Quantitative methods in Finance Flashcards
What is Quantitative Methods?
Quantitative methods is a set of mathematical and statistical tools and techniques that provide a quantitative language to understand, analyse and communicate across a range of practical problems encountered in industry. This module aims to provide students with the knowledge to effectively understand and critically assess a range of problems encountered in Finance and to apply this knowledge in a practical environment.
Importance of Quantitative Methods in Finance
Finance provides the framework for making decisions that involve the allocation of money under conditions of uncertainty.
As such, much of Finance is underpinned by Mathematics, allowing investors to understand the risk (or uncertainty) involved, to optimize their financial allocations and to make considered decisions with regard to investment.
Applications of Quantitative Methods in Finance
- Corporate Financial Management
Project appraisal, Cost of Capital, Debt Policy, Dividend Policy etc. - Capital Markets & Instruments
Valuing Bonds and Stocks, Understanding Risk and Return, Interest Rates - Derivative Securities
Pricing options, futures, swaps etc. - Portfolio & Risk Management
Determining optimal allocation of money, understanding the associated risk of investing. - Fixed Income
Pricing Bonds, Term structure of interest rates etc. - Financial Econometrics
Builds on the tools covered in Quantitative Methods to understand the behaviour of Finance Markets.
What is Money?
Money is anything that is commonly accepted in exchange for goods and services. Money is a measure of value, a store of value and a medium of exchange.
What is financial asset?
Financial Assets, including money, are contracts or certificates indicating financial claims.
What is financial security?
A Security is a financial asset that can be purchased or sold in a financial market.
What is a corporate bond?
A corporate bond is a security that indicates a claim by it’s owner (the investor or bondholder) on a specified series of interest payments and principal repayments by the firm which issued the
bond.
What is a financial stock?
A Stock is a residual claim; That is stockholders have the right to receive all cash flows which remain in a corporation after obligations to all other claims holders have been satisfied.
What is a financial option?
An Option is a type of derivative contract that gives the buyer the right, but not the obligation, to buy or sell an underlying asset at a specific price on or before a certain date.
Prices and returns in terms of Finance
The value of an asset is often referred to in terms of it’s market price, which indicates the last price at which the asset traded.
However, in Financial Economics we usually analyse the asset returns instead of the price.
There are a number of practical reasons for this:
1. Investors are interested in returns not prices, as this is the indicator of whether an investment is successful or not.
2. Time series of prices tend to be non-stationary
3. Time series of prices tend to be cointegrated (they have a unit-root), which means that many standard statistical tools are invalid when used with price series.
Discrete Time Returns
Assuming that an asset has no interim payments such as dividends on stocks or coupons on bonds, the one-period (or holding period)
return on the investment is
Rt = [Pt - P(t-1)] / P(t-1)
where Pt and P(t-1) are the prices of the asset at time t and (t - 1) respectively.
If interim payments such as dividends or coupons are received, then it is added to the one-period (or holding period) return on the investment
Continuous Time Returns
If the value of a portfolio is strictly positive, then the continuous time percentage return over a time interval of length delta t is
S (t + delta t) - S (t)
R (t) =————————
S (t)
where S (t) is used to denote the value of an asset in continuous time.
If the increment delta t is a very small interval of time, then the percentage return is small. We know that for small x,
ln (1 + x) ~ x. Hence, for small t
R (t) ~ ln (1 + R (t))
in other words
R (t) ~ ln S (t + delta t) - ln S (t)
so over small time period, the percentage return is very close to the log return.
Compounding explained
Using compounding framework for discrete and continuous returns, now examine compounding of returns.
In the discrete case, we can express the return as
1 + Rt = Pt / P(t-1)
The left hand side corresponds to the discrete compounding factor, which arises because
Pt = (1 + Rt ) Pt-1
For example, if P0 = 100 and P1 = 105, then R1 = 5% and the compounding factor is 1,05
In the continuous case, we can use log returns and the one-period historical log return is
rt = ln [ Pt / P(t-1)] = ln (Pt ) - ln (P(t-1))
Taking the exponential of both sides,
exp (rt ) = Pt / P(t-1) => Pt = exp (rt ) P(t-1)
where exp (rt ) is the continuous compounding factor.
For example, if P0 = 100 and P1 = 105, then
R1 = ln (1,05) = 4,879% and 105 =
= exp (0,04879)x100
Period Log Returns
The h-period log return is the sum of h consecutive one-period log returns.
