you want to maximize
E Ri – Var(R (^) i)/
Analysis of the dice assignment
- Returns on several “simulated” investments. Avg Annual Return SD of Return Adjusted White 0% 5% 0 Green 7.5% 20% 5.5% Red 71% 130% –13.5%
- What about the mixed investment which is half in Red and half in White? Avg Annual Return SD of Return Adjusted Pink 35.5% 65% 14.4%
Spring, 2000 -4-
Goal with one asset One objective is to maximize your long-term wealth. The average gain with shareγ is E(γ R) – Var(γ R)/2 = γ E(R) – γ 2 Var(R)/ which we can maximize as a function of the share, finding the maximum long- term value is obtained with γ = E(R)/Var(R)
Example for investing in the S&P 500 Using the monthly value-weighted returns over the 19 year period 1976-1994, we get this summary for the S&P 500
0.15 (^) .01 .05 .10 .25 .50 .75 .90 .95.
Normal Quantile
with mean 0.0116 and variance .0019. This return is not adjusted for inflation, however, and removing the risk-free return .0059 gives an inflation adjusted return of .0057. Divided by the variance, you get a net share of
γ = E(R)/Var(R) = .0057/.0019 = 3
Yup, it would have been nice to have been leveraged in the market. (Things are not so impressive, though, if you adjust for the cost of having to borrow that money that you just used to invest further in the market.)
Spring, 2000 -5-
Making a Small Portfolio
Problem We start with initial wealth, say $1 at time 0. We can invest in two securities, with returns R1 and R2. Portfolios with the dice looked pretty good, but those were independent. How much of my current wealth should I put into each investment?
In other words, for the dice, why put half in red and half in white? And how do you decide this when red and white are not independent?
Special trick The investment shares γ and are simply
γ 1 = E(R 1 )/Var(R 1 ) and γ 2 = E(R 2 )/Var(R 2 )
when the investments are uncorrelated. So, how can we make investments uncorrelated? Use regression…
Two investments For an example, consider investing either in Amazon or the S&P 500. Here’s a plot of their value (thanks to Yahoo) for the last couple of years on a log scale
1
1 0
1 0 0
1 0 0 0
1 0 0 0 0
0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 7 0 0
Day
with Amazon in red and the S&P in green at the top (very flat).
Spring, 2000 -7-
Regression Regression allows us to easily construct two investments. Here’s a plot of Amazon on the S&P
R Amzn
- 0 7 - 0. 0 5 - 0. 0 3 - 0. 0 1. 0 1. 0 3. 0 5 R SP
and the associated bivariate (i.e., single predictor) regression results:
R Amzn = 0.00518 + 1.8332 R SP
RSquare 0. Root Mean Square Error 0. Mean of Response 0. Observations (or Sum Wgts) 7 2 5
Parameter Estimates
T e r m E s t i m a t e Std Error t Ratio P r o b > | t | Intercept 0.0052 0.0021 2.44 0. R SP500 1.8332 0.1707 10.74 <.
Residuals are not correlated with the predictor
Synthetic investments
Spring, 2000 -8-
Next Class on Wednesday
Simple regression in finance The term “ beta” in finance refers to a regression coefficient. On Weds, we’ll take a look at what makes this coefficient so interesting.
