What rate is necessary to double an investment of $4000 in 10 years? and IMF Conference: Sustainable Investment scaling up in LIC

What rate is necessary to double an investment of 00 in 10 years?

and also these:


How long will it take to double your investment of $ 1000 at a rate of 2.5% compounding continuously?


How long will it take to earn $ 7000 on a $ 3500 investment at a rate of 7.8% compounding continuously?


i really need help on how to solve these problems i kinda dont understand plz help me


Investment best answer:

Answer by John W
To double an investment in 10 years it's

2X = X * R^10

here the X is $ 4,000 but it doesn't matter what it is, it cancels out from both sides giving you

2 = R^10
.:
ln(2) = 10 * ln(R)
ln(R) = ln(2)/10
R = e^( ln(2) / 10 )
R = 1.071773

Therefore it takes a rate of 7.1773% per annum effective to double your investment in 10 years

Whenever hey have the word "compounded" followed by the compounding interval, it means that the rate is a nominal rate calculated from a true rate of the compounding interval. For example a 1% per month rate is typically annualized to a per year rate by multiplying it with 12 because there are 12 months in a year resulting in 12% compounded monthly but the real rate is 12.6825% because 1.01^12 = 1.126825. Compounded continuously is when this is taken to an extreme with infinitely small intervals and an infinite number of them resulting in the real rate = e^( nominal rate ) - 1. That is 2.5% compounded continuously isn't 2.5% at all but rather it's 2.53% calculated by taking e^0.025 = 1.0253 hence the 2.53%.

The amount of time it takes to double your investment at a ate of 2.5% compounded continuously can be calculated as follows:

2 * X = X * ( e^(0.025) )^t

The X's cancel out on both sides leaving:

2 = ( e^(0.025) )^t

which is the same as:

2 = e^( 0.025 * t )
.:
ln(2) = 0.025 * t

solving for t we have:

t = ln(2) / 0.025
t = 27.73 years

There are two ways to interpret the statement "to earn $ 7,000 on a $ 3,500 investment". One would be to have your $ 3,500 investment grow to $ 7,000 which is essentially $ 3,500 more than what you started with and the other would be to have $ 10,500 from your $ 3,500 investment which would be $ 7,000 more than what you started with. I would say that the latter interpretation is more correct but I suspect that whomever wrote the question was thinking of the former. However the method to solve the problem is the same, just with different numbers so using the latter interpretation we have:

($ 3,500 + $ 7,000 ) = $ 3,500 * ( e( 0.078 ) )^t
.:
$ 10,500 = $ 3,500 * e^( 0.078 * t )
$ 10,500 / $ 3,500 = e^( 0.078 * t )
0.078 * t = ln( $ 10,500 / $ 3,500 )
t = ln( $ 10,500 / $ 3,500 ) / 0.078
t = 14.08 years

With the former interpretation it would be t = ln(2) / 0.078 = 8.89 years which is probably the answer that they want but not what they asked for.

Note: The reason why we annualize interest rates the wrong way is tradition. We simply started doing finance long before mathematicians figured out the math, we know that it's wrong which is why we call such annual rates "nominal rates" and the real rates "effective rates".

Investment

IMF Conference: Sustainable Investment scaling up in LIC


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IMF Conference: Sustainable Investment scaling up in Low Income Countries

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