Compound$/$Present Value$/$Annuity$/$Effective Interest Rate Formulas

Present Value:

Compound Value:

Perpetuity:

Growing Perpetuity:

$P{V}_t=\frac{C_T}{(1+r{)}^T}$

$FVT=C_t(1+r{)}^T$

Perpetuity $t=\frac{C_{T+1}}{r}$

Growth Perpetuity ${?}_{(}()()t)=\frac{C_{t+1}}{r-g}$

Annuity:

Present Value:

$PV$ Annuity ${?}_{(}()()t)=C_{r+1}x[\frac{1}{r}-\frac{1}{r(1+r{)}^T}]$

Future Value:

$FV$ Annuity $T=C_{T+1}[\frac{(1+r{)}^T-1}{r}]$

Growing Annuity:

Present Value:

$PV$ Growth Annuity $t=\frac{Ct+1}{r-g}[1-(\frac{1+g}{1+r}{)}^T]$

Future Value:

Where:

$t=$ initial period

$r=$ interest rate

$T=$ number of periods

$g=$ growth rate

$C_t=$ value in initial period

$C_T=$ value in period T

Conversion of Nominal to Effective Interest Rate:

$e=[(1+(\frac{r}{T}){)}^T]-1$

$e=$ effective annual rate

$r=$ nominal annual rate

$T=$ number of compounding periods

Economic Order Quantity $=[\frac{2x\text{a}\text{n}\text{n}\text{u}\text{a}\text{l}\text{s}\text{a}\text{l}\text{e}\text{s}xcost\text{p}\text{e}\text{r}\text{o}\text{r}\text{d}\text{e}\text{r}}{c}$arrying cost ${]}^5$

Optimal Cash Balance annual cashflow $x$ cost per transaction

Question$-$ How much can be accumulated for retirement if $2000$ is deposited annually, beginning one year from today and the account earns $9$ percent interest compounded annually for $40$ years $?$