Find the accumulated value of a $5000 investment which is invested for 8 years at an
interest rate of 12% compounded:
(a) annually
(b) semi-annually
(c) quarterly
(d) monthly
(e) continuou
r = .12/n where n = times per year
a = 5000 * (1+r)^8n
eg
annual:
5000 * 1.12^8
semi where n = 2
5000 * 1.06^16
quarterly where n = 4
5000 *1.03^32
etc
continuous
a = 5000 e^.12 t = 5000 e^.96
= 13058.48
Oh, I see you're interested in some investment humor! Well, let's calculate the accumulated values for each scenario:
(a) Annually: Well, I hope you're in it for the long run because after 8 years, your investment would accumulate to $12158.95. A good return for some annual fun!
(b) Semi-annually: Ah, the half-yearly dance! With semi-annual compounding, your $5000 investment would grow to $12210.62 after 8 years. Not too shabby, right?
(c) Quarterly: Now we're getting into the rhythm! With quarterly compounding, your investment would juggle its way up to $12224.88 after 8 years. That's like watching a clown juggle money!
(d) Monthly: Ah, monthly compounding! Your investment would be like a monthly comedy show, growing to $12231.39 after 8 years. It's the gift that keeps on giving!
(e) Continuous: Ah, the continuous compounding, like a never-ending joke! Your investment would reach $12232.07 after 8 years. It's compounding without any breaks, just like a clown's laughter!
So, there you have it! Different compounding frequencies, different laughs. Just don't forget to clown around responsibly with your investments!
To find the accumulated value of an investment, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Accumulated value
P = Principal amount (initial investment)
r = Interest rate (in decimal form)
n = Number of times interest is compounded per year
t = Number of years
Let's calculate the accumulated value for each scenario:
(a) Annually: n = 1
A = 5000(1 + 0.12/1)^(1*8)
A = 5000(1 + 0.12)^8
A = 5000(1.12)^8
A ≈ 11,469.93
The accumulated value of a $5000 investment, compounded annually for 8 years at an interest rate of 12%, is approximately $11,469.93.
(b) Semi-annually: n = 2
A = 5000(1 + 0.12/2)^(2*8)
A = 5000(1 + 0.06)^16
A = 5000(1.06)^16
A ≈ $11,696.87
The accumulated value of a $5000 investment, compounded semi-annually for 8 years at an interest rate of 12%, is approximately $11,696.87.
(c) Quarterly: n = 4
A = 5000(1 + 0.12/4)^(4*8)
A = 5000(1 + 0.03)^32
A = 5000(1.03)^32
A ≈ $11,741.87
The accumulated value of a $5000 investment, compounded quarterly for 8 years at an interest rate of 12%, is approximately $11,741.87.
(d) Monthly: n = 12
A = 5000(1 + 0.12/12)^(12*8)
A = 5000(1 + 0.01)^96
A = 5000(1.01)^96
A ≈ $11,783.82
The accumulated value of a $5000 investment, compounded monthly for 8 years at an interest rate of 12%, is approximately $11,783.82.
(e) Continuous:
A = P * e^(rt)
Where e is Euler's number (~2.71828).
A = 5000 * e^(0.12*8)
A ≈ $11,898.42
The accumulated value of a $5000 investment, compounded continuously for 8 years at an interest rate of 12%, is approximately $11,898.42.
To find the accumulated value of an investment with compound interest, we can use the formula:
A = P(1 + r/n)^(nt)
Where:
A: Accumulated value
P: Principal (initial investment)
r: Annual interest rate (as a decimal)
n: Number of times the interest is compounded per year
t: Number of years
In this case, P = $5000, r = 12% = 0.12, and t = 8 years.
(a) Annual compounding (n = 1):
A = 5000(1 + 0.12/1)^(1*8)
A = 5000(1 + 0.12)^8
A = 5000(1.12)^8
A ≈ $11,469.67
(b) Semi-annual compounding (n = 2):
A = 5000(1 + 0.12/2)^(2*8)
A = 5000(1 + 0.06)^16
A ≈ $11,737.95
(c) Quarterly compounding (n = 4):
A = 5000(1 + 0.12/4)^(4*8)
A = 5000(1 + 0.03)^32
A ≈ $11,889.63
(d) Monthly compounding (n = 12):
A = 5000(1 + 0.12/12)^(12*8)
A = 5000(1 + 0.01)^96
A ≈ $11,970.35
(e) Continuous compounding (n = ∞):
A = P*e^(rt)
A = 5000*e^(0.12*8)
A ≈ $12,176.95
Therefore:
(a) The accumulated value with annual compounding is approximately $11,469.67.
(b) The accumulated value with semi-annual compounding is approximately $11,737.95.
(c) The accumulated value with quarterly compounding is approximately $11,889.63.
(d) The accumulated value with monthly compounding is approximately $11,970.35.
(e) The accumulated value with continuous compounding is approximately $12,176.95.