3 Loans

3.1 Supplementary Concepts

The outstanding balance of a loan can be generally calculated by either of these methods:

Prospectively, through the present value of remaining payments. In this case, the outstanding balance of a loan with payments of \(1\) at time \(k\) is \(B^{P}_k=a_{\enclose{actuarial}{n-k}}\), where \(n\) is the term of the loan.
Retrospectively, through the accumulated value of the original loan amount less accumulated value of past payments. In this case, the outstanding balance at time \(k\) is \(B^{R}_k=B_0\,(1+i)^k- s_{\enclose{actuarial}{k}}\).

Amortization tables show principal and interest included in the payments, as well as the outstanding balance of the loan.
A loan can be repaid in many different ways, the most common being:

Level periodic payments that include principal and interest (i.e., amortization method)
Servicing the debt in each period (i.e., paying interest on the outstanding balance) and repaying the full amount of the loan at the end of the loan term (i.e., sinking fund method)

Sinking funds may earn a rate of interest different from the one required by the lender of the loan. Let \(j\) be the interest rate earned by the sinking fund, and let \(i\) be the interest rate required by the lender of the loan. Normally, \(j \leq k\), otherwise the borrower will be able to accumulate money at a higher rate of interest than is being paid on the loan.
The following relationships hold between the amortization method and the sinking fund method, when \(i \neq j\):

\[ \boxed{ \begin{array}{rl} \\ \quad \frac{1}{a_{\enclose{actuarial}{n}}} &= \frac{1}{s_{\enclose{actuarial}{n}}}+i \quad \\ \\ \quad \frac{1}{a_{\enclose{actuarial}{n}\,i\&j}} &= \frac{1}{s_{\enclose{actuarial}{n}\,j}}+i \;=\; \frac{1}{a_{\enclose{actuarial}{n}\,j}}+(i-j) \quad \\ \\ \quad a_{\enclose{actuarial}{n}\,i\&j} &= \frac{a_{\enclose{actuarial}{n}}}{1+(i-j)\,a_{\enclose{actuarial}{n}}} \quad \\ \\ \end{array} } \]

**Entries of an amortization table, with level payments of \(1\)**
Period	Payment amount	Interest paid	Principal repaid	Outstanding balance
0	-	-	-	\(a_{\enclose{actuarial}{n}}\)
\(1\)	\(1\)	\(i \cdot a_{\enclose{actuarial}{n}} = 1-v^n\)	\(v^n\)	\(a_{\enclose{actuarial}{n}}-v^n=a_{\enclose{actuarial}{n-1}}\)
\(2\)	\(1\)	\(i \cdot a_{\enclose{actuarial}{n-1}} = 1-v^{n-1}\)	\(v^{n-1}\)	\(a_{\enclose{actuarial}{n-1}}-v^{n-1}=a_{\enclose{actuarial}{n-2}}\)
\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
\(k\)	\(1\)	\(i \cdot a_{\enclose{actuarial}{n-k+1}} = 1-v^{n-k+1}\)	\(v^{n-k+1}\)	\(a_{\enclose{actuarial}{n-k+1}}-v^{n-k+1}=a_{\enclose{actuarial}{n-k}}\)
\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)	\(\vdots\)
\(n-1\)	\(1\)	\(i \cdot a_{\enclose{actuarial}{2}} = 1-v^2\)	\(v^2\)	\(a_{\enclose{actuarial}{2}}-v^2=a_{\enclose{actuarial}{1}}\)
\(n\)	\(1\)	\(i \, a_{\enclose{actuarial}{1}} = 1-v\)	\(v\)	\(a_{\enclose{actuarial}{1}}-v=0\)

3.2 Exercises

A loan of \(1\,000\) is being repaid with quarterly payments at the end of each quarter for five years at \(6\%\) convertible quarterly.

Find the outstanding loan balance at the end of the second year.

A loan of \(10\,000\) is being repaid by installments of \(2\,000\) at the end of each year, and a smaller final payment made one year after the last regular payment.

Interest is at the effective rate of \(12\%\).

Find the outstanding loan balance remaining when the borrower has made payments equal to the amount of the loan.

A loan is being repaid by quarterly installments of \(1\,500\) at the end of each quarter at \(10\%\) convertible quarterly.

If the loan balance at the end of the first year is \(12\,000\), find the original balance.

A loan is repaid with quarterly installments of \(1\,000\) at the end of each quarter for five years at \(12\%\) convertible quarterly.

Find the amount of principal in the sixth installment.

A loan is being repaid with \(20\) installments at the end of each year at \(9\%\) effective.

In what installment are the principal and interest portions most nearly equal to each other?

A \(35\)-year loan is to be repaid with equal installments at the end of each year.

The amount of interest paid in the \(8^{th}\) installment is \(135\). The amount of interest paid in the \(22^{nd}\) installment is \(108\).

Calculate the amount of interest paid in the \(29^{th}\) installment.

On a loan of \(10\,000\) interest at \(9\%\) effective must be paid at the end of each year. The borrower also deposits \(X\) at the beginning of each year into a sinking fund earning \(7\%\) effective.

