/time-value-of-money

name: time-value-of-money
description: "Calculate present value, future value, NPV, IRR for projects and loans, loan payments, and amortization schedules across all compounding conventions. Use when the user asks about discounting cash flows, valuing an annuity or perpetuity, comparing investments with different timing, building a mortgage amortization table, evaluating whether a project is worth pursuing, or solving for the rate that equates cash flows (project IRR, loan IRR, yield on an investment). Also trigger when users mention 'what is it worth today', 'how much will I have in 20 years', 'monthly payment on a loan', 'discount rate', 'Gordon growth model', 'effective annual rate', 'continuous compounding', or ask how to compare a lump sum versus a stream of payments. For portfolio money-weighted return (dollar-weighted IRR on an investor's contributions and withdrawals), use return-calculations instead."

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: | Frequency | m | |-----------|---| | Annual | 1 | | Semi-annual | 2 | | Quarterly | 4 | | Monthly | 12 | | Daily | 365 | | Continuous | infinity |

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

| Formula | Expression | Use Case | |---------|-----------|----------| | Future Value | `FV = PV * (1 + r)^n` | Compound a lump sum forward | | Present Value | `PV = FV / (1 + r)^n` | Discount a future lump sum | | EAR | `(1 + r_nom/m)^m - 1` | Compare rates across compounding frequencies | | Continuous FV | `FV = PV * e^(r*t)` | Continuous compounding | | Ordinary Annuity PV | `PMT * [1 - (1+r)^(-n)] / r` | Loan payments, lease valuation | | Annuity Due PV | `PMT * [1 - (1+r)^(-n)] / r * (1+r)` | Rent, insurance (paid in advance) | | Growing Annuity PV | `PMT/(r-g) * [1 - ((1+g)/(1+r))^n]` | Salary streams, growing dividends | | Perpetuity PV | `PMT / r` | Preferred stock, consol bonds | | Growing Perpetuity PV | `PMT / (r - g)` | Gordon Growth Model | | NPV | `sum(CF_t / (1+r)^t)` | Project/investment evaluation | | IRR | `solve: sum(CF_t / (1+r)^t) = 0` | Return metric for uneven cash flows |

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