11

I'm not sure about the "CAPM formula" that you are referring to. I assume you are referring to the estimated coefficient of a regression of a security on a market portfolio. That is to say \begin{equation} \beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}} \end{equation} The idiosyncratic risk is the portion of risk unexplained ...


9

I would use the identity and three step process that: $$\textrm{Total Variance} = \textrm{Systematic Variance} + \textrm{Unsystematic Variance}$$ You can calculate systematic variance via: $$\textrm{Systematic Risk} = \beta \cdot \sigma_\textrm{market} \Rightarrow \; \textrm{Systematic Variance} = (\textrm{Systematic Risk})^2$$ then you can rearrange ...


8

By definition, the average investor holds the market portfolio. Risk aversion can be measured as the slope (i.e. ratio of expected returns to volatility) on the efficient frontier. Therefore, the risk aversion of the average investor assuming the S&P500 is the proxy for the market portfolio is the expected returns of the S&P 500 divided by the ...


7

If you have the mathematical sophistication, you should review the original papers referenced on the Equity Premium Puzzle page, particularly Mehra and Prescott (1985). Note, however, that contrary to other opinions on this page, the puzzle is NOT that there is an equity risk premium. On the contrary, the puzzle is that the premium had been so high, at ...


6

This is the equity premium puzzle. (See that article for references.) My thoughts are that individual investors are rational to be risk-averse and demand a premium for bearing a type of market risk that cannot be diversified away. This risk is actually worse and more insidious than it appears, because "personal" circumstances tend to correlate in ...


5

The study you cited seems to be exaggerating slightly. 1) "An interesting fact of returns is that all of the stock returns since 1993 are from overnight returns" -> This is simply factually incorrect. Why don't you pick the S&P 500 names, you calculate the log returns taking into account price changes from the open to the close, then you do the same ...


4

Definitions For fixed $T$ and moving $t \leq T$ then by definition $\color{blue}{(*)}$, forward prices $F(t,T)$ and future prices $\text{Fut}(t,T)$ are both conditional expectations. However, these expectations are not taken under the same probability measure. More specifically: $$ F(t,T) = \Bbb{E}^{\Bbb{Q}^T}\left[ \left. S_T \right\vert \mathcal{F}_t \...


4

All factor returns (including "passive" factors like equity premium, credit premium etc) should be assessed using the fairest possible basis for comparison - self-financing portfolios. For a passive long-only investment, that is equivalent to the total return on the asset class minus the risk-free rate. For a tradable factor premium, that means constructing ...


4

No, it can be negative. The price of risk is what you agree to receive on average in exchange for positive returns when the risk measure is high, and determined by the covariance of the risk measure with your marginal utility of consumption. That said, stochastic volatility risk is negatively priced: you happily agree to a negative return on average in ...


3

Dirty bond price refers to the price of a bond that reflects the interest that has accrued since the issuance of the bond or last coupon payment. It has nothing to do with how you discount cash flows but just whether accrued interest is priced in or not. Thus, dirty and clean bond prices apply to all bonds that pay intermittent cash flows.


3

Another observation that the connection between return and risk is not that straightforward (and in contradiction to modern portfolio theory!) is the low-volatility anomaly. It turns out empirically that stocks that have low-volatility or low-beta show higher returns than high-volatility or high-beta stocks. See also this question and answers: Why does ...


3

If Y is the excess returns of your asset and X is that of the market, then CAPM tells you $Y = \beta X + \epsilon$ Taking the variance of both sides yields $$ \\ \sigma^2_{Y} = \beta^2 \sigma^2_{X} + \sigma^2_{\epsilon} \\ $$ We know that $$\beta = \frac{\sigma_{X,Y}}{\sigma^2_{X}} = \rho_{X,Y}\frac{\sigma_{Y}}{\sigma_{X}}$$ Where $\sigma_{X,Y}$ is the ...


3

do a regression where stock returns is dependent and market return is independent variable. Value of R^2 is Systematic risk and value of 1-R^2 is unsystematic risk...


3

The most rigorous approach I have seen so far eliminating the risk premium is this one: Emanuel Derman: The Perception of Time, Risk and Return During Periods of Speculation (2002) Equation 2.23 on page 11 derives $\mu$ ~ $r$ but it only holds in the limit when you hypothesize countless uncorrelated stocks in a diversifiable market. Still an interesting ...


3

Suppose that you have a model of returns, and a representative agent whose form of utility function you have specified right off the bat. This RA can be constructed, under conditions, from a population that is defined to have heterogeneous utility objectives. This is the problem of aggregation, and it's treated in every good asset pricing theory text (e.g. ...


