12.1 Financial mathematics

Fundamentals of Financial Mathematics

Financial mathematics uses mathematical methods to model and analyze financial markets. Because prices, interest rates, and returns are inherently uncertain, stochastic processes are the natural tool for capturing that randomness. This section covers the core building blocks: time value of money, interest rate mechanics, bond pricing, derivatives valuation, portfolio theory, and stochastic interest rate models.

Time Value of Money

A dollar today is worth more than a dollar tomorrow, because today's dollar can be invested and earn a return. This principle underpins nearly every calculation in financial mathematics.

Present Value vs. Future Value

Present value (PV) is the current worth of a future cash flow, discounted at some rate of return. Future value (FV) is the amount a current sum grows to after earning interest over time. The basic relationship is:

$FV = PV \cdot (1 + r)^n$

where $r$ is the interest rate per period and $n$ is the number of compounding periods. Rearranging gives:

$PV = \frac{FV}{(1 + r)^n}$

These two formulas are the foundation for everything that follows.

Compounding Periods

Compounding means earning interest on previously earned interest. The frequency of compounding matters:

• Annually: interest added once per year
• Semi-annually: twice per year (common for bonds)
• Quarterly/Monthly: four or twelve times per year
• Continuously: the theoretical limit as compounding frequency approaches infinity

More frequent compounding produces a higher effective return for the same nominal rate, because each interest payment starts earning its own interest sooner.

Annuities and Perpetuities

An annuity is a series of equal payments at equal intervals over a fixed period. Mortgages and car loans are classic examples. The present value of an ordinary annuity paying $C$ per period for $n$ periods at rate $r$ is:

$PV = C \cdot \frac{1 - (1 + r)^{-n}}{r}$

A perpetuity is an annuity that never ends. Its present value simplifies to:

$PV = \frac{C}{r}$

This formula works because the payments stretch to infinity, and the geometric series converges when $r > 0$.

Interest Rates

Simple vs. Compound Interest

• Simple interest is computed only on the original principal: $FV = PV \cdot (1 + rt)$
• Compound interest is computed on the principal plus all accumulated interest: $FV = PV \cdot (1 + r)^t$

Over short horizons the difference is small, but compound interest produces exponential growth, so the gap widens dramatically over longer periods. For example, $1,000 at 5% simple interest yields $1,500 after 10 years, while compound interest yields about $1,629.

Nominal vs. Effective Rates

The nominal rate is the stated annual rate, ignoring compounding effects. The effective annual rate (EAR) reflects the actual return after compounding:

$EAR = \left(1 + \frac{r_{nom}}{m}\right)^m - 1$

where $m$ is the number of compounding periods per year. A nominal rate of 12% compounded monthly gives an EAR of about 12.68%. Whenever compounding is more frequent than annual, the effective rate exceeds the nominal rate.

Continuous Compounding

Continuous compounding is the limiting case as $m \to \infty$. The future value formula becomes:

$FV = PV \cdot e^{rt}$

where $e \approx 2.71828$. This formulation is especially important in stochastic finance because it connects naturally to the exponential and logarithmic structures used in models like geometric Brownian motion.

Bonds and Loans

Coupon Payments

A bond is a debt instrument that pays periodic interest (coupons) and returns the face value at maturity. If a bond has face value $F$, annual coupon rate $c$, and pays semi-annually, each coupon payment is $\frac{cF}{2}$. The bond's price equals the sum of all discounted cash flows:

$P = \sum_{i=1}^{n} \frac{C_i}{(1+y)^i} + \frac{F}{(1+y)^n}$

where $y$ is the discount rate per period and $n$ is the total number of coupon periods.

Yield to Maturity

Yield to maturity (YTM) is the single discount rate that equates a bond's market price to the present value of its future cash flows. It represents the total annualized return if the bond is held to maturity and all coupons are reinvested at the same rate.

YTM accounts for the coupon rate, time to maturity, and the difference between the purchase price and face value. There's no closed-form solution for YTM in general; it's typically found by numerical methods (e.g., Newton-Raphson or a financial calculator's iterative solver).

Amortization Schedules

An amortization schedule decomposes each loan payment into interest and principal components.

