12.1 Financial mathematics

Financial mathematics applies mathematical methods to model and analyze financial markets. It uses stochastic processes to represent unpredictable elements like stock prices and interest rates. This field is crucial for understanding investment valuation, derivatives pricing, and portfolio optimization.

Key concepts include the time value of money, interest rates, and risk-return trade-offs. Students learn about present and future values, compounding, bond pricing, and option valuation models like Black-Scholes. Portfolio theory and asset pricing models are also covered.

Fundamentals of financial mathematics

• Financial mathematics applies mathematical methods to financial markets and instruments
• Stochastic processes play a crucial role in modeling financial phenomena such as stock prices, interest rates, and derivatives
• Key concepts include time value of money, interest rates, investment valuation, derivatives pricing, and portfolio optimization

Time value of money

Present value vs future value

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• Present value (PV) represents the current worth of a future sum of money or stream of cash flows given a specified rate of return
• Future value (FV) is the value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today
• The relationship between PV and FV is determined by the interest rate and the number of compounding periods

Compounding periods

• Compounding is the process of earning interest on interest
• The number of compounding periods per year determines how frequently interest is calculated and added to the principal
• Common compounding periods include annually, semi-annually, quarterly, monthly, and continuously

Annuities and perpetuities

• An annuity is a series of equal payments made at equal intervals over a fixed period (mortgages, car loans)
• A perpetuity is an annuity that continues forever, providing an infinite stream of equal periodic payments (consols)
• The present value of an annuity or perpetuity can be calculated using specific formulas that account for the interest rate and number of payments

Interest rates

Simple vs compound interest

• Simple interest is calculated only on the principal amount, ignoring the effect of compounding
• Compound interest is calculated on the principal and the accumulated interest from previous periods
• Compound interest leads to exponential growth of the investment over time

Nominal vs effective rates

• The nominal interest rate is the stated annual rate, not taking into account the effect of compounding
• The effective interest rate is the actual annual rate of return, considering the compounding frequency
• The effective rate is always higher than the nominal rate for compounding periods more frequent than annually

Continuous compounding

• Continuous compounding assumes that interest is compounded infinitely often, leading to the highest possible effective rate
• The future value under continuous compounding is calculated using the exponential function: $FV = PV * e^{rt}$, where $r$ is the annual interest rate and $t$ is the number of years
• The continuously compounded rate is a limit that the effective rate approaches as the compounding frequency increases

Bonds and loans

Coupon payments

• A bond is a debt security that pays periodic interest payments (coupons) and repays the face value at maturity
• Coupon payments are typically made semi-annually, and the coupon rate is expressed as an annual percentage of the bond's face value
• The present value of a bond is the sum of the discounted coupon payments and the discounted face value

Yield to maturity

• Yield to maturity (YTM) is the total return anticipated on a bond if it is held until maturity
• YTM is the discount rate that equates the present value of a bond's future cash flows to its current market price
• YTM considers the coupon rate, the time to maturity, and the market price of the bond

Amortization schedules

• An amortization schedule breaks down the periodic payments of a loan into principal and interest components
• Each payment consists of an interest portion and a principal portion, with the interest portion decreasing and the principal portion increasing over time
• Amortization schedules are commonly used for mortgages, car loans, and other installment loans

Investment valuation

Net present value (NPV)

• Net present value (NPV) is the difference between the present value of cash inflows and the present value of cash outflows over a period of time
• A positive NPV indicates that the investment is expected to be profitable, while a negative NPV suggests that the investment should be rejected
• NPV is used to evaluate the profitability of investment projects and make capital budgeting decisions

Internal rate of return (IRR)

• Internal rate of return (IRR) is the discount rate that makes the net present value of all cash flows from a project equal to zero
• IRR represents the expected rate of return on an investment, considering the time value of money
• Projects with an IRR higher than the required rate of return are considered attractive investments

Payback period

• The payback period is the length of time required to recover the initial cost of an investment
• It is calculated by dividing the initial investment by the annual cash inflow
• While the payback period is simple to calculate, it does not consider the time value of money or cash flows beyond the payback period

