In fixed-capital repayment plans each instalment will repay a fixed principal component, along with a decreasing interest component calculated on the outstanding debt; only the latter component varies over time. This form is very common in residential loans. In other words, instalments remain the same until the end of the plan; the principal component increases over time, whilst the interest component falls in line with the reduction in the outstanding debt (clearly, if a floating-rate rate is chosen, the amount of interest due and hence the amount of each instalment may rise or fall).

EXAMPLE 4.4

A more complex example is presented in Figure 4.4 where a borrower is taking out a loan amount at an annual nominal fixed interest rate, for a 20 year term with a balloon payment. The fixed monthly payment is calculated as follows.

In order to calculate the value of the constant periodical payment (CP), the present value of an annuity formula is used:

• PV is the value of the loan amount

• n is the number of periods per annum

EXAMPLE 4.5

FIGURE 4.5 Loan repayment procedures: Fixed-instalment repayment plan

from which it can be easily derived:

The same formulas applied to monthly constant periodical payments are:

from which it can be easily derived:

At the end of the loan, the original debt amount will be completely amortized and reimbursed.

4.4.2 Floating-Rate Loan Repayment Plan

In the case of a floating-rate loan (interest rate adjusted loan) the outstanding loan balance at the end of each payment period depends on interest rates level in the market (e.g. EURIBOR or LIBOR).

EXAMPLE 4.6

In the example in Figure 4.6 a loan is drawn down at a nominal fixed interest rate for a term of 20 years with monthly payments. A constant monthly payment is calculated together with its amortization pattern.

There are several benefits related to using such an interest based index, that is:

• interest rates are a reflection of investors' future expectations;

• interest rates are thus forward looking;

• adjustments can be more timely.

In the residential loan market there are also Hybrid Adjustable Rate Loans where the loan generally functions as a fixed-rate loan during the initial three, five, or seven years. After this period the loan is adjusted to reflect new prices and interest rates in the economy. Payments after these initial periods are generally adjusted every year.

An example is presented in Figure 4.8. A loan amount of €100,000 for a 20 year term is taken out at a 6% nominal annual interest rate. The initial monthly payment would then be €707.29.

The outstanding loan balance at the end of Period 1 is calculated solving the following formula for FV equal to zero:

with CP = €707.29, PV = €100,000, i = 6%, n = 12

If at the end of Year 1 nominal annual interest rates rise to 6.5%, the new monthly payment would be calculated using the same formula as above but solving for CP with:

• FV = 0

• PV = outstanding loan balance at the end of Year 1

• n = 29

• i = 6.5%

It should be noted that a floating-rate loan does not completely eliminate any risk for the lender. Indeed, it might be the case that the interest rate applicable to the loan is now adjusted to 6.5%, but then during the next two months interest rates rise to 6.75%. The lender will then incur a loss for the remaining time until the next adjustment takes place. This simply means that the shorter the time between adjustments, the lower the risk incurred by the lender. This risk, of course, should be reflected in the level of interest rate settled in the loan agreement.

Figure 4.7 shows the loan pattern for an interest rate adjusted loan. It is similar to what was previously seen in the case of a fully amortizing loan. The difference now is that it is more an exercise of recalculating the periodic interest payments owed to the lender every time the reference interest rate in the market changes, as showed in the example in Figure 4.8.

In this type of loan, in order to limit the upside risk of an elevated payment for the borrower, a Cap (acting as a limit) is imposed on the increase that the interest rate may follow.^{[1]} There could be some loan agreements where such a limit is imposed on the overall amount owed by the borrower to the lender and not simply on the interest rate. Of course, an Interest Rate Cap also acts as a sort of payment Cap limiting the payment amount that the borrower may owe to the lender. In other words, it is simply a matter of different formulations aimed at the same purpose of preserving the ability of the borrower to make repayments.

In case of a loan amount of €100,000 for a 20 year period taken out at a 6% nominal annual interest rate, the initial monthly payment would be €707.2').

The Interest Rate Cap states that 6% is the maximum interest rate applicable, as shown in Figure 4.9 where the interest rate pattern both for the market interest rate and the actual loan interest rate is represented.

FIGURE 4.9 Interest rate with Cap pattern

Basically, whenever prevailing interest rates in the market are below the 6% level, monthly payments are based on that interest level, while in case interest rates are above that level, 6% is the interest rate applied to service the loan.

EXAMPLE 4.8

The following example in Figure 4.10 reports the monthly loan pattern for such an Interest Rate Cap loan agreement. Calculations of monthly payments and end of period balances are performed similarly to the previous example, but now, the interest rates to be used in such calculations should be based either on the composite interest rate or on the lower capped interest rate.

Fully Amortizing, Constant Payment Loan, Floating rate with cap

Loan Amount

€ 100,000

Maturity (years)

20

Initial Interest rate (annual)

6%

Cap rate (annual)

6%

Interest rate (monthly)

0.49%

FIGURE 4.10 Floating rate with Cap

In a loan agreement with this setup, the lender will lose the opportunity to make more money every time interest rates move beyond the fixed Cap level. Compared to loans where there are no restrictions on interest rate movements, the lender should now be remunerated for the risk of losing money; this remuneration for risk should be expressed with a loan agreement that is initially based on an interest rate higher than similar loan contracts with no such Interest Rate Cap restrictions.

[1] Please see also paragraph 3.5 for further detail on the Interest Rate Cap and other hedging techniques.

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