cash flow amount - annuity due - Numbas at mathcentre.ac.uk

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5 years, 9 months ago

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Ben Brawn 5 years, 9 months ago

Created this as a copy of cash flow amount - ordinary annuity.

Name	Status	Author	Last Modified
present value - ordinary annuity	Ready to use	Ben Brawn	21/06/2019 03:29
future value - ordinary annuity	Ready to use	Ben Brawn	21/06/2019 03:29
future value - annuity due	Ready to use	Ben Brawn	26/06/2019 09:26
present value - annuity due	Ready to use	Ben Brawn	21/06/2019 03:24
cash flow amount - ordinary annuity	Ready to use	Ben Brawn	21/06/2019 03:29
cash flow amount - annuity due	Ready to use	Ben Brawn	21/06/2019 03:22
present value - ordinary perpetuity	Ready to use	Ben Brawn	21/06/2019 03:36
cash flow amount - ordinary perpetuity	Ready to use	Ben Brawn	21/06/2019 03:36
interest rate - ordinary perpetuity	Ready to use	Ben Brawn	21/06/2019 03:37
future value - ordinary annuity	draft	Xiaodan Leng	10/07/2019 21:31
present value - deferred annuity	Ready to use	Ben Brawn	11/03/2022 00:42
Patricia's copy of cash flow amount - annuity due	draft	Patricia Cogan	30/01/2024 10:47
Patricia's copy of future value - ordinary annuity	draft	Patricia Cogan	30/01/2024 10:42
Patricia's copy of present value - ordinary annuity	draft	Patricia Cogan	25/01/2021 22:29

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			Name	Type	Generated Value

years	integer	20
period	list	[ "monthly", 12, "month" ]
ipadec	number	0.0369
ipa	number	3.69
n	integer	240
P	integer	178000
C	number	1046.5732785217
Crounded	number	1046.57

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Suppose you borrow $\$\var{P}$ to buy a house. The term of the loan is $\var{years}$ years, and you will need to make {period[0]} repayments on the loan at the beginning of each {period[2]}. Interest is $\var{ipa}\%$ per annum compounding {period[0]}. Calculate the size of your monthly repayment.

$\$$ Unnamed gap (to the nearest cent)

Advice

You are asked to find the cash flow amount of an annuity due (since the payments are at the beginning of each period) and we are given the present value of the annuity. Therefore we will use the present value of an annuity due formula

$\displaystyle P=\frac{C}{i}\left(1-\frac{1}{(1+i)^n}\right)(1+i)$

where $P$ is the present value, $C$ is the cash flow per period, $i$ is the interest rate per period, and $n$ is the number of periods.

In our situation we have,

$P=\var{P}$ ,

$i=\frac{\var{ipa}\%}{\var{period[1]}}=\frac{\var{ipadec}}{\var{period[1]}}$ ,

$n=\var{years}\times \var{period[1]}=\var{n}$ ,

and therefore we have

$\displaystyle \var{P}=\frac{C}{\left(\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)}\left(1-\frac{1}{\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)^\var{n}}\right)\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)$

which we need to rearrange to solve for $C$ .

We want to get $C$ by itself. We start by multiplying both sides by $\left(\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)$ (to remove the division of $C$ by it on the right hand side)

$\displaystyle \var{P}\left(\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)=C\left(1-\frac{1}{\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)^\var{n}}\right)\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)$

Next we will divide both sides by $\left(1-\frac{1}{\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)^\var{n}}\right)\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)$ (to remove the multiplication by it on $C$ )

$\displaystyle \frac{\var{P}\left(\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)}{\left(1-\frac{1}{\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)^\var{n}}\right)\left(1+\simplify[unitDenominator]{{ipadec}/{period[1]}}\right)}=C$

Calculating this we find

$\begin{align}C&\approx \var{C}\\&=\$\var{Crounded}\quad \text{(to the nearest cent)}\end{align}$

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This question is used in the following exams:

Annuity due by Ben Brawn in ACFI1003.
Five randomly selected financial maths questions by Ben Brawn in ACFI1003.
Practice Finance Calculations by Ben Brawn in ACFI1003.

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cash flow amount - annuity due

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Ben Brawn 5 years, 9 months ago

Ben Brawn 5 years, 9 months ago

Ben Brawn 5 years, 9 months ago

Error in variable testing condition

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Generated value: `integer`

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cash flow amount - annuity due

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Ben Brawn 5 years, 9 months ago

Ben Brawn 5 years, 9 months ago

Ben Brawn 5 years, 9 months ago

Error in variable testing condition

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Warning

Generated value: integer

Parts

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Generated value: `integer`