Question 1169507: a firm will need 300000 at the end of 3 years to repay a loan. the firm decided that it would deposit 20000 at the start of each quarter during these 3 years into the account. the account yield 12% per annum compounded quarterly during the first year. what rate of interest should it earn during the remaining 2 years to accumulate enough amount into this account to pay the loan at the end of 3 years.(Answer:13.8%)
Found 2 solutions by math_tutor2020, Theo:
Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!
Consider the first year for now only. One year is 4 quarters.
The 12% annual interest rate converts to the quarterly rate of (12%)/4 = 3%
We'll let i = 0.03 to reflect this.
Compute the future value (FV) of an annuity due with
- deposit of P = 20000
- quarterly interest rate i = 0.03
- n = 4 quarters
We get the following
FV = (1+i)*P*( (1+i)^n - 1)/i
FV = (1+0.03)*20000*( (1+0.03)^4 - 1)/0.03
FV = 86182.7162
FV = 86182.72
After four quarters pass by, we have 86182.72 in the account.
Let x be the quarterly interest rate for the next 8 quarters (2 years).
We'll deposit P = 86182.72 dollars at this rate and time.
At the end of 8 quarters, we'll be left with 86182.72*(1+x)^8 dollars
Note I'm using the compound interest formula here.
Let S = 86182.72*(1+x)^8
We'll come back to this later.
Again keeping x the same, whatever it is, we'll compute the FV of an annuity due like we did earlier. However, this time we don't know the interest rate and the value of n is doubled (n = 8 instead of n = 4).
So we have
FV = (1+i)*P*( (1+i)^n - 1)/i
FV = (1+x)*20000*( (1+x)^8 - 1)/x
Let's call this R
R = (1+x)*20000*( (1+x)^8 - 1)/x
Add the expressions of S and R
S+R = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x
call this sum T. We'll make it a function of x
T(x) = 86182.72*(1+x)^8+(1+x)*20000*( (1+x)^8 - 1)/x
The goal is to solve T(x) = 300000 which is the same as finding the roots of T(x)-300000 = 0
The variable x is buried under an exponent of 8, which suggests that solving this by hand is going to be daunting. Luckily we can use technology.
You can use any technology you like. For me, I prefer GeoGebra since it's a very handy program in many ways. You should find the root on the interval 0 < x < 1 is approximately
x = 0.0345028581
This represents the decimal form of the quarterly interest rate.
So the quarterly rate is about 3.45%
Multiplying this by 4 leads to the annual rate needed
4*x = 4*0.0345 = 0.138 = 13.8%
Answer: 13.8%
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! i did a cash flow analysis using excel, the results of which are shown below.
the time periods are in quarters from time period 0 to time period 12. (3 years * 4 time periods per year = 12 time periods)
twelve investments of 20,000 each are made from time period 0 to time period 11.
for time periods 0 to 4, the investments are made at 3% interest rate per time period (12% per year / 4 = 3% per quarter = 3% per time period).
the total invested in time period 4 is 106,182.72.
it is shown as minus because it's money that is going out.
in effect, you have 106,182.72 in your account in time period 4.
that amount is positive, but you immediately turn around and invest that for the next 8 quarters, so it becomes money going out.
that's why it is shown as negative.
the cash flow for the next 8 time periods is 106,182.72 going out in time period 4 plus 20,000 going out in time period 5 through time period 11.
in time period 12, you need to have 300,000 coming in, so that amount is entered as positive in time period 12.
the internal rate of return analysis is done for time periods 4 through 12 inclusive.
it comes out to be 3.45% rounded to two decimal places.
the equivalent annual rate of return is 3.45 * 4 = 13.8% per year.
here's a reference on how the internal rate of return is calculated manually.
https://www.mathsisfun.com/money/internal-rate-return.html
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