At the end of \(10\) years the sinking fund is exactly sufficient to pay off the loan.

Calculate \(X\).

A borrower is repaying a loan with \(10\) annual payments of \(1\,000\).

Half of the loan is repaid by the amortization method at \(5\%\) effective.

The other half is repaid by the sinking fund method in which the lender receives \(5\%\) effective on the investment and the sinking fund accumulates at \(4\%\) effective.

Find the amount of the loan.

A loan is being repaid with \(10\) payments of \(2\,000\) followed by \(10\) payments of \(1\,000\) at the end of each half-year.

If the nominal rate of interest convertible semiannually is \(10\%\), find the outstanding loan balance immediately after five payments have been made.

A loan is being repaid with \(20\) annual payments of \(1\,000\) each.

At the time of the \(5^{th}\) payment, the borrower wishes to pay an extra \(2\,000\) and then repay the balance over \(12\) years with a revised annual payment.

If the effective rate of interest is \(9\%\), find the amount of the revised annual payment.

A \(10\,000\) loan is to be amortized with quarterly installments of \(1\,000\) for as long as necessary plus a smaller final payment one quarter after the last regular payment.

Interest is computed at \(12\%\) convertible quarterly on the first \(5\,000\) of outstanding balance and \(8\%\) convertible quarterly on any excess.

Find the principal repaid in the fourth installment.

Andrea borrows \(10\,000\) from Barbara and agrees to repay it with equal quarterly installments of principal and interest at \(8\%\) convertible quarterly over six years.

At the end of two years Barbara sells the right to receive future payments to Celia at a price which produces a yield rate of \(10\%\) convertible quarterly.

Find the total amount of interest received by Celia.

Andrea borrows \(10\,000\) from Barbara and agrees to repay it with equal quarterly installments of principal and interest at \(8\%\) convertible quarterly over six years.

At the end of two years Barbara sells the right to receive future payments to Celia at a price which produces a yield rate of \(10\%\) convertible quarterly.

Find the total amount of interest received by Barbara.

A loan is to be repaid with level installments payable at the end of each half-year for \(3.5\) years, at the nominal rate of interest of \(8\%\) convertible semiannually.

After the fourth payment, the outstanding loan balance is \(5\,000\).

Find the initial amount of the loan.

A \(200\,000\) loan is to be repaid with annual payments at the end of each year for \(12\) years.

If \((1+i)^{4}=2\), find the outstanding balance immediately after the fourth payment.

A loan of \(100\,000\) was originally scheduled to be repaid by \(25\) annual payments at the end of each year.

An extra payment \(K\) with each of the \(6^{th}\) through the \(10^{th}\) scheduled payments will be sufficient to repay the loan \(5\) years earlier than under the original schedule.

The effective rate of interest is \(9\%\).

Find \(K\).

A loan of \(200\,000\) is being repaid in \(9\) years at \(i=4\%\).

Find the present value of the interest paid over the life of the loan.

Daniel borrows from Andrew \(12\,000\) for \(10\) years and agrees to make semiannual payments of \(1\,000\).

Andrew receives \(12\%\) convertible semiannually on the investment each year for the first \(5\) years, and \(10\%\) convertible semiannually for the second \(5\) years.

The balance of each payment is invested in a sinking fund earning \(8\%\) convertible semiannually.

\(X\) is the amount by which the sinking fund is short of repaying the loan at the end of \(10\) years.

Find \(X\).

An amount is invested at \(i=5\%\) which is just sufficient to pay \(1\,000\) at the end of each year for \(10\) years.

In the first year, the fund actually earns \(5\%\), and \(1\,000\) is paid at the end of the year.

However, in the second year the fund earns \(8\%\).

Find the revised payment which could be made at ends of years \(2\) to \(10\), assuming the rate earned reverts to \(5\%\) after this one year.

An investor buys a \(7\)-year annuity with a present value of \(1\,000\) at \(8\%\) at a price which will permit the replacement of the original investment in a sinking fund earning \(7\%\), and will produce an overall yield rate of \(9\%\).

Find the purchase price of this annuity.

An actuary buys a vacation home worth \(42\,000\) by paying \(7\,000\) down and the balance at a nominal rate of interest of \(9\%\) convertible monthly, with monthly installments of \(600\) for as long as necessary.

Find the actuary’s equity at the end of five years.

An actuarial student borrows \(15\,000\) to buy a car.

The loan will be repaid over three years with monthly payments at a nominal rate of interest convertible monthly of \(6\%\).

Find the total interest paid in the second year.

A borrower has an \(8\,000\) loan with Bank \(A\), to be repaid over four years at a nominal rate of interest of \(18\%\) convertible monthly.

The contract stipulates a penalty for early repayment equal to three months’ payments.

Just after the \(20^{th}\) payment, Bank \(B\) can lend her the money at a nominal interest rate of \(13.5\%\) convertible monthly.

Determine the proposed monthly repayment to Bank \(B\) and whether the borrower should refinance the loan.

