6

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 ...


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

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 \...


5

As @skoestlmeier and @noob2 commented there's much research going on about the profitability anomaly. Firstly, there are different ways of measuring profitability. Novy-Marx (2013, JFE) uses gross profitability, Fama and French (2015, JFE) total profitability and Hou et al. (2015, RFS) return on equity. The $q$-theory model from Hou et al. claims to explain ...


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 ...


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

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

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 ...


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

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

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

There does not seem to be a clear relationship between interest rates and equity risk premiums. Damodaran (2019) has a great paper that goes into details of equity risk premiums. In this work, he writes: In much of valuation and corporate finance practice, we assume that the equity risk premium that we compute and use is unrelated to the level of ...


3

The CAPM claims that only systematic risk matters (i.e. covariation with the market) to determine an asset's expected return. So the fact that low volatility stocks have returns that are not explainable by market beta is an empirical contradiction of the CAPM to start with. The CAPM is too rigid and performs poorly in explaining the cross section of equity ...


3

What @noob2 said: Actually there is empirical evidence of the opposite, i.e. the existence of a Term Premium. But this is not evidence of arbitrage, just that a more complicated risk model than assumed here is needed. And the simpler theory is still useful in many ways I feel it's helpful to unpack this a little. Let's say you are buying a 10 year Treasury/...


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

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

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

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

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 fair ...


2

In addition to @KeSchn excellent answer i will provide the original intent by Fama/French as they proposed the "Profitability" factor in their 2015 paper "A five-factor asset pricing model". The sources in his answer build up and extend the following economic explanation. Whereas the initial Fama/French (1992/1993) size- and value-factor ...


1

I do not find convincing the argument that the yield curve is upward sloping due to the lack of a secondary market for longer dated securities. In fact, there is a highly liquid market for 2yr, 5yr, 10yr and 30yr Treasuries and yet the yield curve is still biased to be upward sloping. Intuitively I find that the slope is due to the extra yield that an ...


1

You are confusing the underlying index 3M Libor and a 6M loan that pays a compounded 3M interest. A 3M Libor is by definition the (average) rate on an interbank 3M loan. A 6M loan, regardless of its reference interest rate, is subject to counterparty risk up to 6M.


1

The answer given by MJ73550 already covers most if the points in my opinion. I would put it like this: Concerning the drift: if the cost-of-carry relationship is used in your model then this is the correct drift to use for the spot price to derive the price any derivatives (forwards, futures, options) - this is the risk neutral drift. This has nothing to ...


1

Futures prices are drift less under risk neutral measure. In commodities market, it is often Futures. They need to estimate volatility in their model. Since volatilities are not affected by change of probability, you can estimate under real word measure. So what you describe seems correct. # To satisfy commentators. In the continuous time semi martingale ...


Only top voted, non community-wiki answers of a minimum length are eligible