Interest rate risk is associated with the fluctuations in the interest rates. Interest rate risk therefore can be defined as the change in the portfolio value that a bank obtains as a result of the unexpected changes in the interest rates. The interest rate risk can also be termed as one of the ways that an individual or the bank can make profits and the value of the shareholder can gain therefore banks accept interest rate risk as a normal part of their banking business. However, for banks taking an excessive interest rate risk could threaten bank’s earnings and capital base. This is because the fluctuations in the rate of interest normally alter with the earnings of the bank. This means that it changes its net interest income, the level of other incomes that are associated with the interest rate and operating expenses. This will in overall alters the value of the assets as well as liabilities. Often change when interest rates changes. Furthermore interest rate risk affects a bank’ earnings directly as, there may be changes on active and passive interest rates plus changes on market values of assets and liabilities as well as indirectly, due to changes on business volumes. Thus, to effectively manage the risks is very much essential in order to ensure that the earnings and the value of the bank is maintained in its expected position or increases as required.

Therefore, banks typically split interest rate risk into two components: traded interest rate risk and non-traded interest rate risk. The non-traded interest rate risk is often referred to as interest rate risk on the balance sheet or in the banking book and therefore includes all commercial banking activities in the banking sector. Both refer to the potential impact of adverse movements in interest rates but they follow different accounting rules. The underlying principle for separating these portfolios is that while the banking portfolio follows traditional accounting rules of accrued interest income and accrued interest costs, trading on the other hand relies on market values (market-to-market) of transactions.

In short the banking book is generally associated to instruments included in a bank’s commercial portfolio, where exposures are assumed to be non-tradable and held to maturity. All assets and liabilities generate accrued revenues and costs, of which a conspicuous amount is interest rate driven. Accordingly, there are maturity mismatches between assets and liabilities that can lead to excesses or deficits of funds as well as mismatches between interest rate positions, fixed or variable. Banks manage such mismatches through financial transactions (on the capital markets) either by investing excess funds or assuming long-term debt by other banks. There are many studies on interest rate risk on banking accounts including that of Brighouse & Hontoir (2008) which on his study of interest rate risk on banking accounts, determines the banking book excluding investment securities such as bonds and stocks and focusing mainly on loans on the assets side and deposits, financial debentures on the liabilities side.

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The focus of this work is mainly on the major source of market risk for commercial banks which is the risk associated with the interest rate in the banking books. Therefore it includes the measurement of the interest rates risks, leaving out the trading book, overcomes the problem of double counting arising from the presence of a market risk requirement for interest rate sensitive positions held in the trading book. Nonetheless, the interest rate risk exposure of the trading book may compensate partially the exposure of the banking book. Most importantly, the banking book generates liquidity and interest rate risks, on which Assets and Liability Management (ALM) focuses.

There are a number of angles from where interest rate risk exposure can be identified. The primary components the risks that the banks can face are in many categories. These components will therefore include the yield curve risk, the optionality that is the main topic of discussion here, the re-pricing risk and finally the basis risk. The optionality, which will be the object of our study, represents that type of risk that will be faced by the banks from the options that are embedded in the bank’s assets. In our case we will mainly focus on the residential mortgages.

Re-pricing risk: A bank’s income may vary due to different re-pricing periods that will results to assets that are yield sensitive. Considered to be the more often discussed, this risk also is due to the difference in the timings of the maturity that relates to the fixed rate positions. Accordingly, unanticipated fluctuations on bank’s income and economic value can occur in relation to the variation in the interest rates.

As a result, the pricing mismatch between the assets and liabilities are the main reason for the interest rate risk. This is because the changes in the yield curve will impact more quickly on interest paid on liabilities than the interest that is usually earned on the assets. This is as a result of bank’s long term lending and short term borrowing.

Yield curve risk: Another source of interest rate risk is “the slope of the yield curve, which could have differential effects on banks assets and liabilities” (Covello & Hazelgren, 2005). The financing of short and long-term assets with medium-term liabilities exposes the bank to a possible increase curvature of the yield curve.

The basis risk: Furthermore, the interest rate risk exposure can be magnified when imperfect correlation in the adjustment across different interest rate markets of the yields earned and paid, is thus introduced. Hence, even in the relation that is there among the assets, the liabilities that accrue and the off-balance-sheet instruments of similar maturities or re-pricing frequencies, “income may fluctuate due to a lack of co-movement of rates”. For instance, a financial institution can decide to fund a loan whose payments are based on the US Treasury bill rate, with a deposit that re-prices based on Libor rates, exposing itself to the risk that can result henceforth in the changes that are not expected between two index rates.

