Chapter 10 - Capital markets and pricing of risk Flashcards
Define volatility
in finance, volatility is the standard deviation of returns.
Standard deviation, and therefore volatility, is easy to interpret as it is the same unit as the return itself.
What should we be cautious about volatility (std dev) and variance?
They do not make a distinciton between upside and downside. All they care about is changes.
Formula for variance
if we hold stock beyond the first dividend, we need to…
Figure out what to do with the dividend. We can consider investing it all back into shares. This will affect future dividends like this:
Realized returns annual = (1+r_q1)(1+r_q2)(1+r_q3)(1+r_q4) IF DIVIDENDS PAYED QUERTAERYLY.
In general: what we do is compute the return per dividend period, and then compound these periods. If annual dividend payment, we end up with 5 compounding. If first period is 10%, second is 11%, third is 12%,…, we would get total return of 1.101.111.121.131.14 which would give us approx 76% return.
This require the stock price at each time period to be known.
This gives us the annual realized returns of the stock.
In general, how do we calculate realized returns?
Like we did earlier with dividend discount model, but we solve with variables as given, not forecasts.
Will include dividends and the capital gain.
So, the realized return = dividend yield + capital gain rate
Over any particular period of time, we essentially observe one draw from the probability distribution of the returns. How can we get more data here?
if we assume that the distribution remains the same, we can observe multiple draws by observing the returns from multiple periods. We then count the number of times that a return fell within a certain interval, and use this to illustrate the probability distribution.
This is not perfect, but can help understand returns.
What is the average annual return of a security?
Formula for variance based on historical sample?
if we believe investors are neither too optimistic nor pessimistic, what can we say about their expected returns?
Their expected returns should be equal to the average annual return that is recorded on the previous years.
however, there is a difficulty with this assumption: The historical average return is just an estimate of the true expected return, and is therefore subject to estimation error,
elaborate on estimation error
Given volatility of stock returns, which can be quite large, it can be very difficult to accuartely capture anything. As a result, the estimation error can be quite large, even with many years of historical data.
For instance, one stock can produce exact returns equal to the average, while another stock can produce very spread out returns, but with the same average as the other stock. They have the same average, but it is foolish to believe the two estimates are “equal”. The estimation error capture this, by taking a look at how volatile the estimation is.
How is estimation error computed?
we measure the estimation error of a statistical measure by something that we call standard error. The standard error is the standard deviation of the estimated value of the mean of the actual distribution around its true value. That is: the standard deviation of the average return.
Here is the mathematical process:
Suppose we have n independent and identically distributed random variables X_1, X_2, …, X_n that represent stock return for some stock at year “i”.
The random variables are drawn from the same popluaiton (we assume) with the same distributuion. Because of this, eahc of the observations (random variables) has a popluation mean and a popluation variance.
Our goal is to estimate the population mean. We do this by using estimator sample mean.
We have 2 goals. 1) Estimate the true mean/average of the annual stock returns, 2) Quantify the estimation error.
To estimate the populaiton/true mean, we use the estimator called “sample mean”. This estimator is “forventningsrett” (Norwegian), so we know it will tend towards the true mean.
This is fine, but we cannot be sure our estimate is perfect, as it will depend on the samples. Therefore, we quantify the spread of the estimates. Var(estimator sample mean) = (1/n^2)Var(∑Xi) = sigma^2 /n.
We then find standard deviation of this which is sigma/sqrt(n)
The measure in regards to estimating stock returns will measure the spread of our estimate. A low spread indicates that our estimate is likely to be quite accurate, while a larger spread indicates a more error-prone value. Note “estimate” in not plural. We only make one estimate of the mean, and by using math we can find the variance of this estimator, and thereby receiving a value that explain how good the estimate is.
Thus, sigma/sqrt(n) tells us something about the standard error.
REMEMBER: we do not have the sigma value. Therefore, we must use the sample variance, and sqrt it, to get the sample standard deviation.
what is the general conclusion from estimation error part of the chapter?
The estimation error is too high to use past results to estimate expected returns. Even in the case of s and p, it was too difficult to do it. Many stocks have only fractions of that lifetime, and are subject to huge estimation errors.
Name something useful we can use the standard error for+
Establish confidence intervals. The most simple one is the 95% confidence interval. This makes use of the fact that samples will be within 2 standard deviations approximately 95.44% of the time. We can use this to find the interval in which the stock return will be 95.44% of the time:
HistoricalAverageReturn +- 2 x StandardError
HistoricalAverageReturn +- 2 x (std(samples)/sqrt(n))
EX:
12.2% +- 2x(19.7% / sqrt(96)) = 12.2% +- 4%
= [8.2%, 16.2%]
This means: The stock return is inside of this range 95% of the time.
This is the 95% confidence interval for S&P returns:
12.2% +- 2x(19.7% / sqrt(96)) = 12.2% +- 4%
= [8.2%, 16.2%]
what can we conclude from it?
Even with a 96 years of data, we cannot accurately predict the returns. It will vary.
What makes it even more difficult, is that the underlying probability distribution might also change with time, which cause old data to be irrelevant, and only new data to be good.
Define excess return
Excess return is the difference between the average return on an investment, and the average return of the US treasury bills.
excess return measures the average risk premium that investors earned for bearing the risk of the investment.
Riskier investments must offer …
Riskier investments must offer investors higher average returns to compensate for the additional risk they have.
This is because investors are usually risk averse. The loss of X is greater than the profit of X.
We see a positive relaitonship between higher volatility and higher average returns.
NB: This applies for only large portfolios. It is not the case with individual stocks.
Elaborate on the relationship between volatility and average returns for individual stocks
One would think that individual stocks with greater volatility would have greater average returns. But this is not the case.
Elaborate on types of risks
Common risk: Risk that is perfectly correlated
Independent risk: Risks that share no correlation.
Earthquakes and homes are common risks.
Theft and homes are independent risks.
In the case of theft, some home owners are unlucky, others are perhaps lucky. However, the overall number of unlucky home owners is quite predictable.
The averaging out of independent risks in a large portfolio is called diversification.
If we have a stock that pays dividends multiple times a year, how do we find annual realized returns?
