Review How long does it take for an investment to double in value if it is invested at 9% compounded continuously?
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if interest is compounded quarterly?
if interest is compounded continuously?
thanks
Answer by lwsshak3(11628)
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How many years does it take for an investment to double in value if it is invested 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
- Expert Answer You might be interested inNew questionsHow long does it take for an investment to double in value if it is invested 10% compounded monthly compounded continuously?How long does it take for an investment to double in value if it is invested 8 compounded monthly?How long does it take for an investment to double in value if it is invested 12% compounded quarterly?How long does it take for an investment to double in value if it is invested compounded?
waigaK 2022-09-11 Answered
How long does it take for an investment to double in value if it is invested 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 14%
interest compounded quarterly:
The aim is to double the invested or principal amount the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount 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 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 14% interest compounded continuously:
The aim is to double the invested or principal amount the given interest rate.
The future value of the investment loan including the loan should be the double of principal amount
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 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 14%
interest compounded quarterly.
b) In 4.95 years, the investment will be doubled 14% interest compounded continuously.
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New questions
How long does it take for an investment to double in value if it is invested 10% compounded monthly compounded continuously?
In reality, a 10% investment will take 7.3 years to double ((1.107.3 = 2). The Rule of 72 is reasonably accurate for low rates of return.How long does it take for an investment to double in value if it is invested 8 compounded monthly?
The rule says that to find the number of years required to double your money 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 eight percent interest, divide 8 into 72 and get 9 years.How long does it take for an investment to double in value if it is invested 12% compounded quarterly?
A 10% interest rate will double your investment in about 7 years (72 ∕ 10 = 7.2); an amount invested a 12% interest rate will double in about 6 years (72 ∕ 12 = 6).How long does it take for an investment to double in value if it is invested compounded?
The result is the number of years, approximately, it'll take for your money to double. For example, if an investment scheme promises an 8% annual compounded rate of return, it will take approximately nine years (72 / 8 = 9) to double the invested money. Tải thêm tài liệu liên quan đến nội dung bài viết How long does it take for an investment to double in value if it is invested 9% compounded continuously?
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