Introduction

• Financial institutions today are often highly complex organizations, offering multiple financial services through multiple departments, each staffed by specialists in making different kinds of decisions.
• Most of the foregoing management decisions are intimately linked to each other. In a well- managed financial institution, all these diverse management decisions must be coordinated across the whole institution in order to ensure they do not clash with each other, leading to inconsistent actions that damage earnings and net worth.
• Today financial-service managers have learned to look at their asset and liability portfolios as an integrated whole, considering how their institution’s whole portfolio contributes to the firm’s broad goals of adequate profitability and acceptable risk. This type of coordinated and integrated decision making is known as asset-liability management(ALM).
• ALM provides financial institutions with defensive weapons to handle such challenging events as business cycles and seasonal pressures and with offensive weapons to shape portfolios of assets and liabilities in ways that promote each institution’s goals.

Asset-Liability Management Strategies

• There are mainly three asset-liability management strategies –
  • Asset management strategy – The traditional asset management view held that the amount and kinds of deposits a depository institution held and the volume of other borrowed funds it was able to attract were largely determined by its customers. Under this view, the public determined the relative amounts o f deposits and other sources of funds available to depository institutions. The key decision area for management was not deposits and other borrowings but assets. This strategy is not very popular now due to deregulation. The types of deposits, the rates offered, and the non-deposit sources of funds are not closely regulated now. Managers have more limited discretion in shaping their sources of funds.
  • Liability management strategy – The goal of liabilitymanagementis simply to gain control over the bank’s liabilities. The key control lever is the interest rate and other terms offered on deposits and other borrowings to achieve the volume, mix, and cost desired. Financial firms devote greater attention to opening up new sources of funding and monitoring the mix and cost of their deposit and non-deposit liabilities. Interest rate  can be raised or lowered based on the demand for funds. Revenues and costs arise from both sides of the balance sheet (i.e., from both asset and liability accounts). Management policies need to be developed that maximize returns and effectively control costs from supplying services.
  • Funds management strategy – The maturing of liability management techniques, coupled with more volatile interest rates and greater risk, eventually gave birth to the funds management approach, which dominates today. This view is a more balanced approach to asset-liability management that stresses several key objectives –
    • Management should exercise as much control as possible over the volume, mix, and return or cost of both assets and liabilities in order to achieve the financial institution’s goals.
    • Management’s control over assets must be coordinated with its control over liabilities so that asset management and liability management are internally consistent and do not pull against each other, Effective coordination in managing assets and liabilities will help to maximize the spread between revenues and costs and control risk exposure.
    • Revenues and costs arise from both sides of the balance sheet (i.e., from both asset and liability accounts). Management policies need to be developed that maximize returns and effectively control costs from supplying services.

Interest Rate Risk

• One of the toughest and potentially most damaging forms of risk that all financial institutions must face is interest rate risk.
  • When interest rates change, the sources of revenue that financial institutions receive, especially interest income on loans and investment securities change along with their most important source of expenses-interest cost on borrowings.
  • Moreover, changing interest rates also change the market value of assets and liabilities, thereby changing each financial institution’s net worth, and hence, the value of the owner’s investment in the firm.
• Thus, changing interest rates impact both the balance sheet and the statement of income and expenses of financial firms.
• As market interest rates move, financial firms typically face at least two major kinds of interest rate risk –
  • Price risk – Price risk arises when market interest rates rise, causing the market values of most bonds and fixed-rate loans to fall. If a financial institution wishes to sell these financial instruments in a rising rate period, it must be prepared to accept capital losses.
  • Reinvestment risk – Reinvestment risk rears its head when market interest rates fall,  forcing a financial firm to invest incoming funds in lower-yielding earning assets, lowering its expected future income.
  • A big part of managing assets and liabilities consists of finding ways to deal effectively with these two forms of risk.