Financial derivatives

Forwards vs futures contracts

• Forward contracts are customized agreements between two parties to buy or sell an asset at a specified price on a future date
• Futures contracts are standardized agreements traded on an exchange to buy or sell an asset at a predetermined price at a specific time in the future
• Futures contracts are marked-to-market daily, while forward contracts are settled only at expiration

Call vs put options

• A call option gives the holder the right, but not the obligation, to buy an underlying asset at a specified price (strike price) within a specific time frame
• A put option gives the holder the right, but not the obligation, to sell an underlying asset at a specified price within a specific time frame
• The payoff of a call option increases as the price of the underlying asset rises, while the payoff of a put option increases as the price of the underlying asset falls

Black-Scholes pricing model

• The Black-Scholes model is a mathematical framework for pricing European-style options
• It assumes that the underlying asset's price follows a geometric Brownian motion with constant drift and volatility
• The model uses five key variables: current stock price, strike price, time to expiration, risk-free interest rate, and volatility of the underlying asset

Portfolio theory

Risk and return trade-off

• Portfolio theory explores the relationship between risk and expected return in investment portfolios
• Investors seek to maximize their expected return for a given level of risk or minimize the risk for a given level of expected return
• The efficient frontier represents the set of optimal portfolios that offer the highest expected return for a given level of risk

Diversification benefits

• Diversification involves spreading investments across different assets, sectors, or markets to reduce the overall risk of a portfolio
• By including assets with low or negative correlations, diversification can help mitigate the impact of individual asset fluctuations on the entire portfolio
• Diversification allows investors to reduce unsystematic risk, although systematic risk (market risk) cannot be eliminated

Optimal portfolio selection

• Optimal portfolio selection involves identifying the portfolio that provides the highest expected return for a given level of risk
• The Markowitz mean-variance optimization framework is commonly used to determine the optimal weights of assets in a portfolio
• Investors can choose a portfolio along the efficient frontier that aligns with their risk tolerance and investment objectives

Capital asset pricing model (CAPM)

Systematic vs unsystematic risk

• Systematic risk, also known as market risk, refers to the risk inherent to the entire market or economy (interest rates, recessions)
• Unsystematic risk, also known as specific risk, is the risk unique to a particular company or industry (management changes, labor strikes)
• CAPM posits that investors should only be compensated for taking on systematic risk, as unsystematic risk can be diversified away

Beta coefficient

• Beta is a measure of a stock's volatility in relation to the overall market
• A beta of 1 indicates that the stock's price moves with the market, while a beta greater than 1 suggests higher volatility and a beta less than 1 suggests lower volatility
• The beta coefficient is used in the CAPM to calculate the expected return of a security based on its systematic risk

Security market line

• The security market line (SML) is a graphical representation of the CAPM, showing the relationship between expected return and systematic risk (beta)
• The intercept of the SML is the risk-free rate, and the slope represents the market risk premium
• Securities that plot above the SML are considered undervalued, while those below the SML are considered overvalued

Stochastic interest rate models

Vasicek model

• The Vasicek model is a one-factor short rate model that describes the evolution of interest rates over time
• It assumes that the instantaneous short rate follows an Ornstein-Uhlenbeck process, which is a mean-reverting stochastic process
• The model incorporates a long-term mean interest rate, a speed of reversion, and a volatility parameter

Cox-Ingersoll-Ross (CIR) model

• The Cox-Ingersoll-Ross (CIR) model is another one-factor short rate model that extends the Vasicek model
• It assumes that the short rate follows a square-root diffusion process, which prevents negative interest rates
• The CIR model includes a mean-reversion feature and allows for greater flexibility in modeling the term structure of interest rates

Heath-Jarrow-Morton framework

• The Heath-Jarrow-Morton (HJM) framework is a general approach to modeling the evolution of the entire yield curve over time
• It models the forward rate curve as a stochastic process, allowing for a wide range of yield curve dynamics
• The HJM framework is based on the no-arbitrage condition and can incorporate multiple factors, such as short rate, long rate, and volatility