# Quantitative Analysis

It is the analysis of financial instruments

Quantitative analysis is the analysis of financial instruments using mathematical and statistical modeling. It can be used to calculate a financial instrument's value, evaluate performance, and predict future prices.

This is done using various financial techniques/models such as ARCH, GARCH, ARMA, etc. First, let us understand the basic maths behind financial analysis and then move on to the multiple models used.

We must understand why and how investing works, so we must dive into the value created. What is The Time Value of Money?

Money can always be used for investment, producing high returns. Since money has the potential to earn by using it in the present time and generate high returns in the future, money is considered to follow the following:

*Future Money = Present Value + time value*

There are various ways to calculate the value of money. To understand the calculations, let us have a look at the five terms used:

**Present Value:**This is the amount of money present at the current time.**Future Value:**This is the amount of money in the future. Considering it has earned interest rate and grown over time, Future Value should be greater than the Present Value.**The number of periods:**As the name suggests, the period between the end and the start. This is usually in terms of the number of years, quarters, months, etc.**Interest rate:**This is the premium earned over an investment opportunity. It is stated in terms of percentage.**Payment amount:**This is the amount paid in the form of installments over some time

We looked at the definition of Interest Rates. Let us see how it works, its types, and the associated terminology.

There are two main types of interest rates - Simple interest and compound interest.

**1. Simple interest rate** - interest calculated on the original amount.

**SI = Principal * rate (in %) /100**

**2. Compound interest rate** - As the name suggests, interest is compounded, i.e., interest is charged on the total amount (Principal + Previous year's return)

**3. Total amount after compound interest = Principal * (1+r)^n**, where n is the number of periods

**4. **How the above interest rates are charged can be divided into three categories - **Fixed interest rate, Variable interest rate, and Mixed interest rate**.

**5. Fixed interest rate **- The interest rate remains fixed over the entire investment period.

**6. Variable interest rate** - The interest can vary over the investment period based on benchmark rates such as the prime rate.

**7. Mixed interest rate** - As the name suggests, in this case, the interest rate can vary between fixed and variable interest rates.

APR is a commonly used term in financial math, so what is APR?

APR stands for Annual Percentage rate. It is a broader measure of the cost of borrowing. Along with interest rate, it also includes administrative fees.

The value of an asset changes over time. To quantify this, we have amortization and depreciation. There is one more method, a less commonly used one - Depletion.

**What is amortization?**

It is an accounting technique where the company writes off the cost of an intangible asset over its period of life. It is distributed uniformly throughout an asset's lifetime. These assets do not have any resale value.

**What is depreciation?**

Depreciation is used to calculate a decrease in the value of a fixed asset. The distribution is uniform over the lifetime of the asset. Examples of the fixed assets included are Machinery, Equipment, buildings, etc.

Statistical Concepts and Market Returns

- Time-series:
- The random Walk
- Skewness and Kurtosis
- Testing Normality
- Heavy tailed distributions
- ARCH model:

The ARCH model is also known as the autoregressive conditionally heteroscedastic model. This model evaluates the variance of a time series.

- GARCH models
- Value at risk
- Expected shortfall

**Time Series**:

A time series is a set of well-defined data items observed over time through repeated measurements.

The trend (long-term direction), the seasonal (systematic, calendar-related movements), and the irregular (unpredictable movements) can all be dissected from an observed time series (unsystematic, short-term fluctuations).

There are two different forms of time series: stock and flow.

A stock series, often known as "stocktakes," is a measure of various properties at a specific point in time. Flow series are time series that measure activity over some time.

**Random Walk**:

According to random walk theory, stock price changes have the same distribution and are independent. As a result, it assumes that a stock price or market's historical movement or trend cannot be utilized to forecast its future movement.

In a nutshell, the random walk hypothesis states that stocks follow a random and unexpected route, rendering all methods of stock price prediction worthless in the long run.

**Skewness and Kurtosis**:

Skewness is a metric that assesses how lopsided distribution is. Kurtosis measures how to spread out a distribution, especially on the tails.

The state of kurtosis indicates whether the data has a fatter/longer or thinner/shorter tail.

Leptokurtic distribution refers to a distribution with a long tail. The term platykurtic refers to a distribution with a shorter tail.

These concepts are used in risk management. For example, stocks with a high skewness factor exposure have worse returns on average.

The risk premiums on the two market moment factors, volatility, and kurtosis, are smaller in magnitude, according to the authors.

Overall, the findings show that greater market return occasions matter regarding asset pricing.

## Technical Analysis

Technical Analysis is used to predict the stock prices based on past values. The factors taken into consideration are price, volume, and psychological indicators.

The following chart types are used:

- Bar charts
- Japanese Candlestick Charts
- Point and Figure Charts
- Line Charts

Basic Technical Tools:

- Trend Lines
- Candlestick
- Moving Average
- Price Patterns
- Indicators

**Components of Candlestick:**

A candle stick has four levels - High, Close, Open, and Low. A bullish candle will be of green color, and a bearish candlestick will be of the color red. A candlestick has an elongated line and a real body. The elongated body's ends represent the stock's high and low prices.

