Heterogeneous Expectations, Capital Costs, and Asset Pricing

# Answer by Alexander Didenko (score 2)


# Is there a difference between heterogeneous expectations and heterogeneous cost of capital? How are assets priced in these situations?












- How are asset prices set when investors face heterogeneous expectations? Does some form of "negotiation" take place so that the market price is set?

- Can investors face heterogeneous costs of capital (even under homogeneous expectations)? Would this be an explanation for differences across portfolios? For example, investors might have different preferences regarding skewness or kurtosis, or when transaction costs are accounted for in asset pricing models, institutional investors potentially face lower ones as I see it, which should impact their cost of capital and the valuation they assign to risky assets.

- Even in the most basic mean-variance framework, investors have homogeneous expectations but different preferences (and a different risk-aversion coefficient defined by A). Why is the risk-premium the average A x Var(Rm)? Is it because investors agree on this average to set the price for risky assets?

## Answer by Alexander Didenko (score 2)

https://quant.stackexchange.com/a/74859

- Yes, there is a difference between "heterogeneous expectations" and "heterogeneous cost of capital" (of investors). Usually, while the first is about something in future, the second is about something in current situation. Although, some authors may use these terms interchangeably.

- There are many papers, offering varying views on the process. Starting from L. Walras' idea of "tâtonnement" to modern general equilibrium models to agent based model to whatever one may imagine. Just to have a feeling of what I'm talking about:

- https://link.springer.com/chapter/10.1007/978-3-030-66691-0_7

- https://wxiong.mycpanel.princeton.edu/papers/bond.pdf

- Yes they can. Generally speaking, expectations and cost of capital are unrelated concepts, and yes, this would inevitably produce differences in portfolios. Further, you mention heterogeneity in preferences, which, on my mind, is the third kind of heterogeneity. Some investors may face different regimes of taxation in different geographical regions (say, usually dividends, capital gains, and interest are taxed differently - and while some investors might have a preference for outright dividend, others would prefer wealth redistribution through buyout). In this case, heterogeneity in cost of capital (technically speaking investors face differences in that aspect) naturally leads to heterogeneity in preferences. But there is heterogeneity in preferences not connected with heterogeneity in CoC - for example, when we model investors with heterogeneity in parameters of wealth (or gain/loss) utility functions. Cost of capital is unaffected in this case, as well as expectations, which are usually about some economic conditions in future.

- I'm not sure what particular paper you're talking about, but probably it's meant that we have some uniform distribution of A across investors which allows us deriving market risk premium by simply averaging A. The very basic mean-variance framework paper, Markowitz (1952), says nothing about market portfolio.

## Answer by blizzard16 (score 0)

https://quant.stackexchange.com/a/74868

1. If we are navigating in the CAPM world deep enough, we can abandon whether the investors have homogenous expectations or not. It is irrelevant, even in the condition of homogenous investment horizons can be relaxed. Only thing that matters, is that the investors face quadratic utility function AND/OR the returns are normally distributed.

I try to find some online source to cite on the previous claims, but I am 99% certain that I am correct. For the heterogenous/homogenous expectations you can think like this: if some investors are more risk averse than others, then the correct price and the derived expected returns are an aggregate result of the market stance i.e. the equilibrium where the supply of the asset meets the demand for it. And in general, markets are just a place to negotiate the price for the asset.

On this question of preferences of expected returns and risk aversion, I heavily suggest you take a look on Tobin's two fund separation theorem.

This seems fairly reasonable: https://www.academia.edu/30770950/Lecture_8_Relaxing_the_assumptions_Zero_Beta_CAPM_Taxation_and_Borrowing_Lending_constraints_AIM_OF_LECTURE_8

2. One has to very careful when addressing such questions as there is so much going on. Yes, investors can face different costs of capitals, even in the CAPM world. If I am making bets with my own money or with my parents money, there is difference for the cost of capital between the sources of the funds. And of course, investors can and do have preferences that differ on multiple factors. Regarding the institutional investors, I wish that you could provide a bit more specific question. For example, the creditors of the institutional investors demand a discount and coupon that is proportional to the solvency and riskiness of the institutional investor. The rate that the banks are charging is only a result of riskiness of the projects undertaken by the investor, which of course depends on many, many factors.

And pretty much the same goes for the equity side of an institutional investor: if they are able to show a superior track record of returns with very little deviations, and that the returns are not a result of taking systematic risk, then they will be able to charge a lot from their investors, i.e. have low cost of capital. For any active investor to be successful the first thing is to minimize the transaction costs, and without saying, the institutional investors do have efficient means to trade, but that does not yet grant the exceptional returns, it only an important ingredient. Concerning the active investing being successful, and I can promise that there is very little evidence going for it.

Despite the people, also me, of this forum trying to do it:)

The valuation of risky assets should not differ on where you got your funds: valuation of an asset is only a result of the returns (cash flows) and the risk (discount factor). The discount rate is determined by the opportunity cost: what if you were to invest into similar asset risk wise, and then you compare would you be better of investing to the asset you are valuing, to the other risky asset or not investing at all.

3. Nope, investors can have different expectations in the mean-variance framework, it just enough to find the aggregate of the expectations through the markets. There is plenty of math to prove this convergence of expectations, and I will not dwell into it. And the risk premium is defined by: $r_m-r_f=Aσ_m^2$ as you stated. Once again, you can settle with the aggregate as the sample being large enough, all the noise is cancelled and we are left with the average A, average risk aversion.