Answer to Question #193963 in Financial Math for King

Question

PV of annuity due

3) what amount must be deposited now in order to withdraw 2,000 at the beginning of each month for 3 years, if interest is 12% compounded monthly?

4) the bingo bank is paying 9% interest compounded monthly .a) if you deposit 1000 at the beginning of each month into a savings account, how much will it be worth in 10 years?b) how much would the account be worth if the payments were made at the end of each month, rather than at the beginning?

Answer 1

3)

Withdrawal Amount needed (PMT) is 2,000

Time Period of Withdrawal (n) is 3 years

Interest Rate (r) is 12%

Compounding Period (m) is Monthly i.e. 12

Withdrawal will be done at the beginning of each month. Hence, Annuity due concept will be applied.

Calculation of amount must be deposited now is as follows:

$PV=PMT×(\frac{1−(1+\frac{r}{m})^{−n×m}}{\frac{r}{m}})×(1+\frac{r}{m})$

$=2,000×(\frac{1−(1+\frac{0..12}{12})^{−3×12}}{\frac{0.12}{12}})×(1+\frac{0.12}{12})$

$=2,000×30.40858008=60,817.16017=60,817.16$

 Amount must be deposited now is 60,817.16

4)

(a)When an annuity is due 

Annuity Amount=1000

FV of Annuity Due$=(1+r)×P[\frac{(1+r)^{n}−1}{r}]$

Where FV of Annuity due=Future Value of Annuity Due

r=rate per period

p=Periodic Payment

n=number of periods 

FV of Annuity Due $=(1+0.0075)\times1000[\frac{(1+0.0075)^{120}−1}{0.0075}]$

FV of Annuity Due$=(1.0075)\times1000[\frac{(1.0075)^{120}−1}{0.0075}]$

FV of Annuity Due$=(1.0075)\times1000[\frac{2.4513570781−1}{0.0075}]$

FV of Annuity Due$=(1007.5)[\frac{1.4513570781}{0.0075}]$

FV of Annuity Due$=\frac{1,462.2422561857}{0.0075}$

FV of Annuity Due$=194,965.634158093 or 194,965.63$

So It will worth 194,965.63 after 10 years 

(b)

When an Annuity is made at end of the month

Using the Formula,

FV of Annuity $=P[\frac{(1+r)^{n}−1}{r}]$

Where FV of Annuity =Future Value of Annuity 

r=rate per period

p=Periodic Payment

n=number of periods 

FV of Annuity  $=1000[\frac{(1+0.0075)^{120}−1}{0.0075}]$

FV of Annuity $=1000[\frac{(1.0075)^{120}−1}{0.0075}]$

FV of Annuity $=(1000[\frac{2.4513570781−1}{0.0075}]$

FV of Annuity $=1000[\frac{1.4513570781}{0.0075}]$

FV of Annuity $=\frac{1451.3570781}{0.0075}$

FV of Annuity $=193,514.27708 or 193,514.28$

So It will worth 193,514.28 after 10 years