Optionality: Banks are subject to interest rate risk through the use of instruments incorporating embedded options which can drive to loan refinancing or early deposit withdrawal. This may cause an unexpected change in cash flows and asset values for the financial institution. Embedded option risk or optionality is the risk that is caused by the options that are normally embedded in many banks assets and liabilities and also the off-balance-sheet portfolios. An illustration is a mortgage loan in addition to other security instruments such as the bonds that may either have the call option or the put option or both. Therefore proper management is required to avoid the risks that are associated with it . Therefore optionality is considered to be an increasingly important source of interest rate risk.

The Basel Committee on Banking Supervision strongly recommends that “a bank engaging in residential fixed rate mortgage lending, should be aware of the optionality features of the risk embedded in many mortgage products that allow the borrower to prepay the loan at any time with little, if any, penalty (Hussain, 2000).”

The interest rate risk is a very significant risk in the banking books in that the interest rate risk will merit the support for capital Although credit risk is likely to remain the dominant risk to banks, the emergence of new financial products when considered in the context of growing competition in financial services has led to a significant increase of interest rate risk in commercial banking.

For this purpose, ALM is the unit in charge of managing the interest rate risk as well as the liquidity of the bank, focusing essentially on the commercial banking pole. ALM policies use two target variables in order to assess interest rate risk exposure: the interest income determined through the earnings perspective and the net present value of assets minus liabilities through the economic value perspective.

In the earnings approach, it is concerned with the changes that will affect the earnings generally. It therefore deals with the impact analysis of changes in the interest rates that will affect the earnings of the bank in overall.

Thus, the earnings perspective explains the sensitivity of earnings in the short-term to interest rate movements, which in the banking book is captured by accrual accounting and measures such as Earnings at Risk (EaR). In other words, the focus is on a traditional banking book where exposures are not market-to-market and interest rate risk arises due to volatility in the bank’s net interest income, over a given time horizon. Although banks usually adopt this perspective due to the immediate impact interest rate movements have on reported earnings through this variable and also due to the threat, volatility of earnings can pose to capital adequacy, this approach does not completely capture the impact of interest rate shocks on the market value of long term positions. Consequently some banks have moved towards an economic value orientation.

Fluctuations in market interest rates can also impact on the bank’s economic valued assets, the liabilities therein and also the off-balance-sheet positions. Therefore, the sensitivity of the banks interest rate movements is a particularly important consideration for shareholders, management and supervisors alike. This approach examines the economic value of the bank as the value of the banks expected net cash flow today that is its present value. This is in short the expected cash flows plus the expected net of the cash flows on the off-balance-sheet position and finally minus the expected payments that relate to the liabilities.

Although the economic value perspective is more challenging to conduct due to assumptions concerning the behaviour of long-term instruments such as those with embedded options, it provides a more complete outlook of the potential long-term effects of interest rate volatility than is offered by the earnings perspective. The latter examines changes in near term earnings, hence may not provide an accurate sensitivity measure of a bank’s overall positions to interest rate movements.

In the banking books, the risk in interest rates is captured by accrual accounting and measures such as earnings at risk (EaR), the approach banks are lately attracted to, due to its simple implementation. Since EaR relies on existing data, as income that are always available, it provides a quick and easy overview of risks. However the EaR does not measure the economic value effects resulting from interest rate fluctuations but only looks at the impact of the shocks on the cash flows generated by the portfolio (i.e. a bank’s net interest income). Moreover, the major drawback of this simple approach is that does not relate the adverse deviation of earnings to the underlying risks because aggregates the effects of all risks.

The choice of techniques used in assessing interest rate risk depends on the bank’s orientation towards either economic value or earnings and also of the type of business model pursued by the bank. For instance, commercial lending or residential mortgage lending, are managed by a present value approach (economic value). Nonetheless “the economic value of a bank should be equivalent to the discounted sum of all future earnings in a risk neutral world” therefore these two approaches are consistent and both can be useful. Therefore, a focus on cash flows may suggest impending liquidity problems as cash flows drop and alternatively, a sharp decline in economic value may imply that the bank is insolvent, even if operations continue to produce cash flows in the near term (Joseph, 2006).

