This section determines the borrower's payoff at a point in time, which is also the loss for the lender. The gain is valued under immediate exercise if the option is in-the-money. The option value differs from point-in-time payoff because of the time value of American options.

Consider a fixed rate amortizing loan, repaid with constant annuities including interest and capital (Table 25.1). The customer renews the loan at a new fixed rate if he exercises the renegotiation option. The loan might be representative of a generation of loans of a portfolio. It is assumed that the new loan has a maturity equal to the residual maturity of the original loan

TABLE 25.1 Characteristics of original loan

Original loan

1000

Original maturity (years)

5

Original fixed rate

12.00%

Original annuity

277.41

TABLE 25.2 Repayment schedule of the original loan

Dotes

1 2

3 4

5

Annuity before prepayment

-277.4 -277.4

-277.4 -277.4

-277.4

Principal repayment

157.4 176.3

197.5 221.1

247.7

Outstanding amount (end of period)

842.6 666.3

468.8 247.7

0.0

at the exercise date. If the borrower repays the loan, subject to a penalty cost, for instance 3% of the outstanding balance, the penalty cumulates with the outstanding balance of the original loan to make up the new debt. The payoff of prepayment is the present value at the new rate of the differential annuity between the original loan and the new loan. Table 25.2 details the repayment schedule of the loan. The present value of all annuities at the loan rate, 12%, is equal to the amount borrowed, or 1000.

Consider a prepayment at end of period 2, once the annuity due has been paid. The new interest rate decreases to 8%. The amount of the new loan is the outstanding balance of the old loan plus a 3% penalty. The new annuity corresponds to the value of the new loan, for a residual maturity of 3 years. The characteristics of the new loan are shown in Table 25.3. The new loan is the residual principal plus 3%, or 666.29 x (1 + 3%) — 686.28.

The periodical cash savings for the borrower are the differences between the old and new annuities, or 277.41 - 266.30 — 11.11. The present value at the new rate of this differential gain for the borrower is 28.63 (Table 25.4). The payoff from renegotiation, for the borrower, as of the date of renegotiation (f), is equal to the difference between the values of the old and the new debts, calculated at the new loan rate:

valued, new rate) original loan - valued, new rate) new loan = valued, new rate) original loan - outstanding balance (1 + penalty %)

The second part of the equation shows that the renegotiation option has a payoff equal to the value of the original loan calculated at the new fixed rate minus a strike equal to the outstand

TABLE 25.3 Example of a fixed rate loan

Prepayment date (end of period)

2

Principal outstanding

666.29

Residual maturity (years)

3

Penalty (% of outstanding principal)

3%

New debt after prepayment

686.28

New rate of new debt

8%

New annuity

-266.30

TABLE 25.4 Calculation of payoff of renegotiation

New loan

686.28

New loan dates Annuity new loan Principal repayment Outstanding principal

3

-266.30 21 1.40 474.88

4

-266.30 228.3 1 246.57

5

-266.30

246.57

0.00

Cash savings: new annuity - old annuity

1 I.I 1

1 I.I 1

11.11

^(new loan) - V(original loan)

28.63

ing principal plus penalty. The underlying of the implicit option is the value of the original loan calculated at the current customer rate. The breakeven value of the new rate making the borrower's payoff positive is around 10.3%. At this new rate, the net gain for the borrower is zero. The payoff for the borrower is also the cost for the lender. This cost increases when the decline in rates is significant. If the rate moves down to 8%, the present value of the loss for the bank reaches 28.63. This is 4.30% of the original capital outstanding in the example (28.63/666.29). The payoff is not the value of the option. The next section provides an example of the valuation of a prepayment option.

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