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What amount can be paid at the end of every month in perpetuity from an endowment of $350,000 which is earning 5.4% compounded monthly?

Am trying to apply the compound interest formula but it isn't working

Ben
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  • Please elaborate on "trying to apply the compound interest formula". – barak manos Jun 15 '16 at 07:53
  • I thought it was a case of compounding using the formula A=P(1+r/n)^nt but seem not to have time – Ben Jun 15 '16 at 07:55
  • What do you mean "seem not to have time"??? – barak manos Jun 15 '16 at 07:56
  • in the formula, we have A for amount, P for principal, r for rate which is 5.4 n is 12 but time t is not given. I somehow think this is not the appropriate formula for this case – Ben Jun 15 '16 at 07:58
  • @Ben Do you have any additional information that indicates it is required to use devaluation schemes like "present value"? Or is it a simple "sum $x$ generates $p$ percent interest every month" task? – mvw Jun 15 '16 at 09:13
  • @mvw Your statements are not really a contradiction. You sum up the all compounded payments (a). Then you have the future value. But 350.000 is a present value. Thus we have to calculate the summed compounded payments and discount it n times. Then we have an equality. – callculus42 Jun 15 '16 at 09:23
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    You just need the monthly interest rate. Obviously the endowment remains permanently at 350000 and all the interest is paid out monthly. Presumably 5.4% is an annualised rate, so you have to convert to a monthly rate. So $1.054^{1/12}-1=0.439%$ giving a monthly payout of 1538. – almagest Jun 15 '16 at 11:22

2 Answers2

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The present value of the sum of a monthly paid annuity ($a$) after $n$ years is

$PV=a\cdot \frac{(1+0.054/12)^n-1}{0.054/12}\cdot \frac{1}{(1+0.054/12)^n}$

Let $n$ go to infinity

$\lim_{n \to \infty}PV=\frac{a}{\frac{0.054}{12}}=a\cdot222.222$

Thus the equation is $350,000=a\cdot222.222$

$a=1575$

callculus42
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  • Present value seems to be a concept to quantify the idea that getting money today is more useful than tomorrow. I see no such thing requested in the question. – mvw Jun 15 '16 at 09:06
  • 350.000 is a kind of present value. So we have to find the present value of the perpetuity. – callculus42 Jun 15 '16 at 09:09
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It seems that the amount can not be more than the monthly earning of $$ m = 350000 \cdot 5.4/100 = 18900 $$ Otherwise the generating sum would shrink month by month and vanish within finite time.

I would also assume that the monthly payment is assumed to stay the same every month. Otherwise the problem is probably not uniquely solvable.

mvw
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  • Your solution is not right. – callculus42 Jun 15 '16 at 08:35
  • An amazing investment if it is returning 5.4% monthly! The approach is correct, but you have to convert to a monthly return. We have to assume that 5.45% is an annualised figure. – almagest Jun 15 '16 at 11:23
  • I took it because of the "5.4% compounded monthly", It is an unrealistic high rate, indeed. – mvw Jun 15 '16 at 11:26