In the financial field, the value of any asset today is denoted as the present value of the asset and it the value of the cash flows today that are to be obtained in future. Therefore in calculating the present value, there is a discount rate to be used and the cash flows expected in future have to be estimated. Therefore the most challenging issue is to determine the appropriate discount rate to be used and the prediction of the expected cash flows in future which are usually complicated by factors such as the options embedded in the assets and liabilities. The liabilities include the deposits in the banks which are assets to the client, the assets includes loans that are liabilities to those taking up the loans. Therefore deposits and loans are some of the assets and liabilities that contain the embedded options.

An embedded option is one of the components of securities such as the financial bond security, the mortgage security where one of the party or in particular the issuer has the well defined right to take any action against the counter party. Therefore this will complicate the estimation of the cash flows and the interest rates. In fact, there are a number of options that can be embedded into the bonds such as; the puttable bond where there is the put option that is a contract that allows the owner the right but not the obligation to sell a specified amount of the security, callable bond where there is the call option that is a contract that gives the investor the ultimate right to buy a stock, bond or any other security however the investor does not have the obligation to buy the stock, bond or any other security, the convertible bond that can convert into common stocks and others.

For example, a five-year fixed-rate deposit with a financial institution containing an American put option is a bond that the investor has the right to put back to the institution at any time. Similarly, prepayment privileges on loans and mortgages are call options on bonds. A loan commitment made by a bank or another financial institution is a put option on a bond, giving the borrower the right to sell (put) the bond back for its face value any time within the life of the mortgage (Levinson, 2009).

“In practice, banks will generally have a mix of all these types of interest rate risks, with the effects potentially offsetting or reinforcing one another. It is the complexity of the resulting combination of factors that makes interest rate risk difficult to manage”, asserts English (2002) in his study. However he concludes that is improbable that interest rate movements can threaten the stability of a banking system.

The valuations of the securities are done using different approaches such as the Black Scholes model for bonds and other approaches where other securities that are embedded are priced similarly. Therefore to properly value debts with embedded options, one has to create a model that will take into account the probability that any of the options can be exercised also with the behaviour of the borrowers to determine when they will exercise the options. This can easily be illustrated using the options in the residential mortgages by implementing a straight forward valuation model. In addition, the mortgage interest has also decreased encouraging the use of mortgages. When the current mortgage rates are lower on an outstanding mortgage, the benefits of refinancing are thereby enhanced.

Mortgage loan are usually payable monthly and the instalments therein are usually due on the first, seventh, fourteenth or twenty-first calendar day of each month. In addition, each mortgage loan has a repayment schedule which is characterized by the instalments consisting of the payments of interest calculated on the outstanding loan balance and the repayment of the mortgage loan that is the principal portion of the amortization of the mortgage loan (McDowell, 2010). The higher the interest payments, the lower the principal portion repaid and the lower the interest, the higher the principal loan repayment since the instalments amount are fixed.

Mortgagors always have the option to either fully finance the loan before the maturity date or to partially repay the loans before the maturity dates. For the mortgagors, this will depend on their views about the possible opportunities that are there of refinancing the loan. This is because when the mortgagors have acquired the loan and the rates of interest are lower on the outstanding mortgage, the benefits of refinancing the loan are much higher. Normally, the benefits to the borrower who is holding a long-term fixed rate loan is the pay-off of the option under various interest rate levels which is determined as the time profile of the differential annuities savings after and before exercising of the option. Also the benefits to the borrower can be seen as the present value of these cash savings at the date of renegotiation that is when repaying the loan and contracting a new loan at a lower rate, with the same bank (Anderson & Kerr, 2001). Therefore, these loans are usually unexpectedly re-priced.

The uncertainty of the borrower’s time of payment of the loan by the lender such as prepayment of the loan by the borrower is quite a big issue to the lender in order to ensure that the proceedings go as expected. The lender therefore introduces the cash flow and the asset value uncertainty because to them the benefits that are there to a borrower are what make up the cost to them (Bragg, 2006). It is clear that the banks are there to make profits rather than losses and therefore this can be enhanced by ensuring the income they get for example from the loan interest far exceeds the expenses they will have to incur in lending out there money.