3

Maybe it is not exactly what you are looking for, but you can take a look at this paper by Kozak, Nagel and Santosh. Roughly speaking, we know that the first order conditions of arbitrageurs must be satisfied, i.e. the following Euler equation should be satisfied for any gross return $R_{t+1}^i$ $$1 = \widetilde{E}_t[M_{t+1}R^i_{t+1}] = \sum_{\omega\in\Omega}...


3

For clarity, I'll use two expressions, "liquidity premium" and "illiquidity premium": "Liquidity premium" arises when investors value the liquidity profile of an instrument so much that they are willing to pay for the enhanced liquidity, thus pushing the price of the instrument above fair value (and its yield below fair value). "Illiquidity premium" arises ...


3

Well there are two misconceptions in your assessment of how returns behave. 1) Returns can be normally distributed or not; 2) Even if they are normally distributed it does not mean that returns have a mean of zero. In fact the mean as you say is slightly positive. So what can we do? Well we can test the data. I took the SPX returns between 1980 and 2012 ...


3

It's news to me that in today's world anybody really believes that equity returns are normally distributed. For instance in US Senate testimony by a Goldman Sachs CFO, under assumptions of Gaussian normality market returns of, e.g. the drops in the DJIA that presaged the 2008 Downturn were 25 std dev (1 in 3.6 x 10e88) events -- several days in a row. This ...


2

Answering this question is impossible without making many significant assumptions regarding risk preferences, utility functions, a model of returns, etc. One way to measure the risk aversion of the representative investor is to compare the market's expected return to expected volatility. When applied to the allocation decision between equity and fixed ...


2

I have studied unsystematic risk [USR] for more than two decades. In fact, I wrote a book (which is here) whose central focus is how to deal with USR in the valuation of non-public companies. It is a multifaceted, complex, and difficult issue. Modern Portfolio Theory did professionals in my line of work no favors when it assumed away the existence of USR ...


2

Alright, here's the proof (I think): Statement of APT: $$E(r_a)=r_f + \displaystyle\sum_{i=1}^n\lambda_i * cov(r_a, r_i)$$ Expand $E(r_a)$: $$\frac{E(C_1)}{PV_0} - 1 =r_f + \displaystyle\sum_{i=1}^n\lambda_i * cov(\frac{C_1}{PV_0} - 1, r_i)$$ Since $PV_0$ doesn't have any covariance with $r_i$, we can reduce the above to the following: $$\frac{E(C_1)}{...


2

No, the "low-beta" anomaly is not the result of the difference between arithmetic and geometric mean returns. Statistical tests verifying the existence of the anomaly rely on models employing the arithmetic mean returns, $$\mu_a = \mu_g + \frac{\sigma^2}{2}$$, hence the penalty excess volatility incurs when compounding returns over time does not explain the ...


2

This one is far from straight-forward, although bear with me. It is possible to infer from first principles an ERP reasonably close to normative consensus expectations. The attached from Howard Marks at Oaktree is a classic: "Everything you wanted to know about the equity risk premium (and much more)". The simple point is that there are four different ...


2

In general, PPN is the short form for principal protected notes. Here, the principal, or notional, $N$ is generally return in full. I am a little confused why only 80 % is returned. It may be a contractual specification, and it is also called a PPN. Regarding the variable interest, or premium in your term, is the return that the investor will achieve. In ...


2

Assume that under the real world measure $$ dS_t/S_t = (\alpha-\delta) dt + \sigma dZ_t^\Bbb{P} \tag{1} $$ Under the EMM $\Bbb{Q}$ one then needs to have (fundamental theorem of asset pricing: in the absence of arbitrage the discounted value of any self-financing portfolio should be a martingale): $$ dS_t/S_t = (r-\delta) dt + \sigma dZ_t^\Bbb{Q} \tag{2} $$ ...


2

Imagine you hold a zero coupon bond with a certain maturity $T$ and the short rate follows a process like you specified. You might know deterministically what the cash bond pays this period, but you don't know how the interest rate itself is going to change. If the interest rate goes down, then the expectation of future rates goes down and the expected ...


2

A distribution may be normal and have a mean different from zero. For example, IQs, weights, heights and so forth. All normal distributions assume a mean and a standard deviation. These two parameters completely describe the distribution. The standard normal is the special case where the mean is zero and the standard deviation is 1.0. So stocks can be ...


2

Normality does not mean that mean return has be zero. The assumption you are talking about is of standard normal distribution which has mean and SD (0,1) respectively. As your question indicates that why positive returns are higher than negative returns. First of all let us understand the mathematics behind normal distribution which says that distribution ...


2

Within the framework you are proposing, it would make no sense. It would be failing to distinguish noise from signal. Extreme events are rarely triggered by measures of central tendency. It is like flipping heads 20 times in a row, what caused that? From a physicist's, magician or con man's perspective that is a valid question, but from a Frequentist ...


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