1. Calculate the fixed periodic payment using the annuity formula.
2. For each period, compute the interest portion as the outstanding balance times the periodic rate.
3. The principal portion is the total payment minus the interest portion.
4. Subtract the principal portion from the outstanding balance.
5. Repeat until the balance reaches zero.

Early payments are mostly interest; later payments are mostly principal. This pattern arises because the interest charge shrinks as the balance declines.

Investment Valuation

Net Present Value (NPV)

Net present value is the sum of all discounted future cash flows minus the initial investment:

$NPV = \sum_{t=0}^{T} \frac{CF_t}{(1+r)^t}$

where $CF_0$ is typically the (negative) initial outlay. A positive NPV means the project is expected to create value above the required rate of return. A negative NPV means it destroys value. NPV is the standard tool for capital budgeting decisions because it directly measures value creation in today's dollars.

Internal Rate of Return (IRR)

The internal rate of return is the discount rate $r^*$ that sets NPV to zero:

$\sum_{t=0}^{T} \frac{CF_t}{(1+r^*)^t} = 0$

If $r^*$ exceeds the required rate of return, the project is attractive. One limitation: projects with non-conventional cash flows (alternating signs) can have multiple IRRs, making interpretation ambiguous. In such cases, NPV is more reliable.

Payback Period

The payback period is the time needed to recover the initial investment from cumulative cash inflows. It's simple to compute but has two significant weaknesses:

• It ignores the time value of money (a dollar received in year 5 counts the same as one in year 1).
• It ignores all cash flows after the payback date.

The discounted payback period partially addresses the first issue by using discounted cash flows, but the second limitation remains.

Financial Derivatives

Forwards vs. Futures Contracts

Both forwards and futures are agreements to buy or sell an asset at a predetermined price on a future date. The key differences:

Daily marking-to-market means futures holders realize gains and losses incrementally, which affects pricing slightly compared to forwards when interest rates are stochastic.

Call vs. Put Options

• A call option gives the holder the right (not obligation) to buy the underlying asset at the strike price $K$ before or at expiration.
• A put option gives the holder the right (not obligation) to sell at $K$.

The payoff at expiration for a European call is $\max(S_T - K, 0)$ and for a European put is $\max(K - S_T, 0)$, where $S_T$ is the asset price at expiration. The option buyer pays a premium upfront for this right, and the most they can lose is that premium.

Black-Scholes Pricing Model

The Black-Scholes model provides a closed-form price for European options. It assumes the underlying asset price follows a geometric Brownian motion:

$dS = \mu S \, dt + \sigma S \, dW_t$

where $\mu$ is the drift, $\sigma$ is the volatility, and $W_t$ is a standard Wiener process. The European call price is:

$C = S_0 \Phi(d_1) - K e^{-rT} \Phi(d_2)$

where:

$d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T}$

and $\Phi$ is the standard normal CDF. The five inputs are: current stock price $S_0$, strike price $K$, time to expiration $T$, risk-free rate $r$, and volatility $\sigma$. Notice that the drift $\mu$ drops out under risk-neutral pricing, which is a deep consequence of no-arbitrage arguments.

Key assumptions to keep in mind: constant volatility, no dividends (in the basic version), continuous trading, and no transaction costs. Real markets violate all of these, which is why practitioners use extensions and adjustments.

Portfolio Theory

Risk and Return Trade-off

Investors face a fundamental trade-off: higher expected returns generally require accepting more risk. Harry Markowitz's mean-variance framework quantifies this by measuring return as the expected value and risk as the variance (or standard deviation) of portfolio returns.

The efficient frontier is the set of portfolios that offer the maximum expected return for each level of risk. Any portfolio below the frontier is suboptimal because you could get higher return for the same risk, or lower risk for the same return.

Diversification Benefits

Diversification reduces portfolio risk by combining assets whose returns don't move in perfect lockstep. The portfolio variance for two assets is:

$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1 w_2 \sigma_1 \sigma_2 \rho_{12}$

When the correlation $\rho_{12} < 1$, the portfolio standard deviation is less than the weighted average of individual standard deviations. This is the mathematical basis for the "free lunch" of diversification.

