Financial Mathematics: Interest Rates

• Financial mathematics uses mathematical methods to address financial problems.
• Calculus, probability, statistics, and optimization are used to model and solve financial problems.
• Common areas of application include pricing derivatives, managing risk, and portfolio optimization.

Interest Rates

• Interest rates are the cost to borrow money or the return on an investment.
• They are typically expressed as an annual percentage.
• Interest can be categorized as simple or compound.
• Simple interest is calculated solely on the principal amount.
• Compound interest includes interest on the principal and accumulated interest.
• The annual percentage rate (APR) represents the annual interest rate without accounting for compounding.
• The effective annual rate (EAR) factors in the effect of compounding.
• EAR is calculated using the formula: EAR = (1 + (APR/n))^n - 1, where n is the number of compounding periods per year.

Simple Interest

• Simple Interest = Principal x Rate x Time.
  • Principal refers to the initial amount
  • Rate is the annual interest rate
  • Time represents the duration in years
• Future value (FV) with simple interest: FV = Principal + Interest = P(1 + rt).

Compound Interest

• Compound interest involves earning interest on the principal and previously earned interest.
• To find compound interest use this formula: FV = PV(1 + r/n)^(nt).
  • FV = future value
  • PV = present value (principal)
  • r = annual interest rate
  • n = compounding periods per year
  • t = number of years
• Continuously compounded interest occurs when 'n' approaches infinity.
• Continuously compounded interest uses this formula: FV = PV * e^(rt).
  • 'e' is the base of the natural logarithm (≈ 2.71828)

Present Value

• Present value (PV) is the current worth of a future sum of money or cash flow stream.
• An appropriate interest rate is used to discount the future amount back to the present when calculated.
• Present value is calculated as: PV = FV / (1 + r)^t.
  • FV = future value
  • r = discount (interest) rate
  • t = number of years
• Use the formula: PV = FV * e^(-rt) for continuously compounded interest.

Future Value

• Future value (FV) is the value of an asset or investment at a future date, based on a growth rate.
• It is the amount a present sum will grow to, given an interest rate over time.
• The basic future value formula is: FV = PV (1 + r)^t.
  • PV = present value
  • r = interest rate per period
  • t = number of periods
• Continuously compounded interest formula: FV = PV * e^(rt).

Annuities

• An annuity consists of a series of equal payments made at regular intervals.
• Annuities are either ordinary (payments at period's end) or due (payments at period's beginning).
• Present value of an ordinary annuity calculation: PV = PMT * [1 - (1 + r)^-t] / r.
  • PMT = payment amount
  • r = interest rate per period
  • t = number of periods
• Future value of an ordinary annuity calculation: FV = PMT * [(1 + r)^t - 1] / r.
• Present value of an annuity due calculation: PV = PMT * [1 - (1 + r)^-t] / r * (1 + r).
• Future value of an annuity due calculation: FV = PMT * [(1 + r)^t - 1] / r * (1 + r).

Perpetuities

• A perpetuity is an annuity that continues indefinitely.
• Perpetuity present value calculation: PV = PMT / r.
  • PMT = payment amount per period
  • r = interest rate per period

Amortization

• Amortization involves repaying a debt over time through regular payments.
• Each payment covers principal and interest.
• The loan amortization formula determines the payment amount: PMT = PV * [r(1 + r)^t] / [(1 + r)^t - 1].
  • PV = present value (loan amount)
  • r = interest rate per period
  • t = number of periods
• An amortization schedule details the principal and interest components of each payment.

Time Value of Money

• Money available today is worth more than the same amount in the future due to its earning potential, which is the time value of money (TVM).
• Key TVM concepts: Present Value (PV), Future Value (FV), Interest Rate (r), Number of Periods (n), Payment Amount (PMT).
• TVM calculations are fundamental for investment decisions, capital budgeting, and financial planning.

Inflation

• Inflation reflects the increasing general price level of goods/services, reducing purchasing power.
• Real interest rate adjusts for inflation: Real interest rate ≈ Nominal interest rate - Inflation rate.
• Incorporate inflation in a future value calculation: FV = PV * (1 + r)^t / (1 + i)^t.
  • 'i' represents the inflation rate.

Risk and Return

• Higher risk is linked to the potential for higher returns.
• Investors demand higher returns for bearing more risk.
• Common risk measures: Standard deviation, Beta.
• The risk-free rate is the theoretical return rate of a zero-risk investment.
• The Capital Asset Pricing Model (CAPM) relates risk and expected return: Expected Return = Risk-Free Rate + Beta * (Market Return - Risk-Free Rate).

Bonds

• Bonds are debt instruments used by corporations/governments to raise capital.
• Key bond features: Face (par) value, Coupon rate, Maturity date, Yield to maturity (YTM).
• A bond's present value is the sum of the present values of future coupon payments and face value.
• Formula to calculate a bond's value: Bond Value = (C / (1+r)^1) + (C / (1+r)^2) + ... + (C / (1+r)^n) + (FV / (1+r)^n).
  • C = coupon payment
  • r = discount rate (YTM)
  • FV = face value
  • n = number of periods
• YTM is the anticipated total return if a bond is held to maturity.

Stocks

• Stocks signify ownership in a corporation.
• Common stock includes voting rights and potential dividends.
• Preferred stock typically lacks voting rights but offers a fixed dividend.
• Stock valuation methods: Dividend Discount Model (DDM), Free Cash Flow (FCF) Model, Relative valuation.
• The Gordon Growth Model (a DDM): Stock Price = Expected Dividend / (Discount Rate - Dividend Growth Rate).

Derivatives

• Derivatives derive their value from an underlying asset.
• Common derivatives: Options, Futures, Forwards, Swaps.
• Options provide the right (not obligation) to buy (call) or sell (put) an asset at a set price on/before a date.
• Futures contracts represent agreements to buy/sell an asset at a future date/price.

Portfolio Management

• Portfolio management decisions involve investment mix and policy, aligning investments with goals, allocating assets, and balancing risk with performance.
• Key concepts: Diversification, Asset allocation, Risk management.
• Diversification reduces risk by investing in varied assets.
• Asset allocation optimizes risk-adjusted returns by dividing investments among asset classes.