# Tag Info

10

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 $$\beta_{security,market} = \frac{\sigma_{security,market}}{\sigma^2_{market}}$$ The idiosyncratic risk is the portion of risk unexplained ...

7

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 ...

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 ...

5

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 ...

5

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 ...

4

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 ...

4

If you're long the underlying and short the futures contract, then you have no risk and earn the risk-free rate. You get into the position at $S_0$ and will be able to get out of the position at $F_0$ at time $T$. By a no arbitrage argument it must be that $F_0 = S_0 \exp(r T)$. I imagine Hull has a pretty good exposition on this. The risk premium is ...

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

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. ...

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

Short answer yes! It's called VIX. When it goes up, investors are afraid. When it goes down, investors are complacent.

2

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 ...

2

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...

1

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 ...

1

The discount rate that you use to compute the price of the bond is a parameter that you use during the computation of both types of valuations (clean or dirty); it is the rate you use to discount the cash flows. The only difference between clean and dirty price is that the clean price removes the accrued interest since the last coupon. Hence, the discount ...

1

This was documented in this working paper about 6 years ago. I wonder why it was never published... may be some problem with the results.

1

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 ...

1

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 ...

1

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

1

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 ...

1

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 ...

1

I also find the "risk premium" idea unsatisfying, but I don't think your hedging argument works. Because hedging implies correlation, the risk free asset needs to be well correlated with the risky one in order for you to make any spread between the two. Are you saying that someone should be able to earn the risk premium (without taking any risk!) by being ...

1

I second @DirkEddelbuettel here. Measuring aggregate risk aversion implies abstracting away and making an explicitly artificial theoretical construct. There does not 'exist' a representative investor. What we have are investors, traders, with varying risk aversions, utility functions with differing parameters (if not functional forms). The distribution of ...

1

Going out on another limb here: No. First off, there is no representative investor. If we were all the same, the CAPM and the market portfolio would hold. So that part is unobservable. Second, that is why the sell-side strategists are to happy to create things like misery indices (bad example as it's more macro-based in its sum of inflation plus ...

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