Refinancing and Reinvestment Risks and Bond-Like Fixed Income Securities with an Embedded Option

Ketevan Burduli, Teimuraz Toronjadze Georgian American University, Chavchavadze Ave.17a, Tbilisi, Georgia,
Abstract

In the work is considered the refinancing and reinvestment risk exposure of an FI, which occurs when early repayment of an FI’s assets and liabilities take place.

We introduce the method of calculating the amount of commission an FI should charges its customer for the right of early repayment of loan or withdrawal of funds from deposit account.
Key words: Interest rate risk, bond-like fixed income securities, bonds with embedded options.
Financial institutions (FI) are faced to number of risks, namely: interest rate risk, credit risk, off-balance-sheet risk and etc. The effective management of these risks is central to the performance on an FI [2].
Here we will focus on interest rate risk – the risk incurred by an FI when the maturities of its assets and liabilities are mismatched [2].
When FI is holding longer-term assets relative to liabilities, it potentially exposes itself to refinancing risk [2].
When an FI is holding shorter-term assets relative to liabilities it potentially exposes itself to reinvestment risk [2].
FI can approximately hedge or protect against interest rate changes by matching the maturities of its assets and liabilities. However matching the maturities may at the same time reduce profitability of an FI [2].
Thus measuring interest rate risk exposure is important to an FI. There are some models of measuring the assets-liability gap (mismatching maturity) exposure of an FI:
· Reprising model;
· The maturity model;
· The duration model [2].
However all given models, ignore the possibility of early/unscheduled repayment of assets and liabilities (namely, loans and deposits) during calculating interest rate risk exposure of an FI. In fact in banks and other FIs the possibility of unscheduled repayment is quite frequent and increases the exposure to the interest rate risk. It makes more difficult to calculate the risk exposure and at the same time to hedge the risk via matching maturities of assets and liabilities.
To compensate the possible loss that can be incurred because of interest rate risk, when unscheduled repayment occurs, banks charge their customers with some penalties. Namely, in case of loans it is penalty for early repayment and in case of unscheduled withdrawal of customer funds the customer receives significantly less interest that he would receive if holding the deposit to agreed maturity.
Question is whether this penalty is a good hedge of a risk or whether the amount of charge is fair if considered from customer side, maybe FI can hedge the risk with less amount of penalty. How much should an FI charge its customer for granting the right of early/unscheduled repayment of loan or withdrawal of deposit? We have to calculate the dollar value of this right.
To answer the question we suggest considering the deposit or the loan as a bond like fixes income security with an embedded option.
We say bond-like security as in bonds the bondholder receives the coupon payments periodically and the bond’s principal at the end of the life. In case of deposits we may have a bit different situation, namely the deposit may have compound interest rate if the accrued interest is not withdrawn from deposit account and in this case the cash on deposit account will amount to:

where the VD is the deposit amount, r is the interest rate and T number of years.
Accordingly when the deposit is withdrawn before maturity an FI may face refinancing risk not only on deposit amount but on deposit amount plus accrued interest rate.
We have different situation in case of an FI’s assets (loans granted to the customers). The borrower decreases it’s liability towards the FI on every payment (the payment includes accrued interest and some part of the capital) on loan. Therefore the more scheduled payments on loan are done by the borrower the less is the reinvestment risk of an FI.
Now let’s describe the nature of option and the bond with embedded option:
An option (option contract) is an agreement between two partners, the option holder and writer, to buy or sell a specific asset at a specific price.
A call option gives its holder the right to purchase an asset for a specified price, called the exercise or strike price, on or before a specified expiration date.
When the market price exceeds the exercise price, the option holder may “call away” the asset for the exercise price and reap a payoff equal to the difference between the asset price and the exercise price. Otherwise, the option will be left unexercised. If not exercised before the expiration date of the contract, the option simply expires and no longer has value. Calls therefore provide profits when asset prices increase [1][4].
A put option gives its holder the right to sell an asset for a specified exercise price on or before a specified expiration date. Whereas profit on call option increases when the asset value increases, the put is exercised only if its holder can deliver an asset worth less than the exercise price in return for the exercise price.
An option is described as in the money when its exercise would produce profit for its holder. An option is out of the money when exercise would be unprofitable [1][4].
There are two types of options – American and European options. An American option allows its holder to exercise the right to purchase (if a call) or sell (if a put) the underlying asset on or before maturity date. European option allows for exercise of the option only on the expiration date [1][4]. Here we consider an American option.
By embedded option is meant that either the issuer or the bondholder has the right to alter the bond’s cash flows. Bond with embedded option includes callable bonds, putable bonds, mortgage backed securities, convertible bonds. In each case, the cash flow depends on the future level of interest rates [5].
For valuation bond with embedded option, it is necessary to decompose a bond into its component parts. A callable bond, for example, is a bond in which the bondholder has sold the issuer an option that allows the issuer to repurchase the contractual cash flows of the bond from the time the bond is first callable until the maturity date [5].
The owner of the bond is entering into two separate transactions. First, he buys an option-free bond from the issuer for which he pays some price. Then he sells the issuer a call option for which he receives the option price. Therefore, we can summarize the position of a callable bondholder as follows:
Long a callable bond =
Long an option-free bond + sold a call option
In terms of price, the price of a callable bond is therefore equal to the price of the two component parts. That is
Callable bond price =
Option-free bond price – Call option price
Actually the position is more complicated than described. The issuer may be entitled to call the bond at the first call date and any time thereafter, or at the first call date and any subsequent coupon anniversary [5].
The same logic applies to putable bonds. In the case of a putable bond, the bondholder has the right to sell the bond to the issuer at a designated price and time.
A putable bond can be broken into two separate transactions. First, the investor buys an option-free bond. Second, the investor buys an option from the issuer that allows the investor to sell the bond back to the issuer. This type of option as described above is called a put option [5]. Therefore, the position of a putable bondholder can be described as:
Long a putable bond =
Long an option-free bond + Own a put option
The price of a putable bond is then:
Price of a putable bond =
Option-free bond price + Put option price
Again let’s return to an FI and consider its assets and liabilities (namely, loans and deposits):
I. Liabilities (deposit accounts) of an FI
The deposit account opened in an FI can be viewed as the putable bond, where the depositor buys the bond from the bank (puts his funds on the deposit account) and at the same time buys the right to withdraw the funds before maturity (buys the put option).
In given put option:
· The underlying asset is a deposit account.
· The strike price on an underlying asset is the amount placed on deposit account.
· And the expiration date of the put option is the expiration date of deposit.
Buying the put option does not oblige holder to exercise the option. As stated above the holder will choose to exercise the option only if the exercise price is greater than the price of the underlying asset.
Now the question is when the right of unscheduled withdrawing of customer funds from deposit account will be exercised. What happens if the interest rate of deposit decreases or increases? How it influences the price of an underlying asset and the decision of the depositor to exercise or not exercise the option.
1. Decrease of an interest rate – It means that, holding the other factors constant, if the depositor opens new deposit account, with the same expiration date as before, he will need more money to receive the same amount (deposit amount + accrued interest) at the expiration date of the deposit than in case of old higher interest rate. This can be interpreted as follows: when interest rate on deposit decreases the exercise price of the option is less than the price of an underlying asset. Accordingly the option is not in the money and therefore there is no reason for exercising.
2. Increase of interest rate – In this case if new deposit account is opened with the same amount and the expiration date as before the depositor will receive more money (deposit amount + interest rate) at the expiration then under the conditions of less interest rate. Accordingly to receive the same amount at the expiration date the depositor has to open the deposit account with less money then before interest rate was change. It means that the exercise price is greater than the price of the underlying asset. The option is in the money and therefore will be exercised.
The value of a put option at expiration is:
Payoff to put holder = (X – St)+,
Payoff to put writer = -(X – St)+,
where St is the value of underlying assets, X is the exercise price and a+= max(0,a).
II. Assets (loans granted to customers) of an FI
The loans granted to the borrower can be viewed as callable bond from the borrower’s side. Here the borrower sells the option-free bond to the bank and receives its price (the loan amount) and at the same time buys a call option for which he pays option price.
In given call option:
· The underlying asset is the loan value.
· The strike price on an underlying asset is the amount of loan.
· And the expiration date of the call option is the expiration date of loan.
The option may be exercised at any time before loan matures. But how the decision of exercising of an option should be taken? And how the change in interest rate influences this decision?
1. Increase of interest rate – if the interest rate increases the borrower will be granted the less loan amount, holding the installment of loan constant, than before interest rate was changed. This can be interpreted as follows: the market price of an underlying asset is less than the exercise price. Accordingly the call option is out of the money and thus would not be exercised.
2. Decrease of interest rate – if the interest rate decreases the borrower will be able to receive more amount of loan, holding the installment on loan constant, from the bank than before interest rate was changed. It means the market price of the underlying asset exceeds the exercise price of the option, the call option is in the money and the borrower will exercise the option and thus repay the loan in advance.
The value of the call option at expiration equals:
Payoff to the call holder = (St – X)+,
Payoff to the call writer = – (St – X)+,
where St is the value of underlying assets, X is the exercise price and a+=max(0,a).

Conclusion:
For FIs it’s important to calculate the amount of commission which will be both a good hedge of refinancing and reinvestment risks and at the same time will be fair for the customer.
Considering assets and liabilities (namely, deposits and loans granted to customers) of an FI as bond-like fixed income securities with an embedded option gives the possibility to calculate the amount of commission the customer should pay for the right of early repayment of loans or withdrawal of funds.
For that purpose we have to calculate the price of an option. That is possible via using binominal model of option valuation[3].
When opening the deposit account as described above an FI gets involved in trading two securities: 1. bond like fixes income security (deposit) and 2. put option (right of early withdrawal of customers funds).
The same refers to granting the customer with loan. An FI buys the bond and sells the call option to the customer.
In these transactions both the price of bond and the option should be paid. In case of transaction related to bond the price of bond is paid immediately as the transaction is done. The price of bond is the amount deposited or the loan disbursed.
Now concerning the trade with options, an FI may take the option’s price from the customer either when the option is traded or when it is exercised. If the option price is known it’s less important when it is charged.