Diversification eliminates unsystematic risk (company- or industry-specific), but systematic risk (market-wide factors like recessions or interest rate changes) cannot be diversified away.

Optimal Portfolio Selection

The Markowitz mean-variance optimization finds portfolio weights that minimize variance for a target expected return (or maximize return for a target variance). In practice:

1. Estimate expected returns, variances, and covariances for all assets.
2. Solve the constrained optimization problem (often using quadratic programming).
3. Trace out the efficient frontier by varying the target return.
4. Select the portfolio on the frontier that matches your risk tolerance.

Adding a risk-free asset creates the capital market line, and the tangency portfolio (the point where a line from the risk-free rate is tangent to the efficient frontier) becomes the optimal risky portfolio for all investors.

Capital Asset Pricing Model (CAPM)

Systematic vs. Unsystematic Risk

• Systematic risk affects the entire market: interest rate shifts, recessions, geopolitical events. It cannot be diversified away.
• Unsystematic risk is specific to a firm or industry: a CEO departure, a product recall, a labor strike. It can be eliminated through diversification.

CAPM's central insight is that only systematic risk is priced. Since rational investors diversify, the market doesn't reward you for bearing risk that could have been eliminated.

Beta Coefficient

Beta ($\beta$) measures a security's sensitivity to market movements:

$\beta_i = \frac{\text{Cov}(R_i, R_m)}{\text{Var}(R_m)}$

• $\beta = 1$: the stock moves with the market
• $\beta > 1$: more volatile than the market (e.g., tech stocks often have $\beta \approx 1.2 - 1.5$)
• $\beta < 1$: less volatile (e.g., utilities often have $\beta \approx 0.4 - 0.6$)

Security Market Line

The security market line (SML) plots expected return against beta. The CAPM equation is:

$E(R_i) = R_f + \beta_i [E(R_m) - R_f]$

where $R_f$ is the risk-free rate and $E(R_m) - R_f$ is the market risk premium. The SML's intercept is $R_f$ and its slope is the market risk premium.

• Securities plotting above the SML are undervalued (they offer more return than CAPM predicts for their risk).
• Securities plotting below the SML are overvalued.

This relationship holds in equilibrium under CAPM assumptions. In practice, deviations from the SML are the basis for active investment strategies.

Stochastic Interest Rate Models

Interest rates aren't constant; they evolve randomly over time. Stochastic interest rate models capture this behavior and are essential for pricing bonds, interest rate derivatives, and managing fixed-income risk.

Vasicek Model

The Vasicek model describes the short rate $r_t$ as an Ornstein-Uhlenbeck process:

$dr_t = a(b - r_t)\,dt + \sigma\,dW_t$

where $b$ is the long-term mean rate, $a$ is the speed of mean reversion, and $\sigma$ is the volatility. The mean-reversion property means that when rates are above $b$ they tend to drift down, and when below $b$ they drift up. This is realistic behavior for interest rates.

One drawback: the Vasicek model allows negative interest rates, since the noise term is additive and normally distributed. Before the era of negative policy rates, this was considered a significant limitation.

Cox-Ingersoll-Ross (CIR) Model

The CIR model modifies Vasicek by making the volatility proportional to $\sqrt{r_t}$:

$dr_t = a(b - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t$

This square-root diffusion ensures that when $r_t$ approaches zero, the volatility shrinks, preventing the rate from going negative (provided $2ab \geq \sigma^2$, the Feller condition). The CIR model retains mean reversion while producing a more realistic distribution of rates.

Heath-Jarrow-Morton (HJM) Framework

Rather than modeling a single short rate, the HJM framework models the entire forward rate curve $f(t, T)$ as a stochastic process:

$df(t,T) = \alpha(t,T)\,dt + \sigma(t,T)\,dW_t$

The key result is that the no-arbitrage condition pins down the drift $\alpha(t,T)$ entirely in terms of the volatility function $\sigma(t,T)$. This means you only need to specify the volatility structure; the drift follows automatically.

HJM is a very general framework. Both the Vasicek and CIR models can be recovered as special cases with particular volatility specifications. The trade-off is that HJM models the entire curve (infinite-dimensional), which makes implementation more complex. In practice, finite-factor approximations are used.