No. 18 Loan amortizations

Annotation No. 18

LOAN AMORTIZATIONS

Loan amortization. Equal installment loans. Equal principal repayment loans.

Brief overview

Loans can be repaid in various ways over their lifetime as long as what is owed is paid back voluntarily in full with the agreed interest. The loans described as spot loans are repaid in one lump-sum payment. This note looks at loans repaid in installments and compares the virtues of two popular methods of loan amortization that are described commonly as the French and German methods of amortization.

LOAN AMORTIZATION

A loan is a financial debt contract whereby an entity (a bank, a non-financial company, a government agency, or even an individual) grants another entity the use of a given sum of money temporarily and on condition that it will be fully repaid at a later date with the agreed interest. The creditor is the entity providing the loan. The debtor is the entity that received the sum of money at the outset with the understanding that it will repay it with interest. In a bank loan, the creditor is a bank which is often described as the lender; the debtor of a bank loan is known as the borrower. The loan is described as an interbank loan when both entities are banks which are then known in contract terms as the loan counterparties.

In a bank loan, the sum of money initially disbursed is the face value or principal of the loan. It is also called the par value of the loan. The time period between the date when the funds are disbursed and the date when the borrower's repayment obligations are terminated is the original maturity or term of the loan. The London market also describes it as the original tenor of the loan. Long term debts are called loans while short term debts are often described as credits.

The loans that are repaid in one lump-sum amount due on the maturity date are known as spot loans or spot obligations. Spot loans do not carry an explicit interest payment obligation. The lump-sum amount due on the maturity date is described as the face value or principal of the spot loan. The implicit interest on a spot loan is calculated as the difference between the face value and the initial disbursement of funds.

The amortization of a loan is the process of paying off the loan principal regularly over its term. Amortizing loans are normally repaid in equal periodic partial repayments or installments. Each installment amount can be broken down into the payment of the interest on the loan outstanding balance plus a partial repayment of the principal. The

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calendar schedule of installment payments is the amortization schedule. Residential home purchase loans and car purchase loans are typical examples of amortizing loans. The methods used to define an amortization schedule are described as annuity methods. The so-called conventional amortization schedule is typical of most sovereign bond issues: a constant, temporary annuity associated with the interest payments and one lump-sum payment related to the full repayment of the principal. The conventional amortization schedule is also known as the American method of amortization. Two very popular loan amortization methods include a repayment with equal loan installments and the method defined by unequal installments with equal principal repayments. These two methods have typical applications in household finance.

EQUAL INSTALLMENT LOANS The amortization schedule defining equal periodic loan installments is known as the French method of amortization. This sequence of equal installments is a constant, temporary annuity. Each installment includes an interest component and a partial repayment of the principal. The interest component is calculated on the outstanding principal balance given the stated interest rate on the loan. Consider a two-year loan with equal installments X(1) and X(2) which are due at the end of Year 1 and Year 2, respectively.

X (1) X (2) A .

X(1) and X(2) are respectively the loan installment amounts due at the end of Year 1 and at the end of Year 2. A is the value of the annual installments. The principal X(0) is the amount of the funds disbursed initially by the lender. The interest payment due at the end of Year 1 can be described as a proportion of the principal amount X(0).

J (0, 1) r X (0) .

J(0, 1) is the interest payment due at the end of Year 1. X(0) is the principal amount. The annual interest rate measure r is the stated annual interest rate on the loan. The value of the first installment equals the sum of the interest payment due plus a fraction of the principal amount.

X (1) r X (0) X (0) , where 0 1.

The loan balance at a certain date is the value of the outstanding principal amount or the value of the principal after deducting all principal repayments to date. The value of the interest payment due at the end of Year 2 refers to the interest due on the loan balance at the end of Year 1.

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J (1, 2) r (1 ) X (0) .

J(1, 2) is the interest payment due at the end of Year 2, or the interest accrued over the second interest payment period. The value of the second installment equals the sum of the interest due on the loan balance at the end of Year 1 plus its repayment.

X (2) r (1 ) X (0) (1 ) X (0) (1 r) (1 ) X (0) .

We know that both installments are equal in value. A (r ) X (0) (1 r) (1 ) X (0) .

The only unknown value here is the value of the fraction of the principal amount that is due at the end of Year 1.

1 . 1 (1 r)

Consider then a two-year loan of one million euros scheduled to be amortized in two equal annual installments with a stated annual interest rate of 4 percent. We calculate the value of the fraction of the principal amount included in the first annual installment to be equal to 49.0196078 percent, and use the expression of the first installment to calculate the constant value of the annual installment.

A (r ) X (0) (0.040 0.490196078)1,000,000.00 530,196.0784 euros.