Factors Determining Interest Rates

• Demand and Supply – The managers of financial institutions simply cannot control either the level of or the trend in interest rates markets. The rate of interest on any particular loan or security is ultimately determined by the financial marketplace where suppliers of loanable funds (credit) interact with demanders of loanable funds (credit) and the interest rate (price of credit) tends to settle at the point where the quantities of loanable funds demanded and supplied are equal, as shown in this figure.
• In granting loans, financial institutions are on the supply side of the loanable funds (credit) market. Similarly, financial institutions also come into the financial marketplace as demanders of loanable funds (credit) when they offer deposit services to the public or issue non- deposit IOUs to raise funds for lending and investing.

• Inflation – When the inflation rate is high, interest rates tend to increase, and vice versa. Higher interest rates are required to compensate for the decline in purchasing power of money.
• Government actions and Fiscal Deficit – The government actions play an important role in influencing interest rates. Fiscal policies are used by the central government to manage tax revenues and government expenditure. Fiscal deficit occurs when government expenditure exceeds the revenue. To fund this deficit, the governments borrow. Higher the fiscal deficit, higher the government borrowing, higher the interest rates.
• Central Bank Actions – Central banks use monetary policies to regulate the flow of money and the interest rates in an economy. During expansion, the central bank may choose to increase interest rates, and vice versa. Also, interest rates can be used by a central bank for influencing the foreign exchange rate sometimes.
• Since the above factors are not under their control, managers of financial institutions have to be price takers and not price makers. They just have to accept the interest rates changing because o these factors and plan accordingly.

The Measurement Of Interest Rates

• Interest rates are the price of credit, demanded by lenders as compensation for the use of borrowed funds. In simplest terms the interest rate is a ratio of the fees we must pay to obtain credit divided by the amount of credit obtained (expressed in percentage points and basis points). However, over the years, many of interest rate measures have been developed.
• Yield to Maturity (YTM) – It is one of the most popular rate measures. It is the discount rate that equalizes the current market value of a loan or security with the expected stream of future income payments that the loan or security will generate. In terms of a formula, the YTM may be found from

\( \text{Current market price} = \frac{ECF \text{ in Period 1}}{(1 + YTM)^1} + \frac{ECF \text{ in Period 2}}{(1 + YTM)^2} + \dots + \frac{ECF \text{ in Period } n}{(1 + YTM)^n} + \frac{\text{sale or redemption price in period } n}{(1 + YTM)^n} \)

where

𝑛 is the number of years that payments occur, and

𝐸𝐶𝐹 represents expected cash flow

For example, a bond purchased today at a price of $950 and promising an interest payment of $100 each year over the next three years, when it will be redeemed by the bond’s issuer for $1000, will have a promised interest rate, measured by the yield to maturity, determined by –

\$950 = \frac{100}{(1 + YTM)^1} + \frac{100}{(1 + YTM)^2} + \frac{100}{(1 + YTM)^3} + \frac{100}{(1 + YTM)^3} \)

• Bank Discount Rate – This is often quoted on short-term loans and money market securities (such as Treasury bills). The formula for calculating the discount rate (DR) is as follows:

• For example, suppose a money market security can be purchased for a price of $96 and has a face value of $100 to be paid at maturity. If the security matures in 90 days, its interest rate measured by the DR must be

\( DR = \frac{100 – 96}{100} \times \frac{360}{90} = 0.16, \text{ or } 16 \, percent \)

• DR ignores the effect of  compounding of interest whereas assumes that interest income is compounded at the calculated .
• DR is based on a 360-day year, unlike      , which assumes a      -day year.
• DR uses the face value of a financial instrument to calculate its yield or rate of return, a simple approach that makes calculations easier but is theoretically incorrect. The purchase price of a financial instrument, rather than its face amount, is a much better base
• YTM Equivalent Yield – To convert a to the equivalent yield to maturity, the following formula can be used

For the money market security discussed previously,

Apart from these interest rate measures, there are many other measures of “the interest rate” as well.

The Components Of Interest Rates

• The nominal or (published) market interest rate on a risky loan or security is given as the sum of
  • risk-free real interest rate (inflation-adjusted return on government bonds), and
  • risk premiums to compensate lenders who accept risk

NOTE

• The risk-free real interest rate changes with time due to changes in supply and demand of loanable funds.
• The perceptions of lenders and borrowers about each of the risk premiums also change over time.

