Problema Solution
Suppose that you have $10,000 to invest. which investment yields a greater return over 3 years: 4% compounded monthly or 3.5% continuously,
a. write the formula for compound interest for n compounding per year.
b. write the formula for continuously compounded interest.
c. use the formulas and solve for each investment.
d. which investment offers a better return on your money over 3 years.
Answer provided by our tutors
Let the original rate be r, principal be p,and time period be t years.
then
Amount A = P*[{1+(r/100)}^t] if interest is compunded anually.
a.) For 'n' compounding per year, the rate per compounding period r1 = r/n.
Now, A = P*[{1+r1/100}^n]
And C.I. = A-P
b.)For continuously compounded interest,
Amount A = P*[{1+(r/100)}^t]
C.I. = A-P
c.)
I. At 4% p.a. compounded monthly.
1.Compounded monthly means 12 compounding periods per year (since there are 12 months, and the interest in compounded every month)
We are investing for 3 years.Therefore,n=12*3=36
2. r= 4/12 = 0.33% per month (PS : Interest is compounded monthly.So we need the rate of interest per month)
3.P=$10,000
4.Now A= 10,000*[{1+(0.33/100)}^36] = 10,000 * 1.1272 = $11,272 (approx.)
5.C.I. = A-P = $11,272 - $10,000 = $1,272
II. At 3.5% p.a compounded annually(continuously)
1.Compounded annually means only one compounding period per annum
We are investing for 3 years.Therefore, n=1*3 = 3
2.r=3.5/1 = 3.5% per annum
3.P=$10,000
4.Now A = $10,000 * [{1+(3.5/100)}^3] = $10,000 * 1.0350 = $11,087(approx.)
5.C.I. = $11,087 - $10,000 = $1,087
d.) The first investment(4% compounded monthly) is better than the second investment(3.5% compounded annually) because it yields a better interest.
Let the original rate be r, principal be p,and time period be t years.
then
Amount A = P*[{1+(r/100)}^t] if interest is compunded anually.
a.) For 'n' compounding per year, the rate per compounding period r1 = r/n.
Now, A = P*[{1+r1/100}^n]
And C.I. = A-P
b.)For continuously compounded interest,
Amount A = P*[{1+(r/100)}^t]
C.I. = A-P
c.)
I. At 4% p.a. compounded monthly.
1.Compounded monthly means 12 compounding periods per year (since there are 12 months, and the interest in compounded every month)
We are investing for 3 years.Therefore,n=12*3=36
2. r= 4/12 = 0.33% per month (PS : Interest is compounded monthly.So we need the rate of interest per month)
3.P=$10,000
4.Now A= 10,000*[{1+(0.33/100)}^36] = 10,000 * 1.1272 = $11,272 (approx.)
5.C.I. = A-P = $11,272 - $10,000 = $1,272
II. At 3.5% p.a compounded annually(continuously)
1.Compounded annually means only one compounding period per annum
We are investing for 3 years.Therefore, n=1*3 = 3
2.r=3.5/1 = 3.5% per annum
3.P=$10,000
4.Now A = $10,000 * [{1+(3.5/100)}^3] = $10,000 * 1.0350 = $11,087(approx.)
5.C.I. = $11,087 - $10,000 = $1,087
d.) The first investment(4% compounded monthly) is better than the second investment(3.5% compounded annually) because it yields a better interest.