Page Description Equation
1 An amount function $A$ gives the accumulated value $A(t)$ of an investment at time $t.$ \[ A(t) \]
1 The principal of an investment is given by the initial value $A(0)$ of the amount function at time $t=0.$ \[ A(0) \]
1 For each amount function $A,$ we define $a,$ its accumulation function. \[ a(t)=\frac{A(t)}{A(0)} \]
1 The value of the accumulation function at time $t=0$ is $1,$ since $a(0)=\frac{A(0)}{A(0)}.$ That is, the $y$-intercept of $a$ is $1,$ whereas the $y$-intercept of $A$ is its principal $A(0)=k.$ \[ a(0)=1 \]
1 Each amount function $A$ is proportional to its accumulation function $a$ with constant of proportionality $k=A(0),$ the principal. \[ A(t)=k\cdot a(t) \]
2 interest $\textit{Interest }$ $=\textit{Accumulated Value}$ $-\textit{Principal}$
3 effective rate of interest per year (or period) in terms of $a.$ \[ i=a(1)-1 \]
3 effective rate of interest per year (or period) in terms of $A.$ \[ i=\frac{A(1)-A(0)}{A(0)} \]
3 effective rate of interest in the $n$th year (or period) in terms of $a.$ \[ i_n=\frac{a(n)-a(n-1)}{a(n-1)} \]
3 effective rate of interest in the $n$th year (or period) in terms of $A.$ \[ i_n=\frac{A(n)-A(n-1)}{A(n-1)} \]
3 identity for $i$ and $i_1$ \[ i=i_1 \]
4 When $a(t)$ is a straight line, $a(t)=1+bt$ for some $b,$ since $a(0)=1.$ However, $i$ $=a(1)-1$ $=b.$ In this case, $i$ is called simple interest. \[ a(t)=1+it \]
5 simple interest effective rate of interest in the $n$th year (or period) \[ i_n=\frac{i}{1+i(n-1)} \]
5 When $t$ is given in days, you can use the exact simple interest method to calculate $a(t)$ or $A(t).$ \[ t=\frac{\textit{number of days}}{365} \]
6 When $t$ is given in days, you can use the ordinary simple interest method, also known as the Banker's Rule, to calculate $a(t)$ or $A(t).$ \[ t=\frac{\textit{number of days}}{365} \]

Notes
PageNotes
1 An amount of money initially invested is called the principal of the investment.
1 The amount the principal has grown to after a given time period is called the accumulated value.
1 Any function $a(t)$ with $a(0)=1$ can be an accumulation function. It is usually assumed to be increasing, though this is not required. It is also usually assumed to be continuous, but step functions exist in many real-world situations. For example, a savings account whose balance stays constant for periods of time, but then jumps when interest is paid into the account, will be a stepwise accumulation function.
1 Guevara Note. Observe that $a$ itself is the amount function $A'$ whose principal is $A'(0)=1$ and that satisfies \[ a(t)=\frac{A(t)}{A(0)} =\frac{A'(t)}{A'(0)} =A'(t) \] That is, $a=A'$ is the unique, equivalent, normalized form of $A$ such that $A'$ is just like $A$ but with principal $A'(0)=1$ instead of principal $A(0)=k.$ We can display this another way: \[ A(t)=k\cdot a(t) \] and \[ A'(t)=1\cdot a(t) \] so that, \[ A(0)=k\cdot a(0)=k \] and \[ A'(0)=1\cdot a(0)=1 \] For example, if $k=A(0)=100,$ and $a(t)=1.05^t,$ then $A(1)$ $=100\cdot a(1)$ $=100\cdot1.05$ $=105,$ whereas \( A'(1)=1\cdot a(1)=1.05. \)
6 Canada uses exact simple interest.
6 The United States and international markets use ordinary simple interest, aka the Banker's Rule.