If the mortgage rates decline, the individuals will refinance their high fixed-rate or the variable-rate loans to lower fixed-rates loans, the results being margin deterioration for banks. Even though there will be increase in the cash inflow from the accelerated loan run-off and the refinancing fees that will be charged by the banks, in the long term view the cash flows from the loan run-off will decrease since the replacing of the mortgage will carry a lower rate than the original mortgage. Theoretically however, the losses can be limited by making the customers pay for the option that they can exercise which is in short determining the value that the customers should pay to compensate the additional cost of options that the banks will have to bear.

To the borrower, even though the prepayments usually generates a cost to them which is the penalty on the outstanding balance imposed to them, this cost does not at any point offset the benefits that arises by the borrower opting to borrow at a lower interest rates. This is because in the long term view, the lower borrowing interest rates will be advantageous to the borrower compared to the cost there will have to bare for exercising the option and the for refinancing.

In addition, embedded option risk is expected to be particularly important for regional and possibly super-regional banks rather than money-centre banks due to the difference in asset-liability composition and alternative hedging opportunities. Money-centre banks participate in mortgage securitization to a greater extent than regional and super-regional banks in addition to having lower levels of mortgage loans. Thus they are expected to have a lower level of embedded option risk (Brighouse & Hontoir, 2008).

However, if prepayments penalty of the borrower are eliminated or the closing stocks are reduced, the likelihood that the mortgage will be refinanced increases since the overall cost associated with the refinancing will have been reduced. The other way can also be to hedge the option risk using caps and floors and instead pay the cost of hedging the risk which is lower in the overall overview. The risk of the decreasing interest rates can be hedged by use of the floor that ensures there is a minimum fixed return even though the rate of the new loan is lower or by using caps where it is one of the methods of hedging risks that are guaranteed on the insurance contracts where there is a maximum value on the rate guaranteed to the customers in case of a rise in interest rates. Therefore, hedging is one of the risk management tools that are used in controlling the loss associated with the risks in the fluctuations of the prices of the banks interest rates (Dickson, 2006). It is basically the transfer of risks and not necessarily buying the insurance policies.

Therefore because of the option that is available in a fixed-rate home mortgage, it is known as a callable bond where the payments are normally made by the borrower to a bank or another financial institution. A callable bond is that type of a bond that gives the issuer of the bond the privileges of redeeming the bond at some point before the maturity of the bond. Therefore the issuer always has a right although not the obligation to buy back the bonds issued on the call date at a defined call price at that time. The call price is usually higher than the par value of the bond and the issue price and usually, there is a substantial call premium.

McDowell (2010) presents a mortgage pricing model that fully specifies all borrowers’ options with respect to default, since mortgage pricing models recognize two explicit options embedded in mortgage contracts, the right to default and the right to prepay. Regarding the latter, they evaluate a model containing two sources of uncertainty, interest rates and house prices. The sensitivity analysis conducted assuming a mortgage term of 30 years, revealed an increase in the probability of prepayment while increasing interest rate volatility, holding all else constant. This is pretty straightforward, as “higher interest rate volatility creates greater opportunities for interest rate declines to result in refinancing opportunities.

If there is a decline in the market rate of interest by the time of the call date, the issuer will therefore opt to refinance the debt at a cheaper level since the issuer pays the option in form of a higher coupon rate. The mortgages are a good example of callable bonds. The mortgages are usually fixed rate form of bonds. If the rates go down, a lot of home owners will refinance the mortgages. This will become a loss to the bank and advantageous to the house owners. Therefore, in general, the mortgage is usually embedded with the call option.

In practise therefore, the mortgagor’s is treated as formal call option, exercisable at any time and also at par. With the high change in the mortgage rates and the market based interest rate, the borrower can thereby determine the value of the refinancing option in advance and the comparisons are made to the attainable savings. Therefore the bonds such as the mortgage should be refunded only if the savings from the option exercise made represent close to 100% of the option value occurring when the rates are sufficiently low (McGrath, 2006). However, at 100% efficiency, the expected cost of waiting for further interest rate declines exceeds the cost of the new mortgage; the financially sophisticated borrowers will trigger refinancing as soon as the efficiency reaches 100% (McGrath, 2006).

