Notes¶

1. CAPM¶

Market Portfolio: The portfolio of risky assets with the Maximum Sharpe Ratio (Best Allocation of weights of risky assets in the investable universe, excluding the risk-free asset such that you get the highest historical Sharpe Ratio $\Big[\frac{\mathbb{E}[r_{\text{portfolio}}]=w^\top \mu}{\sigma_{\text{portfolio}}}\Big]$)
- We get this by running mean-variance optimization for universe of risky assetsusing historical returns
Key takeaway:

For any security $i$:

\begin{aligned} \mathbb{E}[r_i] &= \beta_i\underbrace{(\mathbb{E}[r_{\text{Market Portfolio}}] - r_f)}_{\text{Market Portfolio’s Expected Excess Return}} + r_f \end{aligned}

2. APT¶

Multi-factor version of CAPM: Instead of assuming that Stock returns are only affected by market exposure, we find other factor exposures
- Fama-French 4 factor model proposes 3 other factors that stock returns are exposed to
- We can create other factors by ourselves such that our portfolio is only exposed to that factor (weather)

3. Black-Litterman¶

Bayesian Framework of updating Portfolio weights given active views: Use Active Views to update the Prior expected return of portfolio to get posterior.

4. Futures¶

Futures contract price $$ will always $=$ Actual spot price when futures contract matures
Current Spot Price $S_{0}$: How much the asset is selling for right now ($t = 0$) with information from $t \leq 0$
Current Futures Contract Price $F_{0,T}$: How much the market thinks ($t = 0$) the asset will sell for with information from $t \leq 0$

Current Spot Price $S_{0}$ Vs. Actual Current Futures Contract Price $F_{0,T}$¶

Intangible Asset Model (Stocks): Link between Current Spot Price $S_{0}$ & Theoretical Current Futures Contract Price $\hat{F_{0,T}}$

\begin{aligned} \hat{F_{0,T}} &= S_0 e^{r_fT} \ r_f &= \text{Risk-free rate / Interest Rate} \ \end{aligned}

Cost of Carry Model (Commodities): Link between Current Spot Price $S_{0}$ & Theoretical Current Futures Contract Price $\hat{F_{0,T}}$

\begin{aligned} \hat{F_{0, T}} &= S_{0} e^{(r_f + u - q - y)T} \ r_f &= \text{Risk-free rate / Interest Rate} \ u &= \text{Cost of Storage} \ q &= \text{Dividends / Cash Benefit from holding the asset} \ y &= \text{Convienience Yield} \ \end{aligned}

Comparing $S_{0}$ and $F_{0,T}$ is seeing whether costs outweigh the benefits of buying the asset right now:

Contango: $S_{0} < F_{0,T}$
- The market thinks that the costs of holding the asset over period $T$ > the benefits
- We can justify this theoretically as $r + u > c + y$ means cost > the benfits of holding the asset over period $T$ which will make $S_{0} < \hat{F_{0,T}}$
Backwardation: $S_{0} > F_{0,T}$
- The market thinks that the costs of holding the asset over period $T$ < the benefits
- We can justify this theoretically as $r + u < c + y$ means cost < the benfits of holding the asset over period $T$ which will make $S_{0} > \hat{F_{0,T}}$

Expected Future Spot Price $\mathbb{E}[S_T]$ Vs. Current Futures Contract Price $F_{0,T}$¶

Link between Expected Future Spot Price $\mathbb{E}[S_T]$ & Current Spot Price $S_{0}$

\begin{aligned} \mathbb{E}[S_T] &= S_{0} e^{k_eT} \ k_e &= \mathbb{E}[R_{stock}], \text{ The Discount Rate / Expected Return of the stock via CAPM} \ \end{aligned}

Link between Expected Future Spot Price $\mathbb{E}[S_T]$ & Theoretical Current Futures Contract Price $\hat{F_{0,T}}$

\begin{aligned} \hat{F_{0,T}} &= \mathbb{E}[S_T] e^{(r_f-k_e)T} \ r_f &= \text{Risk-free rate / Interest Rate} \ k_e &= \mathbb{E}[R_{stock}], \text{ The Discount Rate / Expected Return of the stock via CAPM} \ \end{aligned}

Comparing $\mathbb{E}[S_T]$ and $F_{0,T}$ is seeing how much the longer of the futures contract is compensated for the equity risk premium ($\mathbb{E}[R_{stock}] - r_f$):

Normal Contango: $\mathbb{E}[S_T] < F_{0,T}$
- Less common, market thinks that equity return is < than the risk-free rate
- We can justify this theoretically as $\mathbb{E}[S_T] < F_{0,T}$ happens when $r_f > k_e$
Normal Backwardation: $\mathbb{E}[S_T] > F_{0,T}$
- More common, market thinks that equity return is > than the risk-free rate
- We can justify this theoretically as $\mathbb{E}[S_T] > F_{0,T}$ happens when $r_f < k_e$
- In this case, longer of futures contract pays a price $F_{0,T}$ and is expected to receive the asset of price $\mathbb{E}[S_T]$ at maturity - the expected profit of $\mathbb{E}[S_T] - F_{0,T}$ is compensation for the equity risk premium and the hedger is willing to pay for this in order to lock in the price and limit the losses.

5. Carry¶

6. Risk-parity / Risk-budgetting¶

We start off after finding a set of assets that we want to long on
We place the $w > 0$ constraint in the risk-budgetting optimization because it cannot be $w = 0$ because we already know we want to long it, and it cannot $w \leq 0$ because we want to long it.
Risk-parity means $b = \frac{1}{n}\mathbf{1}$, all risks allocated to each opportunity is equal, risk parity is a special case of risk-budgetting
No one can make a good argument on why you don’t want to allocate equal risk to each opportunity

7. Active Management¶

Fundamental Law of Active Management (Bets must be INDEPENDENT):

\begin{aligned} IR &= IC \sqrt{BR}, \ IC &= 2\text{Accuracy} - 1 \ \end{aligned}

HFT’s secret sauce is making so many bets such that their accuracy only needs to be very low because the BR bumps up their IR
Fundamental investors need to have a very high accuracy in order to get the same IR as HFTs
Quants exist between HFT and Fundamental, higher bets than fundamentals, lower bets than HFT, higher accuracy than HFT, lower accuracy than Fundamental

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