Time Value Of Money

Time Value of Money

Core Concepts

Future Value (FV)

The value of a present sum after earning interest for n periods at rate r per period.

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

Future value grows exponentially with time, which is the mathematical basis of compound interest.

Present Value (PV)

The current worth of a future sum, discounted back at rate r for n periods. This is the inverse of future value.

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

Present value is the cornerstone of all valuation: a dollar today is worth more than a dollar tomorrow because of the opportunity cost of capital.

Compounding Conventions

Interest can compound at different frequencies. The nominal annual rate r_nom compounded m times per year produces different effective yields.

Discrete compounding (m times per year):

$$FV = PV \times \left(1 + \frac{r_{nom}}{m}\right)^{m \times t}$$

Continuous compounding:

$$FV = PV \times e^{r \times t}$$

Effective Annual Rate (EAR):

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

For continuous compounding: EAR = e^(r_nom) - 1

Common frequencies:

Ordinary Annuity

A series of equal payments made at the end of each period for n periods.

Present Value:

$$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$$

Future Value:

$$FV = PMT \times \frac{(1 + r)^n - 1}{r}$$

Annuity Due

A series of equal payments made at the beginning of each period. Each cash flow is one period closer than in an ordinary annuity, so values are scaled by (1 + r).

Present Value:

$$PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$$

Future Value:

$$FV = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$

Growing Annuity

A finite series of payments that grow at a constant rate g per period, where g != r.

Present Value:

$$PV = \frac{PMT}{r - g} \times \left[1 - \left(\frac{1 + g}{1 + r}\right)^n\right]$$

This is widely used in equity valuation (e.g., multi-stage dividend discount models) and salary/pension projections.

Perpetuity

An infinite stream of equal payments.

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

Growing Perpetuity

An infinite stream of payments growing at constant rate g, where g < r for convergence.

$$PV = \frac{PMT}{r - g}$$

This is the Gordon Growth Model when applied to dividends.

Net Present Value (NPV)

The sum of all discounted cash flows, including the initial investment. A positive NPV indicates value creation.

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

Typically, CF_0 is a negative outflow (initial investment), and subsequent CF_t are inflows.

Internal Rate of Return (IRR)

The discount rate r that makes the NPV of all cash flows exactly zero.

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

IRR is solved numerically (Newton-Raphson or bisection) since there is no closed-form solution for general cash flow streams. For conventional cash flows (one sign change), a unique IRR exists.

Amortization

Each payment on an amortizing loan is split into an interest component and a principal component:

• Interest portion: Interest_t = Balance_{t-1} * r
• Principal portion: Principal_t = PMT - Interest_t
• Remaining balance: Balance_t = Balance_{t-1} - Principal_t

Over time, the interest portion decreases and the principal portion increases.

Key Formulas

Worked Examples

Example 1: Monthly Mortgage Payment

Given: A $300,000 mortgage at a 6.5% annual interest rate, fixed for 30 years, with monthly payments (ordinary annuity).

Calculate: The monthly payment amount.

Solution:

First, convert the annual rate to a monthly rate and years to months:

r_monthly = 0.065 / 12 = 0.00541667
n = 30 * 12 = 360 months

Using the ordinary annuity present value formula, solve for PMT:

PV = PMT * [1 - (1 + r)^(-n)] / r

300,000 = PMT * [1 - (1.00541667)^(-360)] / 0.00541667

Compute the annuity factor:

(1.00541667)^360 = 6.99179
(1.00541667)^(-360) = 0.143010
1 - 0.143010 = 0.856990
0.856990 / 0.00541667 = 158.2108

Solve for PMT:

PMT = 300,000 / 158.2108 = $1,896.20

The monthly mortgage payment is $1,896.20.

Over 30 years, total payments = 360 * $1,896.20 = $682,632, meaning total interest paid is $682,632 - $300,000 = $382,632.