Annuity

Annuities are…

• Money you borrowed: mortgages, etc. ⇒ pay regularly to pay off large amount
• Money you lent: social security credit, etc. ⇒ give large money, get regular payments later

Amortization is the same thing but from a different perspective ⇒ You’re spreading a one-time cost into multiple years. Formulae are same if the recurring payments are of the same amount.

Types of Annuities

• Ordinary Annuity: payment occurs at the end of each period
• Simple Ordinary Annuity: $\text{Current balance}=\text{Prev. balance}+\text{Interest on previous balance}-\text{Payment}$

Formulae for Constant Payment $P$ §

• $n$: Term; time from first payment to last payment (in years)
• $A_n$: Present value of annuity of $n$ terms.
• $S_i$: Total payment accrued after $i$ periods (not a present value!)
• $y=1+\frac{r}{k}$ for formula simplification

1. Preset value of annuity (=$A_n$) formula. (see below for more)
   • (i.e. the principal loan amount, mortgage loan amount, etc.)

$$A_n:=\frac{P}{(1+r/k)}+\frac{P}{(1+r/k{)}^2}+\dots \frac{P}{(1+r/k{)}^n}$$

1. Payment accrued after all $n$ periods (=$S_n$ =Future value of annuity after all payments made) (Demonstration, use geometric sum formula)

$$\begin{array}{rlr}{S}_n& =P+\left(1+\frac{r}{k}\right)P+{\left(1+\frac{r}{k}\right)}^{2}P+\cdots +{\left(1+\frac{r}{k}\right)}^{n-1}\cdot P& \\ & =\frac{{\left(1+r/k\right)}^n-1}{r/k}\cdot P& \dots \text{(0)}\\ & =\frac{y^n-1}{y-1}\cdot P& \end{array}$$

1. Simplified formula of PV using 1. and 2.

$$\begin{array}{rlr}{A}_n& =\frac{S_n}{{\left(1+r/k\right)}^n}& \text{time value}\\ & =\frac{1-(1+r/k{)}^{-n}}{r/k}\cdot P& \text{from (0)}\\ & =\frac{1-y^{-n}}{y-1}\cdot P& \end{array}$$

1. Remaining balance after $i$ periods (=$B_i$)

$$\begin{array}{rlr}{B}_i& =\frac{{\left(1+\frac{r}{k}\right)}^n-{\left(1+\frac{r}{k}\right)}^i}{{\left(1+\frac{r}{k}\right)}^n-1}{A}_n& \\ & =\frac{y^n-y^i}{y^n-1}{A}_n& \end{array}$$

Portion of the $i$-th payment $P_i$ that goes to… Principal

$${\mathfrak{B}}_i=\frac{r}{k}\frac{y^{i-1}}{y^n-1}{A}_n$$

• Total Principal ⇒ ${\sum }_{i=1}^n{\mathfrak{B}}_i=A_n$

Interest

1. Every period, interest is fully paid.
2. Interest is applied to the remaining balance $B$ (not to be confused with $\mathfrak{B}$)

$${\mathcal{I}}_i=\frac{r}{k}{B}_{i-1}=\frac{r}{k}\frac{y^n-y^{i-1}}{y^n-1}{A}_n$$

3. Total Interest ⇒ ${\sum }_{i=1}^n{\mathcal{I}}_i=nP-A_n$

⇒ Thus the total payment: ${\sum }_{i=1}^n{P}_i=nP$

• Consider $k$-periodic compounding (=$k$ times every year)
• Cost of Loan: ignoring the time value of money, sum of all interest payments $nP-A_n$
• As montly payment $P$ increases, number of payment periods $n$ decays exponentially.

Formulae for Variable Interest Rates §

Variables are same as above, except:

• $P_i$: payment after period $i$
• $r_i$: interest rate during period $i$

1. Present value of annuity ($A_n$)

$$\begin{array}{rl}{A}_n& =\sum _{l=1}^n\frac{P_l}{{\prod }_{j=1}^l\left(1+\frac{r_j}{k}\right)}\\ & =\frac{S_n}{{\prod }_{j=1}^n\left(1+\frac{r_j}{k}\right)}\end{array}$$