A two-year loan of one million euros with a stated annual interest rate of 4 percent can be amortized in two annual installments of 530,196.08 euros.

Formally, we re-write the expression describing the value of the equal loan installment to obtain a more practically useful version.

A

r

X

(0)

r

1

1 (1

r)

X

(0)

1

r 1

X (0) .

2

1 r

This way of expressing the value of the constant loan installment suggests that we can now consider a general equal installment loan for a maturity of T years with a stated annual interest rate r.

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1

r 1 rT

Ar

X (0)

X (0) .

1 1 T

1 rT 1

1 r

T is the number of years in the maturity or term of the loan. Hence: The value of the constant annual installment A is a function of the principal amount X(0), the level r of the stated annual interest rate, and the number T of years of the term of the loan. The cash flow sequence associated with an equal installment loan can be described as:

{ ? X(0), A, A,... , A }.

The most popular application of the French method of loan amortization is the traditional residential home mortgage loan in the U.S. A mortgage (Law French mort + gage = dead pledge) is a conveyance or pledge of property as a security for the payment of a debt or the discharge of some other obligation for which it is given, in which the security is redeemable on the payment or discharge of such payment or debt.1 The mortgage borrower is the mortgagor. The mortgage lender is the mortgagee. The lender has the right in a traditional U.S. residential home mortgage contract to foreclose the loan and seize the property to ensure that it is repaid. The traditional mortgage is a fixed interest rate loan that is repaid in equal or level monthly installments of principal and interest. We can now calculate the value of the monthly installment of a 30-year traditional mortgage with a stated 8 percent annual interest rate. We use K as the principal of the mortgage loan.

1

8

1

3012

A 8 1 100 12 1

100 8

12 1 3012

K 1

0.007337646 K

.

100 12

We now confirm that the fixed monthly installment is equal to 733.76 euros for each 100,000.00 euros of principal when the stated interest rate is 8 percent per annum. We compare the value of the fixed monthly installment for various annual interest rates.

1 The mortgage contract was first developed in agrarian England to finance the acquisition of farmland as the borrower conveyed the land itself as security for the loan. If all went well, the loan was repaid on the agreed ?law day? and the land was reconveyed back to the borrower.; otherwise, the land and possibly all previous payments were forever gone ?and so [were] dead? to the borrower. Medieval law was written in a peculiar vernacular mixing Norman French and English which was called law French and of which ?mort gage? is one example. Cf. Carol M. ROSE. "Crystals and Mud in Property Law," Stanford Law Review, Vol. 40, No. 3 (February 1988) pp. 577-610; and Andrew R. BERMAN. "Once a Mortgage, Always a Mortgage ? The Use (and Misuse) of Mezzanine Loans and Preferred Equity Investments," Stanford Journal of Law, Business & Finance, Vol. 11, No. 1, (Fall 2005) pp.76-125

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Annual interest rate: 4.0 % Fixed monthly installment = 477.42 euros.

Annual interest rate: 8.0 % Fixed monthly installment = 733.76 euros.

Annual interest rate: 12.0 % Fixed monthly installment = 1,028.61 euros

The relation between the value of the fixed monthly installment and the annual interest rate on the loan is non-linear. Twice or thrice the value of the interest rate is simply not a multiple of the value of the fixed monthly installment. Figure 18.1 illustrates the breakdown of the level monthly payment between principal and interest. Earlier installment payments in the life of the loan are constituted mostly by interest while later payments are almost entirely made of principal repayments. Clearly, the sum of the principal and interest payments is the same in each and every month.

Please insert here: Figure 18-1: Equal installment loan.

The loan amount outstanding at any one time is called the mortgage balance. The mortgage balance can also be viewed as the amount of the property value that the borrower does not yet own. The property value minus the mortgage balance is the owner's equity. As the mortgage balance declines over the life of the loan, the equity value rises. The equity value may also rise because of an increase in the current value of the property or because of improvements or inflation. Sometimes a mortgagor may wish to make a monthly payment in excess of the amount due in order to reduce the mortgage balance. These excess principal payments are called prepayments. When prepayments are allowed, the currency mortgage balance equals the initial property value minus all principal repayments and prepayments done to date.

Example 1

Suppose that the Board of Supervisors of Corporation X has just realized that the retirement packages of its senior management are currently being financed out of its normal activity with an annual cost of 20 million euros. The Board has decided to create a special fund of 400 million euros to fully finance this item after fifteen years. The Corporation's CFO (Chief Financial Officer) is being requested to propose an annual contribution amount.

2.250

A

X (T )

r

1 rT

1

400,000,000.00

1

100

2.250

1

5

1

22,715,410.00 .

100

The CFO uses the current interbank rate for a 15-year term of 2? percent per annum. Using this rate, the annual contribution of Corporation X to this special fund should be

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