Valuing Caps and Floors

The arbitrage-free binomial model can be used to value a cap and a floor. This is because, as previously explained, a cap and a floor are nothing more than a package or strip of options. More specifically, they are a strip of European options on interest rates. Thus to value a cap the value of each period’s cap, called a caplet, is found and all the caplets are then summed. The same can be done for a floor.

To illustrate how this is done, we will once again use the binomial interest-rate tree to value an interest rate option. Consider first a 5.2%, three-year cap with a notional amount of $10 million. The reference rate is the one-year rates in the binomial tree. The payoff for the cap is annual.

Exhibit 25-12 shows how this cap is valued by valuing the three caplets. The value for the caplet for any year, say year X, is found as follows. First, calculate the payoff in year X at each node as either zero if the one-year rate at the node is less than or equal to 5.2%, or the notional principal amount of $10 million times the difference between the one-year rate at the node and 5.2% if the one-year rate at the node is greater than 5.2%

Then, the backward induction method is used to determine the value of the year X caplet.

FundsFor example, consider the year 3 caplet. At the top node in year 3 of Exhibit 25-12c, the one-year rate is 9.1987%. Since the one-year rate at this node exceeds 5.2%, the payoff in year 3 is

$10,000,000 x (0.091987— 0.052) = $399,870

Let’s show how the values shown at the nodes NHH, NH, and the root of the tree, N, are determined. For node NHH we look at the value for the cap at the two nodes to its right, NHHH and NHHL. The backward induction method involves discounting the values at these nodes, $399,870 and $233,120, by the interest rate from the binomial tree at node NHH, 7.0053%, and computing the average present value. That is,

value at NHH = [$399,870/(1.070053) + $233,120/(1.070053)]/2 = $295,775

This is the value reported at NHH.

Now let’s see how the value at node NH is determined. Using the backward induction method, the values at nodes NHH and NHL are discounted at the interest rate from the binomial tree at node NH, 5.4289%, and then the present value is averaged. That is,

value at NH = [$295,775/(1.054289) + $155,918/(1.054289)]/2 = $214,217

This is the value reported at NH.

Finally, we get the value at the root, node N, which is the value of the year 3 caplet found by discounting the value at NH and NL by 3.5% (the interest rate at node N) and then averaging the two present values. Doing so gives

value at N = [$214,217/(1.035) + $96,7261(1.035)]/2 = $150,214

This is the value reported at N.

Following the same procedure, the value of the year 2 caplet is found to be $66,009 and the value of the year 1 caplet is $11,058. The value of the cap is then the sum of the three caplets. That is,

value of cap = value of year 1 caplet + value of year 2 caplet + value of year 3 caplet

Thus the value of the cap is $227,281, found by adding $11,058, $66,009, and $150,214.

Similarly, an interest rate floor can be valued. The value for the floor for any year is called a floorlet. For example, instead of a cap suppose that the contract is a 4.8% floor with a $10 million notional amount. The value at a node is either

  1. zero if the one-year rate at the node is greater than or equal to 4.8%, or
  2. the notional amount of $10 million times the difference between 4.8% and the one-year rate at the node if the one-year rate at the node is less than 4.8%

Applications

To see how interest-rate agreements can be used for asset/liability management, consider the problems faced by the commercial bank and the life insurance company we discussed in demonstrating the use of an interest-rate swap.9 Recall that the bank’s objective is to lock in an interest-rate spread over its cost of funds. Yet because it borrows short term, its cost of funds is uncertain. The bank may be able to purchase a cap, however, so that the cap rate plus the cost of purchasing the cap is less than the rate it is earning on its fixed- rate commercial loans. If short-term rates decline, the bank does not benefit from the cap, but its cost of funds declines. The cap therefore allows the bank to impose a ceiling on its cost of funds while retaining the opportunity to benefit from a decline in rates.

The bank can reduce the cost of purchasing the cap by selling a floor. In this case the bank agrees to pay the buyer of the floor if the reference rate falls below the strike rate. The bank receives a fee for selling the floor, but it has sold off its opportunity to benefit from a decline in rates below the strike rate. By buying a cap and selling a floor, the bank has created a predetermined range for its cost of funds (i.e., a collar).

Recall the problem of the life insurance company that guarantees a 9% rate on a GIC for the next five years and is considering the purchase of an attractive floating- rate instrument in a private placement transaction. The risk that the company faces is that interest rates will fall so that it will not earn enough to realize the 9% guaranteed rate plus a spread. The life insurance company may be able to purchase a floor to set a lower bound on its investment return, yet retain the opportunity to benefit should rates increase. To reduce the cost of purchasing the floor, the life insurance company can sell an interest-rate cap. By doing so, however, it gives up the opportunity of benefiting from an increase in six-month LIBOR above the strike rate of the interest-rate cap.

Usually the money supermarket has strict rules regarding loans. However if you want one for car finance or as a student loan, you can go for one. However make sure that it is not an unsecured loan. Or else it wont take a lot of effort on your part to turn into a bad credit loan.

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