Chapter B, Problem 11E

To determine

Compound Interest is the interest which is calculated on the initial principal amount as well as on the interest that is accumulated. Compound interest can be compounded annually, compounded semi annually, compounded quarterly, monthly, daily etc.

To Determine:-

Here, in the problem provided to us, we are required to ascertain the initial amount that Otto co. can borrow if there are four payments of $13, 000 each and the interest rate is 8% or 12% or 16% compounded semi annually.

Answer to Problem 11E

Solution:-

When the compounding is done semi annually the interest rate is divided by two, since there are two periods in the year when the interest would be charged.

In the given problem, we have been asked to provide the amount which Otto co. can borrow at various interest rates compounded semi annually.

When the interest rate is 8% compounded semi annually, the Otto can borrow $47189.
When the interest rate is 12% compounded semi annually, the Otto can borrow $45046.
When the interest rate is 16% compounded semi annually, the Otto can borrow $43058.

Explanation of Solution

Explanation:-

Following formula has been used:

$\text{Amount}=\text{Principal}{\left(1+\text{Rate}\right)}^{\text{n}}$

, Where n is the number of periods compounded.

Now, when the interest rate is 8% compounded semi annually, then the amount that Otto co. can borrow is as follows:-

$=\frac{13000}{{\left(1.04\right)}^{}}+\frac{13000}{{\left(1.04\right)}^2}+\frac{13000}{{\left(1.04\right)}^3}+\frac{13000}{\begin{array}{c}{\left(1.04\right)}^4\\ \end{array}}$

$=\$47, 189$

It can be proved with an amortisation table:-

Date	Opening balance	Interest Amount @4%	Principal Amount	Closing Balance
01-11-2015	47189	1887.56	11112.44	36076.56
01-05-2016	36077	1443	11557	24520
01-11-2016	24520	981	12019	12500
01-05-2017	12500	500	12500	0

Now, when the interest rate is 12% compounded semi annually, then the amount that Otto co. can borrow is as follows:-

$=\frac{13000}{{\left(1.06\right)}^{}}+\frac{13000}{{\left(1.06\right)}^2}+\frac{13000}{{\left(1.06\right)}^3}+\frac{13000}{\begin{array}{c}{\left(1.06\right)}^4\\ \end{array}}$

$=\$45, 046$

It can be proved with an amortisation table:-

Date	Opening balance	Interest Amount @6%	Principal Amount	Closing Balance
01-11-2015	45046	2702.76	10297.24	34748.76
01-05-2016	34749	2085	10915	23834
01-11-2016	23834	1430	11570	12264
01-05-2017	12264	736	12264	0

Now, when the interest rate is 16% compounded semi annually, then the amount that Otto co. can borrow is as follows:-

$=\frac{13000}{{\left(1.08\right)}^{}}+\frac{13000}{{\left(1.08\right)}^2}+\frac{13000}{{\left(1.08\right)}^3}+\frac{13000}{\begin{array}{c}{\left(1.08\right)}^4\\ \end{array}}$

$=\$43, 058$

It can be proved with an amortisation table:-

Date	Opening balance	Interest Amount @8%	Principal Amount	Closing Balance
01-11-2015	43058	3444.64	9555.36	33502.64
01-05-2016	33503	2680	10320	23183
01-11-2016	23183	1855	11145	12037
01-05-2017	12037	963.0	12037	0

Conclusion

Conclusion:-

As can be seen that with the increase in interest rate the amount that Otto co. can borrow is decreasing, since it has to pay more interest on the same amount.