According to McDowell (2010), a callable bond is the same as a fixed rate home mortgage and the payments that are made by the borrower go to the bank and other financial institutions until the borrower decides to call the bond. A call option can be exercised as soon as the market value of the bond is higher than the value of the original bond as a result of the bond interest rates declining. The price known as the exercise price is also termed as the value of the original loan that the borrower could decide at any time to stop making the payments by paying it off is denoted as. The interest rates and the bond value are correlated, whereby a decrease in the interest rates will cause an increase in the bond value and an increase in the interest rate will decrease the bond value. If the interest rates decrease, the bond value will increase and there will be more money in the call option whereby the price will be denoted as at time t. to calculate the price, ,

Where:

The market value of the remaining payments at date

The mortgage at date

The option price (not-exercised) at date

Assuming the mortgage was a one factor model, the there is a correlation between the zero coupon bonds, interest rates and the prices. In addition, the zero coupon bond will increase in their price as the interest rates decreases and the zero coupon bonds will decrease as the interest rates increases. This means that the coupon bonds and the interest rates have a negative relationship. Exercising an American call option is not advisable especially before the expiration of a dividend paying asset; however there are exceptions to these. The exceptions are when the dividends or monthly payments to the credit institution and it will be appropriate to exercise the option at a time that is immediately before the final ex dividend rate.

For example, assuming the ?t is 0.25 years and the market prices for three months is USD 98.48 and USD 96.96, while the implied volatility which has an expiry of 6 months is 15%, we would first solve for r10 :

98.48 = 100 = 100

1+ r10 ?t 1+0.25 r10

Therefore the three month yield becomes 0.0617

For the six months Treasury bill, the average discounted value for the six months all over the path would therefore include a second step:

96.96 = 1 * « 1 + 1 *100

1+ r10 ?t 1+ r20 ?t 1+ r21 ?t

The right hand side is the discount factor for period one, while the second factor is the average discount factor.

The evaluation of a callable bond is a process. First and foremost one has to determine the monthly payments and the outstanding mortgage payments. That is the payments that are yet to be made between the day of the analysis and the date of the expiration of the mortgage loan. This is the same as coming up with the payment schedule of the loan outstanding.

Using the BDT approach, an interest rate tree is build taking into consideration the mortgage per yield curve and the volatility of the curve. At each node in the tree, the values are compared with each other whereby the value of the existing mortgage is compared against that of a newly financed mortgage. The newly financed mortgage is calculated as the par plus the refinancing costs of the callable bond. If the new mortgage has a lower value, then the value of the existing mortgage is replaced at the node otherwise, no replacements are made.

Comparison of the intrinsic and the time value of the option is done by working backwards whereby we will start with the final payments of the residential loan and the time value of the prepayment call option at each node (Dickson, 2006). Assessment is done at each node of the tree, the option pay off.

The value of the American call option at a specific date is calculated by working backwards through the lattice. Finally, the market value of the callable bond is measured as the difference between the values discounted. However, we would need to consider if the date that corresponds to a node is the call date. If it is, then the bond price in the future of the node is more than the call price. We would therefore have to reset the price to a call price.

This option available to the borrowers has motivated them into obtaining a mortgage. This is because if the mortgages were repayable at their market value instead, the borrowers would instead have no motive to refinance the loan and therefore they will perceive the mortgages to be an unfair deal to them. When considering a fixed rate mortgage, the refinancing ideas are more or less the same. In short, with the decrease in the interest rates, the borrower will always have the right to benefit permanently and in the same cases have the rights to be protected from increase in the interest rates. In this case however, the refinancing would not involve the costly process of re-qualifying the borrower, reappraising the property and paying a variety of loan fees and taxes.

Home mortgage refinancing activity has grown at very high levels during the last several years since 1990 and is an important economic activity that has cropped up. Therefore, the ability to forecast when refinancing occurs, it is of interest to managers of lending institutions and the borrowers themselves.

We assume following (2006) that mortgage holders minimize the market value of their mortgage liabilities. They own a call option that gives them the right to receive an amount equal to each of the remaining mortgage payments in exchange for payment of the remaining principal plus any applicable transaction costs. The mortgage holder has a transaction cost Xi associated with prepayment, representing the fraction of the remaining principal balance that the mortgage holder must pay if he or she decides to prepay.

The following steps would be applicable for measuring the impact that the gradient of the interest rate term curve; the volatility of the natural logarithm of the short interest rate; the credit spread width and the transaction costs level has on the callable bond.

Inputting: this would require one to collect information for input. The two main sources are the provider Reutersand the interest rate term structure in terms of the short rate volatility. The information collected should be relevant and ensures timeliness that is the information should be up to date. In addition there is the developing of a binomial lattice tree by generating the algorithm using the BDT approach; from here we can compose the algorithm so as to know the lattice structure.