An actuarial student is repaying a \(5\,000\) debt with monthly payments over three years at a nominal rate of interest of \(16.5\%\) convertible monthly.

At the end of the first year, she makes an extra single payment of \(500\).

She then shortens the repayment period by one year and renegotiates the loan at the same interest rate.

Find the amount of interest she saves by refinancing.

A loan of \(12\,000\) is being repaid with annual payments over a three-year period at an effective rate of interest \(10\%\).

The payments vary in such a way that the outstanding loan balance is linear.

Find the total amount of interest paid.

A loan is being repaid by \(15\) annual payments at the end of each year.

The first five installments are \(4\,000\) each, the next five are \(3\,000\) each, and the final five are \(2\,000\) each.

Assuming \(i=5\%\), find the outstanding balance immediately afer the second \(3\,000\) installment.

A \(200\,000\) mortgage is being repaid with \(20\) annual installments at the end of each year.

The borrower makes five payments and then is temporarily unable to make payments for the next two years.

If \(i=7\%\), calculate the revised payment to start at the end of the eighth year if the loan is still to be repaid at the end of the original \(20\) years.

A loan is being repaid with a series of payments at the end of each quarter for five years.

If the amount of principal in the third payment is \(1\,000\), find the amount of principal in the last five payments, assuming interest at the rate of \(10\%\) convertible quarterly.

A loan is being repaid with equal installments of \(1\,000\) at the end of each year for \(20\) years.

Interest is at effective rate \(4\%\) for the first \(10\) years and effective rate \(6\%\) for the second \(10\) years.

Find the amount of principal repaid in the fifteenth installment.

A loan of \(150\,000\) is being amortized with payments at the end of each year for \(10\) years.

If \(i=12\%\), find the amount due at the end of \(10\) years if the final five payments are not made as scheduled.

An actuary has borrowed \(10\,000\) on which interest is charged at \(10\%\) effective.

The actuary is accumulating a sinking fund at \(8\%\) effective to repay the loan.

At the end of \(10\) years the balance in the sinking fund \(5\,000\).

At the end of the eleventh year, the actuary makes a total payment of \(1\,500\).

What is the sinking fund balance at the end of the eleventh year?

An actuarial student takes out a loan of \(3\,000\) for \(10\) years at \(8\%\) convertible semiannually.

The student replaces one third of the principal in a sinking fund earning \(5\%\) convertible semiannually and the other two thirds in a sinking fund earning \(7\%\) convertible semiannually.

Find the total semiannual payment.

A payment of \(36\,000\) is made at the end of each year for \(31\) years to repay a loan of \(400\,000\).

If the borrower replaces the capital by means of a sinking fund earning \(3\%\) effective, find the effective rate paid to the lending institution on the loan.

An actuary buys an annuity with payments of principal and interest of \(500\) per quarter for \(10\) years.

Interest is at the effective rate of \(8\%\) per annum.

How much interest does the actuary receive in total over the \(10\)-year period?

A loan of \(100\, 000\) is being repaid with \(10\) annual payments at \(9.128\%\) effective.

The first payment is equal to the interest only, the second payment is twice the first, the third payment is three times the first, and so forth.

Find the total amount of interest paid.

A loan is being repaid with \(20\) payments.

The first payment is \(20\) the second is \(19\), and so forth with then twentieth payment being \(1\).

If \(i=9\%\) effective, calculate the amount of interest included in the tenth payment.

Two loans for equal amounts are amortized at \(4\%\) effective.

Loan \(L\) is to be repaid by \(30\) equal annual payments.

Loan \(M\) is to be repaid by \(30\) annual payments, each containing equal principal amounts with the interest portion of each payment based upon the unpaid balance.

The payment for loan \(L\) first exceeds the payment for loan \(M\) at the end of year \(k\).

Find \(k\)

To pay a \(200\,000\) loan, payments of \(P\) are made at the end of each quarter.

Interest on the first \(50\,000\) of the unpaid balance is at rate \(i^{(4)}=16\%\), while interest on the excess is at \(i^{(4)}=14\%\).

If the outstanding loan balance is \(100\,000\) at the end of the first year, find \(P\).

A loan of \(100\,000\) is to be amortized with quarterly installments of \(10\,000\) for as long as necessary plus a smaller final payment one quarter after the last regular payment.

Interest is calculated at \(12\%\) convertible quarterly on the first \(50\,000\) of outstanding debt and at \(8\%\) convertible quarterly on any excess.

Find the principal repaid in the fourth installment.

An investor is making level payments at the beginning of each year for \(10\) years to accumulate \(100\,000\) at the end of the \(10\) years in a bank which is paying \(5\%\) effective.

At the end of five years the bank drops its interest rate to \(4\%\) effective.

Find the annual deposit for the second five years.

A loan is being repaid with equal installments of \(1\,000\) at the end of each year for \(20\) years.

Interest is at effective rate \(4\%\) for the first \(10\) years and effective rate \(6\%\) for the second \(10\) years.

Find the amount of interest paid in the fifth installment.