Time Value of Money

The foundation: a rupee today is worth more than a rupee tomorrow because it can be invested to earn a return.

Basic Formulas

  • FV = PV × (1 + r)^n
  • PV = FV / (1 + r)^n

Annuity Formulas

  • PV of ordinary annuity = PMT × [1 − (1+r)^(-n)] / r
  • FV of ordinary annuity = PMT × [(1+r)^n − 1] / r
  • Annuity Due: Payments at beginning of period. Multiply ordinary annuity by (1 + r).
  • Perpetuity: PV = PMT / r (payments continue forever)

Effective Annual Rate (EAR)

EAR = (1 + r/m)^m − 1, where m = compounding periods per year.

Example: 12% nominal, monthly compounding: EAR = (1 + 0.12/12)^12 − 1 = 12.68%

Continuous compounding: EAR = e^r − 1

Statistics

Measures of Central Tendency

  • Arithmetic Mean: Simple average — use for cross-sectional averages
  • Geometric Mean: For compounded returns over time. G = (∏(1+Rt))^(1/n) − 1
  • Harmonic Mean: For averaging price-paid at different times (dollar-cost averaging)
  • Rule: Arithmetic Mean ≥ Geometric Mean ≥ Harmonic Mean (for non-negative returns)

Dispersion Measures

  • Range: Max − Min (simple but sensitive to outliers)
  • Mean Absolute Deviation (MAD): Average of |Xi − X̄|
  • Variance (σ²): Average of (Xi − X̄)² — uses squared deviations; population vs sample (÷n vs ÷n-1)
  • Standard Deviation (σ): √Variance — in same units as original data
  • Coefficient of Variation (CV): σ / X̄ — relative risk per unit of return

Covariance and Correlation

Covariance measures how two variables co-move: Cov(A,B) = E[(RA − E(RA)) × (RB − E(RB))]

Correlation = Cov(A,B) / (σA × σB). Range: −1 to +1.

  • +1: Perfect positive correlation — assets move in lockstep
  • 0: No linear relationship — diversification benefit
  • −1: Perfect negative correlation — maximum diversification

Probability and Distributions

Normal Distribution

Bell-shaped, symmetric. Mean = Median = Mode. Fully described by mean (μ) and standard deviation (σ).

Key intervals: 68% of observations within ±1σ; 95% within ±1.96σ; 99% within ±2.58σ

Standard Normal (Z): Z = (X − μ) / σ. Use Z-table for probability lookups.

Lognormal Distribution

If X is normally distributed, e^X is lognormally distributed. Stock prices are often modelled as lognormal (because prices can't be negative).

t-Distribution

Used when population standard deviation is unknown (replaced by sample std dev). Heavier tails than normal. As degrees of freedom increase, t-distribution → normal distribution.

Hypothesis Testing Framework

  1. State H₀ (null) and Hₐ (alternative)
  2. Choose significance level α (typically 5% or 1%)
  3. Calculate test statistic (t = (X̄ − μ₀) / (s/√n) for mean tests)
  4. Determine critical value from t-table or z-table for given α and degrees of freedom
  5. Decision: Reject H₀ if |test statistic| > critical value
  • Type I Error (α): Rejecting a true H₀ (false positive)
  • Type II Error (β): Failing to reject a false H₀ (false negative)
  • Power of a test = 1 − β = probability of correctly rejecting a false null