These changes cause market interest rates to move up or down, often erratically.

The Components Of Interest Rates – Risk Premiums

• The default-risk premium component is the extra amount paid to compensate a lender for assuming default risk.
• The inflation-risk premium component is the amount lenders require for the possibility that inflation may rise or fall more than they expect over the reference period.
• Many loan and security interest rates also contain a liquidity risk premium, because some financial instruments are more difficult to sell quickly at a favorable price to another lender.
• Another rate-determining factor is call risk, which arises when a borrower has the right to pay off a loan early, possibly reducing the lender’s expected rate of return if market interest rates have fallen. Financial instruments with a greater call risk will carry higher interest rates because of higher call-risk premium, other factors held equal.
• Yield Curves – Another key component of each interest rate is the maturity, or term, premium. Longer-term loans and securities often carry higher market interest rates than shorter-term loans and securities due to maturity risk because of greater opportunities for loss over the  life of a longer-term loan. The graphic picture of how interest rates vary with different maturities of loans viewed at a single point in time (and assuming that all other factors, such as credit risk, are held constant) is called a yield curve.
• Yield curves constantly change their shape because the yields of the financial instruments included in each curve change all the time. Moreover, different yields tend to change at different speeds with short-term interest rates tending to rise or fall faster than long-term interest rates.
  • In periods of recession, short-term interest rates tended to fall relative to long-term interest rates and the yield “spread” or “gap” between short and long maturities often tended to widen.
  • In contrast, a period of economic prosperity usually begins with a fairly wide gap between long- and short-term interest rates, but that interest-rate gap tends to narrow and sometimes becomes negative.
• Yield curves display an upward slope (rising yield curve) when long-term interest rates exceed short-term interest rates. This often happens when all interest rates are rising but short-term rates have started from a lower level than long-term rates. Modern financial theory tends to associate upward-sloping yield curves with rising interest rates and with expansion in the economy.
• Yield curves can also slope downward (i.e., a falling yield curve) with short-term interest rates higher than long-term rates. Such a negatively sloped yield curve suggests that interest rates will begin falling and the economy may soon head into a recession.
• Finally, horizontal yield curves prevail when long-term interest rates and short-term rates are approximately at the same level so that investors receive the same yield to maturity no matter what maturity of investment security they purchase. A horizontal yield curve suggests that interest rates may be stable for a time with little change occurring in the slope of the curve.

Interest Rates Risk Challenges

• The Maturity Gap and the Yield Curve – The upward-sloping yield curve is usually more favorable for the profitability of lending institutions because their loans and security holdings on the asset side of their balance sheet tend to have longer maturities than their sources of funds (liabilities). In this case, the revenues from longer-term assets are more than the expenses from shorter-term liabilities. Thus, most lending institutions experience a positive maturity gap between the average maturity of their assets and the average maturity of their liabilities. This results in a positive net interest margin (𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑒𝑣𝑒𝑛𝑢𝑒𝑠 > 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑒𝑥𝑝𝑒𝑛𝑠𝑒𝑠), which tends to generate higher earnings. In contrast, a relatively flat (horizontal) or negatively sloped yield curve often generates a negative maturity gap and may lead to a small or even negative net interest margin, putting downward pressure on the earnings of financial firms that borrow short and lend long.

