Prices, returns, holdings, and portfolios

all of the techniques we will discuss are actually used in most of the largest investment banks, AM, HF.
changes in portfolio holdings are called trades
although it wasn’t emphasized in classical studies of portfolio theory, trading itself is a complex process with many decision variables tha need to be considered and potentially optimized.
all trading is costly and some putative arbitrage opportunity doesn’t exist at all once costs are properly accounted for.
I’ll wager that most quant-oriented hedge funds have only a vague idea of their capacity.

Definition 1.1
Bid, Ask.

trading equities, cost includes
- borrow fees if one is selling short,
- financing costs of leverage.
- taxes on transaction
- taxes on dividends
- commissions
- small order charges
- …

costs also include things you can not know

temp or permanent price impact.
one implication of all these cost is that strategies with high turnover and/or requiring leverage must also have extremely accurate forecasts

$p_t^i$ is the price for the i-th asset
$r_{t+1} = p_{t+1} / p_t$ is the return over the interval [t, t+1]
VWAP = volume-weighted average price

Definition 1.2
A stock split is an issue of new shares in a company to existing shareholders in proportion to their current holdings. A dividend is a sum of money paid by a company to its shareholders out of its profits or reserves.

some measures such as revenues are typically specified as whole-com measure
but earnings etc. are often reported as forecasts on a per-share basis those measure are related to split

Definition 1.3
bold face notation without a superscript $r_{t+1}$ denotes as the vector of return.

Any structure inherent in the random process driving $r_{t+1}$ potentially gives rise to analogous structure in the portfolio’s return.

Definition 1.4
$h_t^i$ is holding at time $t$ . $h$ will always have units of US dollars. $h_t \in R^n$

$h_t \cdot r_{t+1}$ is the P&L Information about $cov(r_{t+1})$ gives information about $h_t$ .

var(h_t \cdot r_{t+1}) = h_t' cov(r_{t+1}) h_t

$r^{rf}_{t+1}$ is risk-free rates. If the asset returns one is modeling are local currency returns, be careful to use different risk-free rate for diff cur

Definition 1.5
Excess returns is
$r_{t+1} - r^{rf}_{t+1}$
which is $n\times 1$ matrix

normality is usually the assumption behind model.

Definition 1.6
Sharpe ratio is
$S = \mathbb{E} \left[ \frac{r-r_f}{\sqrt{var(r-r_f)}} \right]$
universal convention: whatever frequency, always use annualized returns and annualized volatility.

Sharpe ratio is not the best way for long only fund.

Definition 1.7 benchmark
long only strategy usually use S&P500 as benchmark

Hence a manager’s skill is closely related to performance relative to the appropriate benchmark. For this reason, one often considers the information ratio (IR), which is the same formula as the Sharpe ratio, but with the risk-free rate being replaced by an appropriate benchmark.

Lotteries and Rewards

A lottery is a probability distribution over the reward space.

u(w) = \frac{1-exp(-κw)}{κ}

where $κ > 0$ is some positive scalar, then supposing $w$ is normal,

\begin{aligned} \mathbb{E}[u(w)] &= \frac{1-exp(\mathbb{E}[-κw] + \mathbb{V}[-κw])}{κ} \\ &= \frac{1-exp(-κ\mathbb{E}[w] + \frac{κ^2}{2}\mathbb{V}[w])}{κ} \\ &= \frac{1-exp\{-κ(\mathbb{E}[w] - \frac{κ}{2}\mathbb{V}[w])\}}{κ} \\ &= u \left( \mathbb{E}[w]- \frac{κ}{2} \mathbb{V}[w] \right) \\ \end{aligned}

This implies that maximizing $\mathbb{E}[u(w)]$ is equivalent to maximizing $\mathbb{E}[wT]- \frac{κ}{2} \mathbb{V}[w]$ since $u$ is monotone. It turns out this is true for many fat-tailed distributions as well.

Utility

Definition 1.8
An agent is risk-averse if, at any wealth level w, she dislikes every lottery with an expected payoﬀ of zero: $∀w,∀\tilde{z}$ with $\mathbb{E}[\tilde{z}] = 0$ ,
$\mathbb{E}[u(w + \tilde{z})] ≤ u(w).$

Any lottery ˜ z with non-zero expected payoff can be decomposed into its expected payoff E[˜ z] and a zero-mean lottery.

Proposition 1.1 An agent is risk-averse, ie. inequality
$\mathbb{E}[u(w + \tilde{z})] ≤ u(w + \mathbb{E}[\tilde{z}])$
holds for all $w$ and $\tilde{z}$ , if and only if $u()$ is concave.

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ECONOMIC AND FINANCIAL DECISIONS UNDER UNCERTAINTY