CFA - Quantitative analysis.

Let me start by saying time value of money. A $ today is not worth the same $ tomorrow. Hence someone not spending that $ today invests in something that can bring a reasonable additional return tomorrow. The more the risk in investing the more expected, though not guaranteed, return.
This forms the fundamentals of the Quant section leading to Present value (PV) and Future value FV) of money.

Present Value and Future value:
-------------------------------
Few formula to calculate future value

FV = PV (1+i)^n
where
FV = Future value
PV = PResent value
i = Expected interest rate or RAte of return
n = Time period

On a basic level, the principal or initial amount is compounded by the interest received at the end of every time period.
eg: $1000 today at 10% interest rate for 5 years compounded annually will be

FV = 1000 * (1+ 0.1)^5
However if the above was compounded semi annually, it would be

FV = 1000 * ( 1 + 0.05) ^ 10
As the frequency of compounding increases, the net FV increases. If the amount is compounded continuously, the equation would become

FV = PV * (e) ^ nr
= PV * (e) ^ (0.1 * 5)

Though knowing the above concepts is good, from a CFA point of view, it is more important that one knows how to use the financial calculator. The key is to identify and make sure that the n values and r values to be input are correct based on the frequency of compounding. A lot of practice is needed so as to not go wrong in this simple area.

Annuity:
--------
Annuity in simple terms mean a series of periodic payments. Unlike PV where the entire principal is a one time lump sum, Annuity is same amounts applied in equal time periods. Hence common sense is that, with all factors being equal, FV for sum of annuity amounts will be lower than the FV for the same amount when invested as a lump sum today.
Annuity normal or regular is when the payments are invested or applied at the end of every time period whereas Annuity due is when the payments are invested or applied at the begining of the time period. Use Begin mode to calculate PV or FV for Annuity due and End mode for ordinary annuity. A special case of Annuity is perpetuity where PV is CF/discount rate. Perpetuity in layman terms mean that the periodic payments run till end of one's life period.


Probability:
------------
Another area to focus is the chance of something happening. Typically in financial world, this is applied to the chance or probability that a particular asset/security/portfolio will result in a particular return for a given risk.
The Expected return for a security is given by the sum of (probability of something happening* expected return if that happens)
Eg:
If consumer sentiment optimistic, expected return for a particular stock is $1.25 per share
Let us say that the probability of that happening is 80%.
If consumer sentiment optimistic, expected return for a particular stock is $0.25 per share
Let us say that the probability of that happening is 20%.
From the above, if someone invests in the above stock, the expected return would be the weighted average, ie (0.8 * 1.25) + ( 0.2 * 0.25) = 1.05

Assuming there is another stock with the same return, but a different combination of probabilities and return attributes, then one needs to weigh in the risk of putting the money in one vs another. This is calculated through the Standard deviation, which is square root of variance.
In our example,
variance = (1.25-1.05)^2 * 0.8 + (0.25-1.05)^2 * 0.2
= .04*0.8 + (.64 * .2)
= .032 + .124
= .156
SD = sqrt(var) = .39

Hence the expected return can vary from .75 to 1.44 (Exp return +_ SD)
If we did a similar calculation for a different stock and end up with a range 1 to 1.1, then we know that the risk of the other stock is less since the expected returns oscillate in a narrow range.

The above analysis works fine if the expected returns are same but SD's are different. What if a stock with higher Expected return has more SD compared to another with lower return but a lower risk. To make apple to apple comparison, we calculate Coefficient of variation which is defined as risk per unit of return

Hence CV = Return/SD.

Hypothesis testing :
------------------
When someone tells you that they can earn you $5 dollars in 1 yr for every $ you invest, how can you weigh in on their claim. Well, hypothesis testing comes to the rescue. All you need is a sample of their performance that contains the number of sample, mean return of the sample and the SD of the sample. Using the above data, one can tell with a certain % of confidence that their claim is valid or not valid.
Formula:

CI = Mean(sample) +- t value * std error

If sample size small, t value need to be used. Note that degree of freedom should be used to look up t value. df = sample size - 1
std error = SD of sample/ sqrt(sample size)

This is a two tailed model since the Null hypothesis where claim = $5 needs to be rejected or not rejected and the alternative hypothesis is everything but that.
There may be cases when the claim could be >= $5 in which case, the analysis would be one-tailed analysis.

( All the above notes have been without looking at any notes. I will take additional time tomorrow to provide better examples and also revisit the sequence for better readability).

Done for the day :)