# The Market Value of Loan

At any intermediate date, the value of the debt discounts future annuities. At maturity, the value of debt at date 6 would be zero under all scenarios and equals the constant annuity at date 5 (Table 25.10).

# The Exercise Price and the Payoff of the Option

The market value of the original loan varies with interest rates and is the underlying of the prepayment option. It is the present value of all subsequent annuities, given the market rate. The

TABLE 25.10 Market value of loan

 Dates 0 1 2 3 4 5 Flows 0.00 263.80 263.80 263.80 263.80 263.8 1000.00 1061.72 867.37 676.15 479.88 263.8 1134.86 920.66 709.02 493.69 263.8 961.11 733.82 503.99 263.8 751.89 511.42 263.8 516.67 263.8 263.8

Values include the annuity paid immediately after each date.

rate values of each node served above to calculate the time paths of the value of the loan. The gain from renegotiation is the difference between the current value of the loan and the current strike price. The strike price of the debt is the amount of outstanding loan plus the penalty. The strike price of the option derives from the repayment schedule of the debt and adding 3% to outstanding capital due (Table 25.11).

Once we have all values of the loans and the strike price, the immediate payoff of the option is the positive difference between the value of loan and the strike price, without considering further opportunities of exercise. For making the calculation consistent, the value of loan should be net of principal repayment as well, whether it was calculated above before such repayment. The immediate payoff at any date is equal to the market value of loan minus principal times (1 +3%) (Table 25.12).

The payoff at date 1, after a downward move of interest rate is, 1134.86 - 861.29 — 9.77, and so for other dates. There is no positive payoff at inception, because the strike price is above the value of the loan at those dates.

TABLE 25.11 Outstanding capital due and penalty for early repayment

 Dotes 0 1 2 3 4 Outstanding capital Capital (1 +3%) 1000.00 1030.00 836.20 861.29 656.03 675.71 457.83 471.57 239.82 247.01

TABLE 25.12 Payoff of options under immediate exercise

 Dotes 0 1 2 3 4 5 0.00 0.00 0.00 0.00 0.00 0 9.77 0.00 0.00 0.00 0 21.60 0.00 0.00 0 16.52 0.61 0 5.86 0 0

Payoff = loan value - principal ( I + 3%).