Finance

Important Concepts

  • Interest is money paid at a regular rate in order to use lent money, or to delay the repayment of debt.

  • Inflation is the increase in the price of goods and services with respect to time. Inflation means that a fixed amount of capital will be able to purchase fewer goods and services as prices increase.

  • Depreciation is the decrease in the value of an asset with respect to time.

  • The principal is the original loan amount that is borrowed.

  • The principal portion of a payment is the portion that reduces the outstanding loan balance (principal).

  • The interest portion is the payment covering the interest charged for that period.

  • Amortisation is a process whereby a loan is repaid through a series of regular payments. In an amortised loan, interest for each period is calculated on the outstanding loan balance, therefore as regular payments reduce the principal, the interest portion of each payment decreases over time, whilst the principal portion increases.

Parameters and Conventions

In order to work with financial formulas, it is crucial to first understand the standard notation on which they depend.

| Calculator Label | Symbol in This Book | Meaning | | —————- | ——————- | ————————————– | | N | $N$ | Total number of compounding periods | | I% | $r$ | Nominal annual interest rate (%) | | | $r_{\text{eff}}$ | Effective annual interest rate (%) | | PV | $V_0$ | Present value of the loan/investment | | PMT | $M$ | Payment per period | | FV | $V_f$ | Future value of loan/investment | | P/Y | $n_p$ | Number of payments per year | | C/Y | $n_c$ | Number of compounding periods per year |

Note: This topic is primarily solved using calculators or software tools which allow for the automation of iterative calculations. Regardless, it is important to recognise the meaning of each parameter and their relation to the relevant formulae.

Interest Rates

Interest rates are a measure of the cost of borrowing money, or the return on investment over a set period of time.

\[\boxed{\text{Interest Paid = Total Repayments - Amount Borrowed}}\]

Nominal and Effective Rates

The nominal annual interest rate ($r$) is the yearly rate of interest which does not consider compounding. Financial calculators often denote this as I%.

The effective annual interest rate ($r_{\text{eff}}$) considers the effect of compounding throughout the year. It reflects the percentage increase in value, or cost of debt, over one year.

The formula for the effective annual interest rate is given as follows:

\[\boxed{ r_{\text{eff}}=\left(1+\frac{r}{n_c}\right)^{n_c} - 1 }\]

Compound Interest

Compound interest is when the interest is calculated using any accumulated interest from previous periods in addition to the original principal.

The general formula for compound interest is:

\[\boxed{ V_f = V_0 \left(1 + \frac{r}{n_c}\right)^{n_{c}t} }\]

where $t$ is the time in years.

Continuous Compounding

When compounding occurs infinitely often (as $n_c \to \infty$), the formula becomes:

\[\boxed{ V_f = V_0 e^{rt} }\]