Responses To Interest Rate Risk

• Recent decades have witnessed a period of volatile interest rates, confronting financial managers with a more unpredictable environment to work in. Hence, financial institutions have tried to figure out ways to insulate their asset and liability portfolios and their profits from the dangers of changing interest rates.
• Asset-Liability Committee (ALCO) – Many banks now conduct their asset-liability management strategies under the guidance of an asset-liability committee, or ALCO.
  • The committee is expected to have a firm grasp of the organization’s principal goals, usually centered on the maximization of shareholder wealth, maintaining adequate profitability, and achieving sufficient capitalization. These goals are often stated as specific numerical targets (e.g., an ROA of percent and an equity-to-assets ratio of at least percent).
  • A well-run ALCO meets regularly (quarterly, monthly, or even more frequently) to manage the financial firm’s interest rate risk (IRR) and other risk exposures as well.
  • The ALCO regularly conveys to the board of directors the firm’s financial condition plus suggestions for correcting identified weaknesses.
  • The committee lays out a plan for how the firm should be funded, the quality of loans that should take on, and the proper limits to its off-balance-sheet risk exposure.
  • The ALCO estimates the firm’s risk exposure to its net interest margin and net worth ratios, develops strategies to keep that risk exposure within well-defined limits, and may employ simulation analysis to test alternative management strategies.
• In order to protect profits against adverse interest rate changes, then, management seeks to hold fixed the financial firm’s net interest margin ( ), expressed as follows –

For example, suppose a large international bank records $4 billion in interest revenues from its loans and security investments and $2.6 billion in interest expenses paid out to attract borrowed funds. If this bank holds $40 billion in earning assets, its net interest margin is

This net interest margin (which is fairly close to the average for the banking industry) is at just 3.5 percent, which is not this bank’s profit from borrowing and lending funds because non-interest  expenses have not been considered (such as employee salaries, taxes, and overhead expenses). Once these expenses are also deducted, the manager of this bank generally has very little margin for error against interest rate risk.

The bank will use a variety of interest-rate risk hedging methods to protect this NIM value, thereby helping to stabilize net earnings.

If the interest cost of borrowed funds rises faster than income from loans and securities, a financial firm’s will be squeezed, with likely adverse effects on profits. If interest rates fall and cause income from loans and securities to decline faster than interest costs on borrowings, the will again be squeezed. In other words, yield curves do not usually move in parallel fashion over time, so that the spread between borrowing costs and interest revenues is never constant. Management must struggle continuously to find ways to ensure that borrowing and threaten the margin of a financial firm.

Interest Sensitive Gap Management

• A popular interest rate hedging strategy is interest-sensitive gap management. Gap management techniques require management to perform an analysis of the maturities and repricing opportunities associated with interest-bearing assets and with interest-bearing liabilities. If management feels its institution is excessively exposed to interest rate risk, it will try to match as closely as possible the volume of assets that can be repriced as interest rates change with the volume of liabilities whose rates can also be adjusted with market conditions during the same time period.

For example, a financial firm can hedge itself against interest rate changes (no matter which way rates move) by making sure for each time period that the
Dollar amount of repriceable (interest-sensitive) Assets = Dollar amount of repriceable interest-sensitive) Liabilities

In this case, the revenue from earning assets will change in the same direction and by approximately the same proportion as the interest cost of liabilities.

• Repriceable Assets and Liabilities – The most familiar examples of repriceable assets include loans that are about to mature or will soon to come up for renewal or repricing, such as variable- rate business and household loans (including credit card accounts and adjustable-rate  home mortgages (ARMs)). If interest rates have risen since these loans were first made, the lender is likely to renew them only if it can get a rate of return that approximates the higher yields currently expected on other financial instruments of comparable quality. Similarly, loans that are maturing will provide the lender with newly released funds available to reinvest in new loans at today’s interest rates.
• In contrast, repriceable liabilities include a depository institution’s CDs about to mature or eligible to be renewed, where the financial firm and its customer must negotiate new deposit interest rates that capture current market conditions. Other examples include floating-rate deposits whose yields move up or down automatically with market interest rates; savings accounts that may be likely to be withdrawn at any time to seek out more favorable returns; interest-bearing checkable deposits (such as NOW accounts); and non-deposit money market borrowings whose interest rates are often adjusted several times daily to reflect the latest market developments.
• When the amount of repriceable assets does not equal the amount of repriceable liabilities, there is a gap which is the portion of the balance sheet affected by interest rate risk:

Interest-sensitive gap = Interest-sensitive assets – Interest-sensitive liabilities

