Σ Quantitative Methods

Time Value of Money Formula — PV, FV, Annuity & Perpetuity

Master the Time Value of Money formula. Present value, future value, annuity, and perpetuity calculations explained with examples. Essential for CFA Level 1.

Key Concepts

Present Value (PV)

The current worth of a future cash flow discounted at the appropriate rate. Foundation of all valuation.

Future Value (FV)

The value of a current amount after earning interest over a specified period.

Annuity

A series of equal cash flows at regular intervals. Ordinary annuity: payments at END. Annuity due: payments at BEGINNING.

Perpetuity

An annuity that pays forever. PV = PMT/r. Growing perpetuity: PV = PMT/(r - g).

Cash Flow Additivity

PV of a series of cash flows equals the sum of the PVs of individual cash flows. Basis for no-arbitrage pricing.

Implied Return

The discount rate that equates the PV of future cash flows to the current market price. For bonds: YTM; for stocks: implied return from DDM.

Implied Forward Rate

Future interest rate implied by current spot rates via no-arbitrage. (1+S<sub>2</sub>)² = (1+S<sub>1</sub>)(1+f<sub>1,1</sub>).

Formulas

From this module

Future Value

FV = PV × (1 + r)n

Where: r = interest rate per period, n = number of periods

Present Value

PV = FV / (1 + r)n

Where: r = discount rate, n = periods

PV of Ordinary Annuity

PV = PMT × [(1 - (1+r)-n) / r]

Where: PMT = periodic payment

PV of Perpetuity

PV = PMT / r

Where: PMT = periodic payment, r = discount rate

PV of Growing Perpetuity

PV = PMT / (r - g)

Where: g = growth rate (must be < r)

EAR (Effective Annual Rate)

EAR = (1 + rperiodic)m - 1

Where: m = compounding periods per year

Implied Forward Rate

(1 + Sn)n = (1 + Sn-1)n-1 × (1 + fn-1,1)

Where: S = spot rate, f = forward rate

Master Formula Sheet -- Quantitative Methods

Future Value

FV = PV × (1 + r)n

Single lump sum compounding

Present Value

PV = FV / (1 + r)n

Discounting future cash flows

Annuity PV

PV = PMT × [1 - (1+r)-n] / r

Equal periodic payments

Effective Annual Rate

EAR = (1 + r/m)m - 1

m = compounding periods per year

Holding Period Return

HPR = (P₁ - P₀ + D) / P₀

Total return including income

Population Variance

σ² = Σ(Xᵢ - μ)² / N

Divide by N for population

Sample Variance

s² = Σ(Xᵢ - X̄)² / (n-1)

Divide by n-1 for sample (Bessel's correction)

Coefficient of Variation

CV = σ / μ

Risk per unit of return

Sharpe Ratio

Sharpe = (R̄ₚ - Rᶠ) / σₚ

Excess return per unit of total risk

Bayes' Formula

P(A|B) = P(B|A) × P(A) / P(B)

Updating probabilities with new info

Correlation

ρ = Cov(X,Y) / (σₓ × σᵧ)

Standardized covariance, -1 to +1

Linear Regression

Y = b₀ + b₁X + ε

b₁ = Cov(X,Y)/Var(X)

Confidence Interval

X̄ ± z(α/2) × (σ/√n)

z = 1.96 for 95% CI

Test Statistic

t = (X̄ - μ₀) / (s/√n)

Compare to critical value

Decision Frameworks

Ordinary Annuity vs Annuity Due?

Use when:

  • Ordinary annuity: most bonds (coupons at end of period), most loans
  • Annuity due: lease payments at start, insurance premiums

Avoid when:

  • Using ordinary annuity formula when payments occur at beginning of period

When to use Cash Flow Additivity?

Use when:

  • Pricing bonds by discounting each coupon separately at spot rates
  • No-arbitrage pricing of derivatives
  • Implied forward rate calculations

Avoid when:

  • When cash flows are not independent or when there are embedded options

Test Your Understanding

What is the present value of $10,000 to be received in 5 years at a discount rate of 8%?

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