The upper end of the elongated line is called the upper shadow, and the lower end is called the lower shadow. The real body ends represent the close and open price of the stock.

Let us understand how these technical tools are used for price prediction:

**1. Trend Lines**: Trend Lines show the direction of movement (trend) of the stock prices. These lines represent support and resistance to prices. Trendlines must be adjusted a lot and are not used for advanced analytics.

**2. Japanese Candlesticks**: Japanese CandleSticks is a technique that uses candlestick charts and evaluates the price movement based on past high, low, open, and close.

Hence it is one of the most comprehensive analysis techniques.

- Bullish Engulfing Pattern
- Bearish Engulfing Pattern
- Dark Cloud Cover
**Doji**: Open and Close are very close. Indecisive**Dragonfly Doji:**It is a bullish reversal pattern. It has a long lower shadow.**Gravestone Doji**: It is a bearish reversal pattern. It has a long upper shadow.- Evening star
- Morning star
**Hammer**: It is a bullish reversal pattern. It has a long lower shadow twice the length of the upper body.**Hanging Man**: Similar to Hammer, it has a long lower shadow more than twice the length of the upper body. Again, it indicates a bearish sign.- Harami
- Inverted Hammer
**Piercing Line Pattern:**It is a bullish reversal pattern. There are two candles in this candlestick pattern on day 1, a Bearish candle and a Bullish candle on day 2. It reverses minor bearish trends.**Shooting Star**: Unlike the Hanging Man, the Shooting star has a long upper shadow, twice the length of the lower body. It indicates a bearish reversal pattern.

**3. Simple Moving Averages**

Simple moving averages are the average of price past days (no. of days is the input), i.e., a simple moving average (14) will be equivalent to the average (unweighted) of the past 14 days.

It helps identify trends, trend reversals, support, and resistance levels. The direction of the moving average is the direction of the trend followed.

Another way to use moving averages is to use a pair of moving averages (of different numbers of days) and check the stock's momentum. When a shorter moving average is above the longer moving average, there is an upward trend and vice versa.

**Price and moving average crossover:**

We try to find the entry and exit points at the crossover between the price of a stock and the moving average or the crossover of two moving averages.

**4. Indicators**:

Some commonly used indicators are - Moving Average Convergence/Divergence, Relative Strength Index, and Bollinger bands.

Moving average convergence/Divergence or MACD is a modified version of moving average crossovers. Instead of finding intersection points, it finds the difference between the two.

Hence the level of concern is 0. If the MACD moves from negative to a positive value, then a buy signal is generated and vice versa. MACD is useful for sideways markets. However, it helps generate little profits.

The Relative Strength Index is used to identify overbought and oversold levels. If the RSI level is above 70, the stock is overbought, and if it is below 30, it is oversold.

A stock can remain below 30 or above 70 for an extended period. Hence this tool doesn't account for timing.

Bollinger bands use moving averages. These bands are two standard deviations above and below the moving average. When the stock price goes below the lower band, a buy signal is generated, and if it crosses the upper band, a sell signal is generated.

## Models

**ARCH Models:**

ARCH models are time series models used for conditional variance. So why do we use conditional variance? The higher the volatility, the higher the risk associated with the financial instrument. Hence, we are more focused on variance.

Especially conditional variance, as it helps understand the trend in volatility. The ARCH model captures the volatility clustering. The concept of volatility clustering suggests that large changes follow significant changes. These large changes can have different directions.

**ARCH Process:**

Let it be today's return, and r(t-1) be yesterday's return. Now, rt and r(t-1) are correlated to each other in the following way:

**Rt = white noise * (sqrt (w + a1*(r(t-1))^2)**

White noise is assumed to have zero mean and one variance.

Since the mean of white noise is zero, the conditional mean is also zero. So, the ARCH model has zero mean. Similarly, using the Law of Iterated expectation (LIE), the covariance between rt and r(t-1) is also 0.

Because of no correlation with the historical values, it cannot be predicted from past values; this is called the efficient market hypothesis. However, the conditional mean and variance are non-zero for rt^2.

For* rt^2, var(rt) = w/(1-a1)*, where a1 and w1 are coefficients used in rt calculations.

For the ARCH model, several historical values are used for modeling the coefficient of white noise in rt^2 prediction. Similar to the ARCH model, we have the GARCH model. GARCH model or Generalized ARCH Model.

The generalized GARCH model follows the following relation:

*rt^2 = white noise * (kt^2)*

Where

*kt^2 = w + a*(r(t-1))^2 + b*(k(t-1))^2*

For this model, variance* *

*(rt) = w/(1-a-b)*

**Value at Risk (VaR):**

Value at risk is a statistic that calculates the maximum possible loss of a portfolio or a position. VaR is used to control risk exposure.

**Expected Shortfall:**

Expected shortfall is used to calculate the market or credit risk of the portfolio. Expected shortfall is also known as CVaR or Conditional Value at risk.

### Everything You Need To Master Valuation Modeling

To Help You Thrive in the Most Prestigious Jobs on Wall Street.

*Researched and authored by Punit Manjani *| **LinkedIn**

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