Example 1: Calculating Loan Repayments
Imagine you take out a loan of $10,000 with an annual interest rate of 5%, to be repaid over 5 years. Using financial mathematics, you can calculate the monthly repayment amount. The formula for the monthly payment on an amortizing loan is P = [r*PV] / [1 – (1 + r)^-n], where P is the monthly payment, r is the monthly interest rate, PV is the loan amount, and n is the total number of payments. In this case, r = 0.05/12, PV = 10,000, and n = 60. Plugging these values into the formula gives a monthly payment of approximately $188.71.
Example 2: Evaluating Investment Projects
Suppose you are considering two investment projects. Project A requires an initial investment of $100,000 and is expected to generate $30,000 per year for 5 years. Project B requires an initial investment of $120,000 and is expected to generate $35,000 per year for 5 years. Using the Net Present Value (NPV) method, you can evaluate which project is more profitable. Assuming a discount rate of 10%, the NPV of Project A is $11,434 and the NPV of Project B is $8,029. Therefore, Project A is the better investment.
Example 3: Pricing Options with Black-Scholes Model
Consider a European call option with a strike price of $50, expiring in 6 months. The current price of the underlying stock is $55, the risk-free interest rate is 2%, and the volatility of the stock is 20%. Using the Black-Scholes model, you can calculate the price of this option. The Black-Scholes formula for a call option is C = S0N(d1) – Xe^(-rT)*N(d2), where C is the call option price, S0 is the current stock price, X is the strike price, r is the risk-free rate, T is the time to expiration, N is the cumulative distribution function of the standard normal distribution, d1 = [ln(S0/X) + (r + σ^2/2)T] / (σ√T), and d2 = d1 – σ√T. Plugging in the values gives an option price of approximately $7.45.
Example 4: Using Value at Risk (VaR) for Risk Management
Suppose you manage a portfolio worth $1 million. You want to calculate the Value at Risk (VaR) at a 95% confidence level over a 1-day period. If the standard deviation of daily returns is 1.5%, VaR can be calculated using the formula VaR = Zσ√t, where Z is the Z-score for the confidence level (1.645 for 95%), σ is the standard deviation of returns, and t is the time period. In this case, VaR = 1.6450.015√1 = $24,675. This means there is a 5% chance that the portfolio will lose more than $24,675 in a single day.
Example 5: Calculating Insurance Premiums
An insurance company wants to set premiums for a new health insurance policy. Based on statistical analysis, they estimate that the average cost of claims per policyholder per year is $500, with a standard deviation of $100. If they want to ensure that they cover 95% of all claims, they can use the normal distribution to set the premium. Using the formula Premium = Average Cost + ZStandard Deviation, where Z is the Z-score for the desired confidence level (1.645 for 95%), the premium would be $500 + 1.645$100 = $664.50. This ensures that the premium covers the claim costs for 95% of policyholders.