2. Market Efficiency

Suppose we have two things to choose from

1. 1% from a government treasury
2. 1% from a tech startup

Hypothetically, both give the same return, but we choose the first option as it’s more reliable. For the tech startup to get our money, it needs to increase our incentive, say to a 1.5% return in the same period. It’s still risky, but it may be worth it (depends), as the reward is higher. The extra 0.5% return is called the risk premium, and the default 1% we get is the risk-free rate (we get it anyway; it’s simply the baseline interest)
There is another component called the liquidity premium. It is the extra return or yield investors demand for holding an asset that is hard to sell quickly at its fair value, compensating them for liquidity risk and inconvenience. This premium is not explicitly paid by anyone; it is embedded in prices and required returns

Equity Risk Premium:
there are two broad categories of risk-related payoffs

1. risk positive / premium / harvesting — this is like buying a bond: you give up your money at the start in exchange for periodic coupon payments, and you receive your principal back at maturity.
2. risk negative / insurance / hedging — this is like buying health insurance: you pay premiums regularly (a small expected loss each period), and in adverse states (when health deteriorates) you receive a large payout, transferring risk rather than earning a risk premium.

Capital Asset Pricing Model:

1. say a global index fund makes a 10% return — this is the case where we invest passively, keep the money invested, and earn the market return.
2. say we give the money to a fund manager and he gets us a 12% return — in this case we delegate investment decisions to the manager (ignore fees for now)

Does this mean we are in profit by giving money to the fund manager? Did he actually perform well?

$$R_a=R_{fr}+{\beta }_a(R_m-R_{fr})$$

Here, we calculate the expected return given the asset’s exposure to market risk. The key parameter is $\beta$, which measures sensitivity to market movements. Higher exposure to volatile, market-sensitive assets implies a higher $\beta$, while safer, diversified portfolios tend to have lower $\beta$

If the fund manager has high market exposure, say ${\beta }_a=1.5$, then

$$E(R_a)=1\%+1.5\times (10\%-1\%)=15\%$$

If the manager delivered only 12%, then the manager underperformed relative to risk taken

If instead ${\beta }_a=0.7$ (the fund manager invested more in safer assets), then

$$E(R_a)=1\%+0.7\times (10\%-1\%)\approx 7\%$$

If the manager delivered 12%, then the manager outperformed on a risk-adjusted basis

We can also extend this to multi-factor models:

$$E(R)=R_{fr}+{\beta }_a(R_a-R_{fr})+{\beta }_b(R_b-R_{fr})+{\beta }_c(R_c-R_{fr})$$

Here, returns are explained by exposure to multiple systematic risk factors (for example, market, size, value, momentum). So even if a fund manager delivers a 17% return, once we account for all relevant factor exposures, the manager may still have underperformed on a risk-adjusted basis
At the end of the day, performance is evaluated relative to the risk taken, as captured by $\beta$ (or factor loadings), not by raw returns alone
Q What makes a factor a factor ?
A

1. Should be independent of time(persistent over time)
2. Should be present or noticeable irrespective of geography
3. Robust to Alternative Specification
4. It should be investable

In Sample vs Out-of-Sample:
Suppose an HFT found a factor and traded on it and earn profit and published a research paper, now the whole world knows about this factor, if it’s behavioral factor, people will change and not react to that, if it’s pattern based, it might not work as the whole market knows it, so before publishing it works that is called In Sample and the world knows it, it might not work it’s Out of Sample

Cash flows are time-valued, as the $100todaymorevaluablewrt$100 after 10 years(given that U.S still exists as a country after 10 years and we don’t face any global breakdown and inflation of course)
So let’s determine the Price of an asset

$$DCF=\frac{C{F}_1}{(1+r{)}^1}+\frac{C{F}_2}{(1+r{)}^2}+\cdots +\frac{C{F}_n}{(1+r{)}^n}$$

The formula represents the discounted cash flow and is the summation of individual cash flows in different time periods, where “r” is the discount rate and the powers in the denominator represent the time period
To understand this, let’s take an example. Let’s say we bought a $1000governmentbondanditpays$50 every year for 5 years and the discount rate is 5% (0.05)
Here the formula is

$$DCF=\frac{50}{(1+0.05{)}^1}+\frac{50}{(1+0.05{)}^2}+\frac{50}{(1+0.05{)}^3}+\frac{50}{(1+0.05{)}^4}+\frac{1050}{(1+0.05{)}^5}$$

As we can see, the present value of the money we get every year decreases (see powers in the denominator), and we will get the whole amount in the last year, this is the total cash flow
And suppose rather than investing in the government bond, we invested in some highly volatile, less secure tech startup. The formula might be the same, but the discount rate in the denominator increases. This reduces the present value and reflects a higher required return due to increased risk. This is how we price an asset

Let’s understand the closed form formula of price of an asset

$$Price=\frac{C{F}_1}{r-g}$$

Here “r” is the discount rate and “g” is the growth rate. We use this formula when we assume perpetual cash flows with a constant growth rate. Also, another thing: Expected Return = Discount Rate. Now let’s understand with an example
Say we have two companies with a stock price of $100 payoff in one year and discount rates of 5% and 12% (higher the discount rate, lower the value, as the discount rate is in the denominator in the formula). Now when they need to raise money, as per the discount rates and the formula (not the closed form, the upper one):

• company 1 will sell its shares for 95.24 (discount rate 5%)
• company 2 will sell its shares for 89.29 (discount rate 12%)

At last,

$$StockPrice=\frac{CF}{r}$$

This holds only for zero-growth perpetual cash flows. So to earn profit, investors look for assets where the realized cash flows are large relative to the market-implied discount rate (i.e., lower risk or higher cash flows)

The best approach is to diversify investments via different factors, like invest in the stocks which follow different factors like momentum, value, … such that all the factors will not fall at a time, if one does other covers the risk and make sure the fall is as less as possible