Introduction to Financial Mathematics

Fundamentals of Time Value of Money Why the Present Matters More The time value of money is a foundational principle in finance: a dollar you receive today is worth more than a dollar you receive in the future. Why? Because today's dollar can be invested to earn interest and grow. This simple insight is the basis for understanding all the financial calculations that follow. Imagine you're offered two choices: receive $100 today or $100 in one year. You should always prefer the $100 today, because you could invest it at, say, 5% interest and have $105 in one year—better than the $100 you'd get by waiting. Simple Interest Simple interest is the most basic way to calculate earnings on an investment. It's calculated on the principal amount only, with no compounding. $$I = Prt$$ Where: $I$ = interest earned $P$ = principal (starting amount) $r$ = annual interest rate (as a decimal) $t$ = time in years Example: If you invest $1,000 at 6% annual simple interest for 3 years, you earn $I = 1000 \times 0.06 \times 3 = $180$, giving you a total of $1,180. The key limitation of simple interest is that it only applies to the original principal—the interest you earn doesn't earn interest itself. Compound Interest In reality, most financial products use compound interest, where accumulated interest is reinvested and earns interest itself. This is more realistic and leads to faster growth. $$A = P\left(1 + \frac{r}{n}\right)^{nt}$$ Where: $A$ = final accumulated amount $P$ = principal $r$ = annual interest rate $n$ = number of compounding periods per year $t$ = time in years Compounding frequency matters. If interest compounds annually ($n=1$), semi-annually ($n=2$), quarterly ($n=4$), monthly ($n=12$), or daily ($n=365$), you get different results. More frequent compounding means faster growth. Example: $1,000 invested at 6% annual interest, compounded quarterly for 3 years: $$A = 1000\left(1 + \frac{0.06}{4}\right)^{4 \times 3} = 1000(1.015)^{12} \approx $1,195.62$$ Compare this to the simple interest result of $1,180—compound interest earned you an extra $15.62. The difference grows larger with longer time periods and higher rates. Future Value of a Lump-Sum Investment The future value (FV) of a single lump-sum investment is found by applying the compound interest formula. This answers the question: "If I invest a dollar amount today, how much will I have at a specific date in the future?" Using the compound interest formula, if you invest $P$ dollars today at rate $r$, compounded $n$ times per year, for $t$ years: $$FV = P\left(1 + \frac{r}{n}\right)^{nt}$$ This is the same as calculating accumulated amount $A$—we're just using "FV" to emphasize we're thinking about the future value of today's money. Key insight: The longer the time period or the higher the interest rate, the larger the future value. This is why starting to save early is so powerful—time is your greatest asset. Present Value of a Future Amount Often, we need to work backwards. If you know you'll receive a certain amount in the future, what is it worth in today's dollars? This is present value (PV), calculated by discounting a future amount back to today. $$PV = \frac{FV}{(1 + r)^{t}}$$ Where: $FV$ = future amount you'll receive $r$ = discount rate (interest rate per period) $t$ = number of periods until receipt The discount rate reflects your cost of capital or the return you could earn elsewhere. For example, if you could earn 5% in a savings account, a 5% discount rate is reasonable. Example: You're promised $1,500 in 3 years. What is it worth today if the discount rate is 6%? $$PV = \frac{1500}{(1.06)^3} = \frac{1500}{1.191} \approx $1,259.71$$ This means that receiving $1,500 in 3 years is equivalent to receiving about $1,260 today. The difference ($240) reflects the interest you could have earned over three years by investing $1,260. Important concept: Present value is the inverse of future value. If you discount a future amount back to today and then compound it forward, you get back your original future amount. This relationship is fundamental to all financial valuation. Annuities and Related Cash Flow Concepts Most real financial situations don't involve single lump-sum payments. Instead, you have a series of payments over time—rent, loan payments, investment contributions, or retirement income. These recurring cash flows are called annuities. Ordinary Annuities An ordinary annuity is a series of equal payments made at the end of each period. Examples include: Annual loan payments (payments typically made at period end) Bond coupon payments Savings plans where you deposit money at the end of each period The timing matters. When payments occur at the end of the period, less time is available for early payments to grow compared to when they occur at the beginning. Present Value of an Ordinary Annuity If you're evaluating what a stream of future payments is worth today, you calculate the present value of an ordinary annuity: $$PV = P \times \frac{1 - (1 + r)^{-n}}{r}$$ Where: $P$ = payment amount each period (must be equal) $r$ = interest rate per period $n$ = total number of payments Example: What is the present value of receiving $500 at the end of each year for 5 years, at a 6% discount rate? $$PV = 500 \times \frac{1 - (1.06)^{-5}}{0.06} = 500 \times 4.212 \approx $2,106$$ This means five $500 payments in the future are worth about $2,106 today. Each payment is discounted back by one more period than the previous payment. Future Value of an Ordinary Annuity The future value of an ordinary annuity answers: "If I make regular equal payments, how much will I have accumulated at the end?" $$FV = P \times \frac{(1 + r)^{n} - 1}{r}$$ Where the variables have the same meaning as above. Example: You deposit $500 at the end of each year for 5 years in an account earning 6% annually. How much will you have? $$FV = 500 \times \frac{(1.06)^{5} - 1}{0.06} = 500 \times 5.637 \approx $2,818.55$$ Note that your total deposits were $2,500, but compound interest added $318.55—the benefit of letting each payment earn returns for the remaining time periods. Annuities Due An annuity due is a series of equal payments made at the beginning of each period. This timing difference matters significantly. Examples include: Lease or rent payments (typically due at the start of the period) Insurance premiums Savings plans where you contribute at the beginning of each period Because payments are made earlier, they have more time to earn interest. This means annuities due are always worth more (both present and future values) than ordinary annuities. Present and Future Values of Annuities Due The relationship is straightforward: PV of annuity due = PV of ordinary annuity × $(1 + r)$ FV of annuity due = FV of ordinary annuity × $(1 + r)$ The $(1 + r)$ factor represents one additional period of earning for all payments since they occur one period earlier. Example: Using the previous example, if the $500 payments are made at the beginning of each year instead of the end: $$PV{\text{due}} = 2,106 \times 1.06 \approx $2,232.36$$ You get about $126 more in present value because each payment is received and can start earning interest one period earlier. Perpetuities A perpetuity is a special type of annuity that continues indefinitely—it's an infinite series of equal payments with no end date. Examples include: Preferred stock dividends (theoretically forever) Some types of bonds Endowment payments For a perpetuity, the present value formula simplifies dramatically: $$PV = \frac{P}{r}$$ Where: $P$ = periodic payment $r$ = discount rate Notice there's no $n$ (number of payments) because payments never end. Why this works: In the ordinary annuity formula $PV = P \times \frac{1 - (1 + r)^{-n}}{r}$, as $n$ approaches infinity, $(1 + r)^{-n}$ approaches zero, leaving us with $\frac{P}{r}$. Example: A preferred stock pays $4 per year forever. If the appropriate discount rate is 8%, what is it worth? $$PV = \frac{4}{0.08} = $50$$ So the perpetuity is worth $50. As a sanity check: 8% of $50 equals $4, which is exactly the annual payment. This makes intuitive sense—if you have $50 earning 8%, you generate exactly $4 per year indefinitely. Amortization and Amortization Tables When you take out a loan, you typically repay it with equal periodic payments that cover both interest and principal. Amortization is the process of paying down a loan through regular payments. An amortization table (or amortization schedule) tracks your loan repayment over time, showing for each payment: The payment amount (constant throughout) The portion going to interest The portion going to principal reduction The remaining balance How it works: Early payments are mostly interest because the outstanding balance is large. As you pay down principal, later payments have less interest and more principal. Example structure: For a $10,000 loan at 5% annual interest, paid monthly for 3 years: The monthly payment is calculated using the ordinary annuity formula (solving for $P$), which equals about $299.71. In month 1: Interest charged: $10,000 × 0.05 ÷ 12 = $41.67 Principal paid down: $299.71 - $41.67 = $258.04 Remaining balance: $10,000 - $258.04 = $9,741.96 In month 36 (the final payment): Interest charged: much smaller Principal paid down: most of the payment Remaining balance: $0 The key insight is that amortization tables show how every payment is split between interest and principal—they don't change the total payment, but show what portion actually reduces debt. Risk and Return Fundamentals An investment's return (how much money you make) depends on its risk (how uncertain that return is). Understanding the relationship between risk and return is crucial to making sound financial decisions. Expected Return of a Portfolio When you own multiple investments, your overall return depends on: The expected return of each individual investment How much money you have in each one (the weight) The expected return of a portfolio is the weighted average of individual expected returns: $$E(Rp) = w1E(R1) + w2E(R2) + \ldots + wnE(Rn)$$ Where: $E(Rp)$ = expected return of the portfolio $wi$ = weight of asset $i$ (proportion of total portfolio value) $E(Ri)$ = expected return of asset $i$ Weights sum to 1 (or 100%) Example: You invest $6,000 in Stock A (expected return 10%) and $4,000 in Stock B (expected return 15%). Total portfolio is $10,000. Weights: $wA = 0.6$ and $wB = 0.4$ $$E(Rp) = 0.6(0.10) + 0.4(0.15) = 0.06 + 0.06 = 0.12 \text{ or } 12\%$$ Your portfolio's expected return is 12%—higher than Stock A's 10% but lower than Stock B's 15%, weighted by how much you own of each. Variance and Standard Deviation as Risk Measures Variance measures how spread out possible returns are around the expected return. If returns are tightly clustered around the average, variance is low (less risky). If returns scatter widely, variance is high (riskier). Standard deviation is the square root of variance and is the more commonly used risk metric. It's measured in the same units as returns (percentages), making it easier to interpret. Think of it this way: A savings account has very low standard deviation—returns barely fluctuate A small-cap stock has high standard deviation—returns jump around significantly The higher the standard deviation, the more uncertain the return. For investors, higher uncertainty usually means higher required returns to compensate for the extra risk. Why this matters on an exam: You'll likely need to understand that standard deviation measures risk and that higher standard deviation means more risk. You may need to calculate weighted portfolio risk (which is more complex due to correlation effects), but the fundamental concept is that variance and standard deviation quantify