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Calculates present value, future value, NPV, IRR, loan payments, and amortization schedules across all compounding conventions. Useful for financial modeling, investment valuation, and cash flow analysis.
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This skill enables Claude to perform present value, future value, and discounted cash flow calculations across all standard compounding conventions. It covers annuities, perpetuities, amortization schedules, NPV, and IRR, providing the foundational building blocks for virtually all financial valuation and planning tasks.
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This skill enables Claude to perform present value, future value, and discounted cash flow calculations across all standard compounding conventions. It covers annuities, perpetuities, amortization schedules, NPV, and IRR, providing the foundational building blocks for virtually all financial valuation and planning tasks.
0 — Mathematical Foundations
both (retrospective for valuing past cash flows, prospective for projecting future values)
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.
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.
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 |
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:
$$FV = PMT \times \frac{(1 + r)^n - 1}{r}$$
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:
$$FV = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$$
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.
An infinite stream of equal payments.
$$PV = \frac{PMT}{r}$$
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.
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.
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.
Each payment on an amortizing loan is split into an interest component and a principal component:
Interest_t = Balance_{t-1} * rPrincipal_t = PMT - Interest_tBalance_t = Balance_{t-1} - Principal_tOver time, the interest portion decreases and the principal portion increases.
| 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 |
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.
Given: A project requires an initial investment of $50,000 and produces the following cash flows:
The required rate of return (discount rate) is 10%.
Calculate: The NPV and whether the project should be accepted.
Solution:
Discount each cash flow to present value:
PV(CF_0) = -50,000 / (1.10)^0 = -50,000.00
PV(CF_1) = 12,000 / (1.10)^1 = 10,909.09
PV(CF_2) = 15,000 / (1.10)^2 = 12,396.69
PV(CF_3) = 18,000 / (1.10)^3 = 13,524.21
PV(CF_4) = 22,000 / (1.10)^4 = 15,026.30
PV(CF_5) = 25,000 / (1.10)^5 = 15,523.03
Sum all present values:
NPV = -50,000.00 + 10,909.09 + 12,396.69 + 13,524.21 + 15,026.30 + 15,523.03
NPV = +$17,379.32
Since NPV is positive ($17,379.32), the project creates value and should be accepted. It earns more than the 10% required rate of return.
To find the IRR, we would solve for the rate where NPV = 0. Numerically, the IRR for this cash flow stream is approximately 21.0%, well above the 10% hurdle rate.
(1 + annual rate) unless converting from EAR.(1 + r) adjustment factor will undervalue annuity-due streams.See scripts/time_value_of_money.py for computational helpers.