The Core Concept of Time Value of Money (TVM)
The Time Value of Money (TVM) is the foundational principle of finance: a dollar today is worth more than a dollar tomorrow. Why? Because money you have now can be invested to earn interest, dividends, or capital gains.
This principle underpins nearly every financial decision — from saving and borrowing to valuing stocks and bonds.
Present Value (PV) vs. Future Value (FV)
- Future Value (FV): What a sum of money today will grow to in the future, given a rate of return. This process is called compounding.
- Present Value (PV): What a future sum of money is worth today. This is called discounting, and the rate used is the discount rate.
Basic TVM Formula (Single Cash Flow)
PV = FVₜ / (1 + r)ᵗ
Where:
- r = interest rate per period
- t = number of periods
Example: Future Value
You invest $1,000 at 6% annual interest for 3 years.
FV = 1000 × (1.06)³ = 1000 × 1.191 = $1,191
Example: Present Value
You need $5,000 in 5 years. Discount rate = 4%.
PV = 5000 / (1.04)⁵ = 5000 / 1.2167 ≈ $4,109
Continuous Compounding
Used in advanced finance (e.g., derivatives), this assumes interest is compounded continuously.
PV = FVₜ × e^(-r×t)
Where e ≈ 2.71828 (Euler's number)
Example: Continuous Compounding
$1,000 invested at 5% continuously compounded for 2 years:
FV = 1000 × e^(0.05×2) = 1000 × e^0.10 ≈ 1000 × 1.1052 = $1,105.20
TVM in Fixed-Income and Equity Valuation
The value of any financial asset is the present value of its expected future cash flows. Let's see how this applies to bonds and stocks.
Valuing Bonds
A bond's value is the sum of the present values of all its future cash flows: coupon payments and principal repayment.
- Zero-Coupon Bond (Discount Instrument): Pays no coupons. Bought at a discount, redeemed at face value.
- Coupon Bond: Pays periodic interest (PMT) and returns face value (FV) at maturity.
(Sum from i=1 to n)
Annuities: Regular Payments Over Time
An annuity is a series of equal payments made at regular intervals (e.g., mortgage, lease, retirement payout).
- Present Value of Ordinary Annuity (payments at end of each period):
- Present Value of Annuity Due (payments at beginning of each period):
- Future Value of Ordinary Annuity (payments at end of each period):
- Future Value of Annuity Due (payments at beginning of each period):
- Solve for Payment (A): Use when calculating loan payments or savings goals.
Example: Car Loan Payment (Ordinary Annuity)
You take a $20,000 car loan at 6% annual interest (0.5% monthly) for 5 years (60 months).
A = (0.005 × 20000) / [1 - (1.005)⁻⁶⁰] ≈ 100 / 0.2586 ≈ $386.66 per month
Perpetuity: Infinite Payments
A stream of equal payments that never ends (e.g., perpetual bond).
Example: Valuing a Perpetual Bond
A bond pays $50 annually forever. Required return = 8%.
PV = 50 / 0.08 = $625
Valuing Stocks Using TVM
A stock's value is the present value of all expected future dividends. Since stocks have no maturity, we use models that project dividends into the future.
Dividend Discount Models (DDM)
- Zero Growth Model: Dividends are constant forever → perpetuity.
- Constant Growth Model (Gordon Growth Model): Dividends grow at a constant rate g forever.
- Two-Stage Growth Model: Realistic for companies with high initial growth that later stabilizes.
- Stage 1 (High Growth): Calculate PV of dividends during high-growth phase (growth rate gₛ).
- Stage 2 (Stable Growth): Use Gordon model to find terminal value at end of Stage 1, then discount it back to today.
Where D₁ = D₀ × (1 + g)
⚠️ Must have r > g, or the model breaks down.
Total value = Sum of Stage 1 PVs + PV of terminal value.
Example: Gordon Growth Model
Next year's dividend D₁ = $2, required return r = 10%, growth rate g = 5%.
P₀ = 2 / (0.10 − 0.05) = 2 / 0.05 = $40
Implied Return and Growth
We can rearrange TVM formulas to find the implied return or implied growth rate based on current market prices.
For Bonds: Yield-to-Maturity (YTM)
Yield-to-Maturity (YTM) is the internal rate of return (IRR) on a bond — the discount rate that makes the PV of all future cash flows equal to the bond's current price.
- For Zero-Coupon Bond:
r = (FV / PV)^(1/t) − 1
- For Coupon Bonds: No closed-form solution. Use financial calculator or iterative method.
Example: YTM of a Zero-Coupon Bond
Face value = $1,000, price = $800, maturity = 5 years.
r = (1000 / 800)^(1/5) − 1 = (1.25)^0.2 − 1 ≈ 1.0456 − 1 = 4.56%
For Stocks: Implied Return and Growth
- Implied Required Return (r):
r = (D₁ / P₀) + g
- Implied Growth Rate (g):
g = r − (D₁ / P₀)
Link to Price-to-Earnings (P/E) Ratio
The P/E ratio reflects investor expectations about growth, payout, and risk.
Where D₀/E₀ = dividend payout ratio.
Forward P/E Ratio: Uses next year's earnings E₁
✅ Higher P/E if: High payout, high growth, low required return
❌ Lower P/E if: High risk (high r)
The No-Arbitrage Principle and Cash Flow Additivity
The No-Arbitrage Principle states that two assets with identical cash flows must have the same price — otherwise, investors could make a riskless profit (arbitrage).
This leads to the Cash Flow Additivity Principle: The PV of a series of cash flows is the sum of the PVs of each individual flow.
Applications of No-Arbitrage
- Implied Forward Rates: The rate that must prevail in the future to prevent arbitrage between bonds of different maturities.
- Interest Rate Parity (IRP): Links interest rates and exchange rates to prevent arbitrage in currency markets.
- Option Pricing: An option's price must equal the cost of a replicating portfolio (a combination of the underlying asset and risk-free borrowing/lending) that produces the same payoffs.
Where f₁,₁ = one-year forward rate one year from now.
Example: Calculating Forward Rate
1-year spot rate = 3%, 2-year spot rate = 4%.
(1.04)² = (1.03) × (1 + f)
1.0816 = 1.03 × (1 + f)
1 + f = 1.0816 / 1.03 ≈ 1.0501
f ≈ 5.01%
Where F = forward exchange rate, S₀ = spot rate, r_d = domestic rate, r_f = foreign rate.
The hedge ratio (number of shares to buy per option sold) ensures the portfolio is riskless.