Interest rate?

If interest is compounded continuously and the interest rate is tripled, what effect will this have on the time required for an investment to double?|||The equation for continuous compounding is e^ rt. So the value of an investment can be found out by making this equation i.e.


P(e^ rt)= ? where P is the value of investment and ? is the value the investment will become.


For example $100 at a rate of 10% after one year under continuous compounding will be worth Rs. 110.516 approximately


As you are asking it to be doubled it means we have to make the equation like this


P(e^ rt)= 2P ( Future value is not denoted by P)





So it means


$100(e^rt)= $200, as we have assumed the rate of 10% and time one year therefore


$100(2.718^0.1*t)= $200


simplifying (dividing both sides by $100


2.718^0.1*1 = 2


As rate and time are in the exponents therefore to solve it you have to take the logs





rt *log of 2.718 = log of 2





on further simplifying


time = log of 2 / y (log of 2.718 * rate)------------(a)





Now putting the values


log of 2 = 0.693147181


log of 2.718 = 0.999896316


Rate = 10%





time = 6.932 years approximately





As you are talking about tripling the rate it means in equation (a) you will have to multiply the rate by 3 so it will become





time = log of 2 / by (log of 2.718 * 3*rate)


so denominator is increased by three times so your answer of 6.932 years will reduce by 3 times to 2.31 years|||by tripling the interest rate, the time to double will be cut to 1/3 the time.





Example: interest rate of 5%: Years = ln2/.05 = 13.86 years Example: interest rate of 15%: Years = ln2/.15 = 4.62 years





Continuopusly compounded formula-


FV = P*e^(Yr) where FV = future value, P = principal, Y = years, and r = rate.|||since it is compounded continuously u have to use p(t) = Pe^rt


e is inverse of ln( natural log).


2p=Pe^3t


divide by p, they cancel


u have


2 = e^3t


use ln


ln2 = lne^3t


ln and e cancel.


u have


ln2= 3t


divide by 3


(ln2)/3 = time


use a calulator


t = 0.23104906018664842


or t = .23 years of


23/1000 = x/12


= 3 months to double