Example of Repriceable (Interest-Sensitive) Assets and (Interest-Sensitive) Liabilities and Non-repriceable Assets and Liabilities
Repriceable (Interest-Sensitive) Assets	Repriceable (Interest-Sensitive) Liabilities	Non-repriceable Assets	Non-repriceable Liabilities
Short-term securities issued by governments and private borrowing (about to mature)	Borrowings from the money market (such as federal funds or RP borrowings)	Cash in the vault and deposits at the Central Bank (legal reserves)	Demand deposits (which pay no interest rate or a fixed interest rate)
Short-term loans made to borrowing customers (about to mature)	Short-term saving accounts	Long-term loans made at a fixed interest rate	Long-term saving and retirement accounts
Variable-rate loans and securities	Money-market deposits (whose interest rates are adjustable frequently)	Long-term securities carrying fixed rates, building, and equipment	Equity capital provided by the financial institution’s owners

• If interest-sensitive assets in each planning period (day, week, month, etc.) exceed the volume of interest-sensitive liabilities subject to repricing, the financial firm is said to have a positive gap and to be asset sensitive. Thus –
• 𝐴𝑠𝑠𝑒𝑡-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 (𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒) 𝑔𝑎𝑝 = (𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑎𝑠𝑠𝑒𝑡𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 > 0). For example, a bank with interest-sensitive assets of million and interest-sensitive liabilities of million is asset sensitive with a positive gap of million.
  • If interest rates rise, this bank’s will increase because the interest revenue generated by assets will increase more than the cost of borrowed funds. Other things being equal, this financial firm will experience an increase in its net interest income.
  • If interest rates fall when the bank is asset sensitive, this bank’s will decline as interest revenues from assets drop by more than interest expenses associated with liabilities. The financial firm with a positive gap will lose net interest income if interest rates fall.

• In the opposite situation, suppose an interest-sensitive bank’s liabilities are larger than its interest-sensitive assets. This bank then has a negative gap and is said to be liability sensitive. Thus –

𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 (𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒) 𝑔𝑎𝑝 = (𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑎𝑠𝑠𝑒𝑡𝑠 − 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡-𝑠𝑒𝑛𝑠𝑖𝑡𝑖𝑣𝑒 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 < 0)

• For example, a financial institution holding interest-sensitive assets of $150 million and interest-sensitive liabilities of $200 million is liability sensitive, with a negative gap of $50 million.
  • Rising interest rates will lower this institution’s , because the rising cost associated with interest-sensitive liabilities will exceed increases in interest revenue from interest-sensitive assets.
  • Falling interest rates will generate a higher and probably greater earnings as well, because borrowing costs will decline by more than interest revenues.

• The above method of measurement is simply called simply the Dollar IS GAP. For example, as we saw above, if interest-sensitive assets (ISA) are $150 million and interest-sensitive liabilities (ISA) are $200 million, then

Dollar IS GAP = ISA – ISL = $150 million – $200 million = -$50 million

• The Relative IS GAP ratio is defined as

A Relative IS GAP greater than zero means the institution is asset sensitive, while a negative. Relative IS GAP describes a liability-sensitive financial firm.

• Finally, the Interest Sensitivity Ratio (ISR) can be defined as the ratio of ISA to ISL. Based on the figures in the previous example,

In this instance an ISR of less than 1 tells us we are looking at a liability-sensitive institution, while an ISR greater than unity points to an asset-sensitive institution.

An Asset-Sensitive Financial Firm Has	A Liability-Sensitive Financial Firm Has
Positive Dollar IS GAP	Negative Dollar IS GAP
Positive Relative IS GAP	Negative Relative IS GAP
Interest Sensitivity Ratio greater than one	Interest Sensitivity Ratio less than one