the dispersion of returns. Concept of Diversification Here's a powerful insight: you can reduce risk by combining investments whose returns don't move together perfectly. Suppose you own only a boat rental company. When it rains, you lose money. Now suppose you also own an umbrella company that makes money when it rains. Your two businesses' returns are negatively correlated—when one does poorly, the other does well. Your combined portfolio is less risky than either alone, even though each individual business is risky. This is diversification—combining assets whose returns are not perfectly correlated to reduce overall portfolio risk. Key principles: Combining two perfectly correlated assets (correlation = +1) provides no risk reduction Combining uncorrelated assets (correlation = 0) provides meaningful risk reduction Combining negatively correlated assets (correlation = -1) provides the most risk reduction The more assets you add, the more diversification benefits you gain Practical example: Stock and bond returns often move in different directions. In recessions, stocks fall but bonds often rise. A portfolio of both stocks and bonds is less volatile than 100% stocks, even though stocks offer higher long-term returns. The crucial takeaway: Diversification reduces risk without necessarily reducing expected return. This is why investors diversify rather than putting all money in the single "best" investment. Capital Asset Pricing Model (CAPM) The Capital Asset Pricing Model is a framework for understanding expected returns based on systematic risk. The model states: $$E(R) = Rf + \beta(Rm - Rf)$$ Where: $E(R)$ = expected return of the asset $Rf$ = risk-free rate (return on safe investments like Treasury bonds) $\beta$ = beta coefficient (measures systematic risk) $(Rm - Rf)$ = market risk premium (expected excess return of the overall market) Understanding beta: $\beta = 1$ means the asset moves exactly with the market $\beta > 1$ means the asset is more volatile than the market (higher risk) $\beta < 1$ means the asset is less volatile than the market (lower risk) Example: Suppose: Risk-free rate = 3% Market risk premium = 7% (market expected to return 10% vs. 3% risk-free rate) Stock has beta = 1.2 $$E(R) = 0.03 + 1.2(0.07) = 0.03 + 0.084 = 0.114 \text{ or } 11.4\%$$ The stock should earn 11.4% because it's 20% more volatile than the market, so investors require 20% more additional return (1.2 × 7% premium = 8.4%). Key insight: CAPM separates risk into two parts: Systematic risk (measured by beta)—market-wide risk you can't diversify away, so you need compensation Unsystematic risk—company-specific risk that diversification eliminates, so you don't need compensation Introduction to Option Pricing An option is a financial contract that gives you the right—but not the obligation—to buy or sell an underlying asset at a predetermined price. Definition and Types of Options There are two basic types: Call option: Gives the holder the right to buy the underlying asset at the strike price Put option: Gives the holder the right to sell the underlying asset at the strike price Key terminology: Strike price (or exercise price): The predetermined price at which you can buy/sell Underlying asset: The stock, bond, commodity, or other item the option is based on Expiration date: The deadline to exercise (use) the option Example: You buy a call option on Apple stock with a strike price of $150 and expiration in 3 months. This gives you the right (but not obligation) to buy Apple stock at $150 anytime before expiration, regardless of the actual market price. Why would this be valuable? If Apple stock rises to $160, you could exercise your option, buy at $150, and immediately sell at $160 for a $10 profit (minus the option cost). But if Apple falls to $140, you'd simply let the option expire worthless—you're not forced to buy. Why "not an obligation" matters: This is what makes options different from stocks or bonds. You have the choice to use it or not, and this flexibility has value. Factors Influencing Option Value Option value depends on five main factors: 1. Price of the underlying asset For call options: Higher asset price = higher option value (more likely to be in-the-money) For put options: Lower asset price = higher option value 2. Strike price relative to current price How far "in-the-money" or "out-of-the-money" is the option? An in-the-money call has higher value than an out-of-the-money call 3. Time remaining to expiration More time = higher option value (more opportunities for favorable price movements) This is why options lose value as expiration approaches 4. Volatility of the underlying asset Higher volatility = higher option value (more chance for large favorable moves) A stable stock option is worth less than a volatile stock option 5. Risk-free interest rate Higher rates = higher call option values (affects the present value of paying the strike price) Effects are typically small compared to other factors These five factors determine an option's theoretical value, and understanding their relationships is crucial. Black-Scholes Model Overview <extrainfo> The Black-Scholes formula is a mathematical model that calculates the theoretical price of European-style options (options that can only be exercised at expiration, not before). While you don't need to memorize the formula for most exams, you should understand: It uses all five factors mentioned above in a continuous-time framework It assumes certain market conditions (no transaction costs, efficient markets, lognormal price distributions) It provides a theoretical benchmark for option pricing Real market prices may differ from Black-Scholes predictions due to market frictions and psychology The model's primary contribution was showing that option values depend on volatility and time in quantifiable ways, revolutionizing options trading. </extrainfo>