The parameters are then estimated so as to know of the short rate dynamics. This is where the use of retours comes in so as to estimate the input parameters for the BDT lattice structure. From these we can describe the dynamics in the stochastic of the interest rates.

Using the American call option, with a three year fixed mortgage then the amount of debt would equal to 1,000,000 Euros with payments made in every quarter. The payment schedule is important since it identifies the sequence of each payment between the date of analysis and the mortgage contract. The transaction costs level help to identify the strike prices of the call option.

The final step involves getting the short term interest rates by going back to the previous steps, two and three. To get the dynamics of obtaining the cash flow’s present value, a structure known as the lattice structure has to be constructed with the cash flow series of the mortgage and credit spread. To identify the price of the prepayment option, we would use the cash flows and the strike prices. The market value of the loan (bond) and the prepayment option are both used at the date of analysis. The market value of the mortgage is the price of the bond less the prepayment option.

In conclusion, when the short term structure is independent an increase in the volatility would cause an increase in the call option which would result to a decrease in the mortgage market value. If the positive gradient decreases, the negative gradient would result to a price increase. Therefore the prepayment option prices signify volatility changes with respect to the mortgage market value. The credit spread is related to the prepayment option price since it is a decreasing function. With high volatility levels, then the price sensitivity is high as well (Bragg, 2006). The transaction cost and the prepayment option price is negatively related. Considering that transaction costs exist, if the volatility increases the mortgage market value will decrease as well.

The binomial tree and the option adjusted spread can be used to evaluate the adjusted spread. The option adjusted spread is the same as the binomial tree only that there is a constant amount that is added to or subtracted at each rate so that the end value is the actual market price. The amount that is constantly added is known as the option adjusted spread value. The two options may conflict when it comes to the credit risk of the issuers, the liquidity risk that is exhibited by the markets and the different tax status payments. A callable security always yields more than a non callable issue. The differences in the nominal yields are eliminated by the OAS option.

It is useful and increasingly popular to quantify the spread between two rate structures. One structure is the binomial tree, appropriately calibrated to give a theoretical price. The other structure is the same tree but with a constant amount added to (subtracted from) each rate, such that the resulting value of the bond is the actual market price. The constant amount added (subtracted) is termed the option adjusted spread (OAS).

The on any two financial instruments may differ due to differences in:

credit risk of the issuers’

liquidity risk exhibited by the markets in which the instruments are traded

differential tax status of payments made on the issues and

optionality.

For instance, the yield on a BB-rated straight corporate bond will exceed that on a straight treasury issue due to credit risk, whereas the yield on a straight municipal bond will be less than that n a similarly rated straight industrial bond due to tax differences.

The yield on a callable security will exceed that on another wise identical non callable issue. The OAS removes such differences from the nominal yields of two financial instruments, hence the term, option-adjusted. Thus the OAS reflects differences only in credit risk, taxability and liquidity risk. The authors illustrate this by supposing that the benchmark interest tree is for an on the run treasury issue, and calculate the theoretical price of a BB-rated, callable corporate issue using this structure. Not surprisingly, they find that the market price is substantially below the theoretical value, thus the rates that would deliver a price equal to market value would be greater than the benchmark, say by 210 basis points. Assuming we have priced the callable corporate issue taking the call feature carefully into account at each node, then the 210 basis point spread is option adjusted. In this case the spread represents differences in credit risk between the corporate issuer and the Treasury Department, and possibly differences in liquidity between the corporate and government bond markets.

In conclusion therefore, in order to evaluate the callable bond, we will need to adopt one of the commonly used steps in order to be successful in the process. Generally, in order to find the price of any bond we must determine the cash flows and discount them at an appropriate rate. Once we’ve grown and calibrated our BDT interest-rate tree, we may then price these securities by inserting cash flows at various nodes reflecting option payoffs. By then averaging and discounting, we can price a security with all of its option features taken into account.

First and foremost for each payment date, we will determine the payment schedule of the residential loan that is the mortgage that one is working on. The payment schedule is revised from the date of payment to the day that the mortgage contract comes to expiration.

Secondly, we will build an interest rate tree from the BDT model as seen earlier. The model should be build considering a bullet mortgage par yield curve and the volatility that is calibrated to the data.

