Lesson Annual depositing and semi-annual compounding in ordinary annuity saving plan

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<H2>Annual depositing and semi-annual compounding in ordinary annuity saving plan</H2>

<H3>Problem 1</H3>Lady Lee needs &nbsp;$57,500,000 &nbsp;at the end of the year &nbsp;2029, &nbsp;and her only investment outlet 
is a &nbsp;9 &nbsp;percent long term certificate of deposit from &nbsp;Puzzo &nbsp;Bank, &nbsp;compounded semi-annually. 
What amount could &nbsp;Lady &nbsp;Lee invest at the end of each year annually up to the year &nbsp;2029 &nbsp;to achieve this objective?

<B>Solution</B>

<pre>
The situation is close to ordinary annuity account, but there is some difference.
Since the account is replenished once per year, but is compounded twice per year (semi-annually),
we should and we can use the scheme of the annually deposited and annually compounded ordinary annuity
with the EFFECTIVE annual multiplicative growth rate of  {{{(1 + 0.09/2)^2}}} = 1.092025.

So, we use the formula for the future value of the ordinary annuity

FV = {{{P*(((1+r)^n-1)/r)}}},

where  FV is the future value of the account;  P is annual payment (deposit); 
r is the annual effective percentage rate presented as a decimal; 
n is the number of deposits (= the number of years, in this case).

From this formula, you get for for the annual payment

P = {{{FV*(r/((1+r)^n-1))}}}.     (1)

Under the given conditions, FV = 57,500,000;  r = 0.092025;  n = 7.  
So, according to the formula (1), you get for the annual payment

P = {{{57500000*(0.092025/((1+0.092025)^7-1))}}} = 6,211,008.91.

<U>Answer</U>.  The necessary annual deposit value is TZS 6,211,008.91 .
</pre>

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