How long does it take for an investment to double in value if it is invested at 9% compounded continuously?

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How many years does it take for an investment to double in value if it is invested at 6%
if interest is compounded quarterly?
if interest is compounded continuously?
**
compound interest formula: A=P(1+r)^t, P=initial investment, r=interest rate per period, t=number of periods, A=amount after t periods.
For continuous compounding: A=Pe^rt
..
Quarterly compounding: A/P=2, r=.06/4=.015, t=quarters
A=P(1+r)^t
A/P=(1+.06/4)^t
A/P=(1+.015)^t
2=(1.015)^t
take log of both sides
log(2)=t*log(1.015)
t=log(2)/log(1.015)
t≈46.55 qtrs≈11.64 years
..
Continuous compounding: A/P=2, r=.06, t=years
A=Pe^rt
A/P=e^rt
2=e^rt
2=e^.06t
take log of both sides
ln2=.06t*lne
lne=1
.06t=ln2
t=ln2/.06
t≈11.55 years

Have you always wanted to be able to do compound interest problems in your head? Perhaps not... but it's a very useful skill to have because it gives you a lightning fast benchmark to determine how good (or not so good) a potential investment is likely to be.

The rule says that to find the number of years required to double your money at a given interest rate, you just divide the interest rate into 72. For example, if you want to know how long it will take to double your money at eight percent interest, divide 8 into 72 and get 9 years.

Y   =   72 / r   and   r   =   72 / Y

where Y and r are the years and interest rate, respectively.

Compound Interest Curve

Suppose you invest $100 at a compound interest rate of 10%. The rule of 72 tells you that your money will double every seven years, approximately:

If you graph these points, you start to see the familiar compound interest curve:

Practice using the Rule of 72

It's good to practice with the rule of 72 to get an intuitive feeling for the way compound interest works. So...

Why Stop at a Double?

There's nothing sacred about doubling your money. You can also get a simple estimate for other growth factors, as this calculator shows:

Why Does the Rule of 72 Work?

If you want to know more, see this explanation of why the rule of 72 works. (Brace yourself, because it's slightly geeked out.)

How long does it take for an investment to double in value if it is invested at 14 % compounded quarterly and compounded continuously?
a) At 14% compounded quarterly, the investment doubles in how many years?
b) At 14% compounded continuously, the investment doubles in how many years?

You can still ask an expert for help

Expert Answer

Step 1
Compound interest:
In compound interest, interest is added back to the principal sum so that interest is earned on that added during the next compounding period. That is, compound interest will give an interest on the interest. The interest payments will change in the time period in which the initial sum of money stays in the bank or with the barrower.
The general formula for compound interest is,
A=P⋅(1+r n)nt
Where:
A is the future value of the investment loan including the loan,
P is the principle amount,
r is the annual interest rate in decimals,
n is the number of times interest is compounded per year,
t is the time of years the money is invested or borrowed.
Step 2
a) Find the number of years in which the investment will be doubled at 14% interest compounded quarterly:
The aim is to double the invested or principal amount at the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount at 14 % interest compounded quarterly.
Here,
Let the principal amount or the invested amount is P.
The future value of the invested amount including the amount is A= 2P
Annual interest rate is r=14% =0.14
The number of times interest is compounded per year is quarterly. That is, n=4.
The number of years required to double the invested money is invested t.
The number of years in which the investment will be doubled at 14% interest compounded quarterly is obtained as 5.04 years from the calculation given below:
A=P×(1+rn)nt
2P=P×(1+0.144)4t
2=(1+0.035)4t
2=(1.035)4t
Take natural logaritm on both sides
ln⁡(2)=ln⁡(1.0354t)
=4 tln⁡(1.035)
t=ln⁡(2 )4×ln⁡(1.035)
=5.04
Step 3 Continuous compound interest:
In compound interest, interest is added back to the principal sum so that interest is earned on that added during the next compounding period. That is, compound interest will give an interest on the interest.
In continuous compound interest, the principal amount will be constantly earning interest and the interest keeps earning on the interest earned.
The general formula for continuous compound interest is,
A=P⋅ ert
Where:
A is the future value of the investment loan including the loan,
P is the principle amount,
r is the interest rate in decimals,
t is the time of years the money is invested or borrowed.
Step 4
b) Find the number of years in which the investment will be doubled at 14% interest compounded continuously:
The aim is to double the invested or principal amount at the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount at 14% interest compounded continuously.
Here,
Let the principal amount or the invested amount is P.
The future value of the invested amount including the amount is A=2P
Annual interest rate is r=14%=0.14,
The number of years required to double the invested money is invested t.
The number of years in which the investment will be doubled at 14% interest compounded continuously is obtained as 4.95 years from the calculation given below:
A =P× ert
2P=P× ert
2=ert
Take natural logarithm on both sides
ln⁡(2)=ln⁡(e rt)
ln⁡ (2)=rt
t=ln⁡(2) r
=ln⁡(2)0.14; [ ∵ r=14%=0.14]
=4.95
Step 5
Answer: a) In 5.04 years, the investment will be doubled at 14% interest compounded quarterly.
b) In 4.95 years, the investment will be doubled at 14% interest compounded continuously.

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