The third step is to compute the value of the mortgage at each node. This is done using the back induction starting from the final scheduled payments. At each node, the value of the existing mortgage is compared with the value of a newly refinanced mortgage. The newly refinanced mortgage is assumed to be par value plus the refinancing costs. Therefore the value of the existing mortgage will be replaced at the node only if the mortgage has a lower value than the initial value.

Next, we determine, starting with the final scheduled payments of the residential loan, the time value of the prepayments call option at each node and then we work backwards. At this point, we will compare the intrinsic and the time value of the option. In the final step, we will access the option pay off at each node of the tree.

We will continue working backwards through the lattice while determining the value of the American call option at the date of analysis that is the first node of the lattice.

In fact, using numerical procedures, Chen et al. (2009) demonstrate how risk factors, such as interest rate volatility, house value return volatility, the loan to value ratio, and prepayment penalties, impact the value of a fixed-rate mortgage. In their numerical approximation, they consider a callable and default fixed-rate mortgage, including prepayment penalties. They employ an option-based approach which is derived from the contingent claims analysis of Cox, Ingersoll and Ross (1985), which models derivative securities based on a partial differential equation. The results show that the mortgage value is lower to the lender and greater to the borrower than the value of an equivalent option free mortgage, even at origination

Of course actual mortgagors prepay for a variety of reasons, not just because they want to refinance the same principal when interest rates drop. For example, some might choose to keep the principal amount unchanged while taking advantage of lower mortgage rates by paying the same monthly coupon amount as before, thereby paying off the loan more quickly than before. Others might choose to take advantage of increased property values by withdrawing equity and increasing the size of the loan. And many others prepay because, they sell their property.

The simplest measure of interest income exposure to interest rate risk is the GAP concept which has a central place in Asset-Liability Management. There are different types of interest rate gaps that can be broadly be classified into two as

- The variable interest rate gap. The variable interest rate gap is basically a difference of the assets and the liabilities that are interest sensitive. Therefore this is the defined difference between the interest sensitive assets and the interest sensitive liabilities.
- The fixed interest rate gap. This is usually for a certain time period and is also a difference between interest rate assets and liabilities but this time the interest rate is fixed. In short it is the difference between fixed rate assets and fixed rate liabilities.

When the variable rate gap is positive, the base of assets that are rate-sensitive is larger than the base of liabilities that are rate sensitive. If the index is common to both assets and liabilities the interest income increases mechanically with interest rate. The opposite happens when the variable rate gap is negative. When the interest rate gap is zero, the interest income is not sensitive to changes in interest rates. This approach is called gap analysis. How large the gap is that is its size for a specified time bucket, which is assets minus liabilities that re-price or mature within that time bucket, indicates the bank’s re-pricing risk exposure.

Although straightforward, gap analysis provides only a rough approximation of the actual change in net interest income because it does not take account of the changes that occur within a specified time while it ignores the results of change in the market rate which always cause a difference in spread between the interest rates (Bragg, 2006). The gap analysis also fails to relate the income and option related positions that are it does.

In order to conclude about the effects of fluctuations in interest rates on the economic value, sensitivity weights can be applied to each time bucket. This approach is based on the duration as such that the percentage change in economic value will be obtained and related to the percentage change in the level of interest given a certain fixed duration of time. To add onto that such weights use the estimates that relate to assets and liabilities that fall within a specified time bucket. A combination of maturity/re-pricing schedule with sensitivity weights can be used to provide a rough measure of the volatility in the economic value of a bank that would result in case of interest rate movements..

Institutions with complex interest rate risk profiles primarily use simulations to assess the impact of changing rates on earnings. However, these simulation models typically entail detailed assessments of the potential impact of interest rate volatility on the earnings of the firm and also the economic value that bit gains by trying to simulate the interest rate that will be in the future and the effects that it will be imposed on the cash flows. In reality, this simulation models can handle the more the interest rates that change and varies a lot in the interest rate environment. This varied and refined change in the interest environment includes the changes in the slope and the shape of the yield curve to other interest rates as derived from the famous Monte’s simulation. For example in the statistic, a simulation is dealt with current exposures and a constant balance sheet with no new growth is assumed. In order to know how much the earnings are exposed, the simulations conducted on one or more interest rate scenarios, over a specific period. Dynamic simulations are considered to rely on detailed assumptions regarding changes on existing business lines, new businesses and changes in management and customer behaviour.

Such simulations can be useful in order to estimate the cash flows that are expected in the future (project the cash flows) and also any expected earnings and the economic value outcomes. Even though, dynamic simulation is highly dependent on key variables and assumptions that are extremely difficult to project with accuracy over an extended period.

It is further apparent that the mortgage loans give the borrowers of the loan the option to prepay the loan at book value and the refinancing option. If mortgages were repayable at their market value instead, borrowers would in this case have no financial motive to refinance, even though they might perceive market-value prepayment to be unfair. The same result can be obtained considering a fixed rate mortgage that would refinance itself. In other words, the borrower would always have the right to benefit permanently from reductions in interest rates while being protected against interest rate increases. In this case, the refinancing would not involve the costly process of re-qualifying the borrower, reappraising the property and paying a variety of loan fees and taxes.

In the market today, refunding efficiency is used as a tool by many corporate and municipal bond issuers in their refunding decisions

Refunding Efficiency = Present Value of Cash-flow Savings / Value of Call Option

It is therefore practical that the mortgagor’s right to refinance is now treated as a formal call option. The call option is exercisable at any time at par. Given the prevailing mortgage rates and market based interest rate volatility, the borrower can therefore determine the value of the refinancing option and compare it to the attainable savings.

The figure bellow illustrates how mortgagors can approach the refinancing decision using the notion of refunding efficiency. The new mortgage is assumed to be option-less and matching the amortization schedule of the outstanding one, for the remaining 25 years. (In reality the matching does not occur, however, as long as the new mortgage is fairly priced, its precise amortizing structure is irrelevant).

The two options may conflict when it comes to the credit risk of the issuers, the liquidity risk that is exhibited by the markets and the different tax status payments. A callable security always yields more than a non callable issue. The authors illustrate this by supposing that the benchmark interest tree is for an on the run treasury issue, and calculate the theoretical price of a BB-rated, callable corporate issue using this structure. Not surprisingly, they find that the market price is substantially below the theoretical value, thus the rates that would deliver a price equal to market value would be greater than the benchmark, say by 210 basis points. Although straightforward, gap analysis provides only a rough approximation of the actual change in net interest income because it does not take account of the changes that occur within a specified time while it ignores the results of change in the market rate which always cause a difference in spread between the interest rates (Bragg, 2006). Dynamic simulations are considered to rely on detailed assumptions regarding changes on existing business lines, new businesses and changes in management and customer behaviour.

In conclusion, seeking the optimal refinancing strategy, mortgagors might be too hasty to refinance accounting only for their monthly coupon payments while ignoring the expected present value of the new mortgage. As a result, the mortgagor might force an equilibrium solution that has a higher expected present value (Bragg, 2006). Conversely, employing the notion of refunding efficiency (Bragg, 2006) the borrower calculates the savings as the difference between the present values of the existing mortgage and the new one. From above, it is clear that banks typically split interest rate risk into two components: traded interest rate risk and non-traded interest rate risk. The non-traded interest rate risk is often referred to as interest rate risk on the balance sheet or in the banking book and therefore includes all commercial banking activities in the banking sector. Both refer to the potential impact of adverse movements in interest rates but they follow different accounting rules.

Indeed, when the short term structure is independent an increase in the volatility would cause an increase in the call option which would result to a decrease in the mortgage market value. If the positive gradient decreases, the negative gradient would result to a price increase. Therefore, the prepayment option prices signify volatility changes with respect to the mortgage market value. The credit spread is related to the prepayment option price since it is a decreasing function. With high volatility levels, then the price sensitivity is high as well (Bragg, 2006). Therefore, in order to evaluate the callable bond, we will need to adopt one of the commonly used steps in order to be successful in the process. Generally, in order to find the price of any bond we must determine the cash flows and discount them at an appropriate rate. Once we’ve grown and calibrated our BDT interest-rate tree, we may then price these securities by inserting cash flows at various nodes reflecting option payoffs.

Fig. 1. Refinancing decision for 25-year mortgages

The authors note that efficiency depends not only on the refinancing rate but also on the factors that affect the yield curve and thus the shape of the yield curve. Also the interest rate volatility affects the efficiency which we have assumed to be 16%. At a higher volatility the option value would increase and therefore the 100% efficiency level would require a lower rate (Anderson & Kerr, 2001).

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