• Gapping methods used today vary greatly in complexity and form. All methods, however, require financial managers to make some important decisions –
  • Management must choose the time period during which the net interest margin ( NIM ) is to be managed (e.g., six months or one year) to achieve some desired value and the length of subperiods (“maturity buckets”) into which the planning period is to be divided.
  • Management must choose a target level for the net interest margin-that is, whether to freeze the margin roughly where it is or perhaps increase the NIM.
  • If management wishes to increase the NIM, it must either develop a correct interest rate forecast or find ways to reallocate earning assets and liabilities to increase the spread between interest revenues and interest expenses.
  • Management must determine the volume of interest-sensitive assets and interest-sensitive liabilities it wants the financial firm to hold.
• Computer-Based Techniques – Many institutions use computer-based techniques in which their assets and liabilities are classified as due or repriceable today, during the coming week, in the next 30 days, and so on. Management tries to match interest-sensitive assets with interest- sensitive liabilities in each of these maturity buckets in order to improve the chances of  achieving the financial firm’s earnings goals. For example, a financial firm’s latest computer run might reveal the figures as shown in the table. It is obvious from this table that the time period over which the gap is measured is crucial to understanding this financial institution’s true interest-sensitive position.
  • For example, within the next 24 hours, the institution in this example has a positive gap; its earnings will benefit if interest rates rise between today and tomorrow.
  • However, a forecast of rising money market interest rates over the next week would be bad news because the cumulative gap for the next seven days is negative, which will result in interest expenses rising by more than interest revenues. If the interest rate increase is expected to be substantial, management should consider taking countermeasures to protect earnings. These might include selling longer-term CDs right away or using futures contracts to earn a profit that will help offset the margin losses that rising interest rates will almost surely bring in the coming week.

• Overall, net interest margin of a financial-service provider is influenced by multiple factors –

1. Changes in the level of interest rates, up or down.
2. Changes in the spread between asset yields and liability costs (often reflected in the changing shape o f the yield, curve between long-term rates and short-term rates).
3. Changes in the volume of interest-bearing (earning) assets a financial institution holds as it expands or shrinks the overall scale of its activities.
4. Changes in the volume of interest-bearing liabilities that are used to fund earning assets as a financial institution grows or shrinks in size.
5. Changes in the mix of assets and liabilities that management draws upon as it shifts between floating and fixed-rate assets and liabilities, between shorter and longer maturity assets and liabilities, and between assets bearing higher versus lower expected yields (e.g., a shift from less cash to more loans or from higher-yielding consumer and real estate loans to lower-yielding commercial loans).

• The table in the next page provides a more detailed example of interest-sensitive gap management techniques as they are applied to asset and liability data for an individual bank. In it, management has arrayed (with the help of a computer) the amount of all the bank’s assets  and liabilities, grouped by the future time period when those assets and liabilities will reach maturity, or their interest rates will be subject to repricing. This bank is liability sensitive during the coming week and over the next days and then becomes asset sensitive in later periods.
• Net interest income can be derived from the following formula:

• For example, suppose the yields on
  • rate-sensitive assets is percent
  • fixed assets is percent,
• And the cost of
  • rate-sensitive liabilities is percent
  • non-rate-sensitive liabilities is percent,
• As given in the previous table, during the coming week the bank holds
  • $1,700 million in rate-sensitive assets (out of an asset total of million), and
  • $1,800 million in rate-sensitive liabilities.
• Then this institution’s net interest income on an annualized basis will be

0.10 × $1,700 + 0.11 × 4,100 − 1,700 − 0.08 × $1,800 − 0.09 × 4,100 − 1,800 = $83 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

However, if the market interest rate on rate-sensitive assets rises to percent and the interest rate on rate-sensitive liabilities rises to percent during the first week, this liability-sensitive institution will have an annualized net interest income of only $81 𝑚𝑖𝑙𝑙𝑖𝑜𝑛, calculated as

0.12 × $1,700 + 0.11 × 4,100 − 1,700   − 0.10 × $1,800 − 0.09 × 4,100 − 1,800   = $81 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

Therefore, this bank will lose million in net interest income on an annualized basis if market interest rates rise in the coming week. Management must decide whether to accept that risk or to counter it with hedging strategies or tools.

• A useful overall measure of interest rate risk exposure is the cumulative gap, which is the total difference in dollars between those assets and liabilities that can be repriced over a designated period of time. For example, suppose that a bank has $100 million in earning assets and $200 million in liabilities subject to an interest rate change each month over the next six months. Then its cumulative gap must be

$100 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑖𝑛 𝑒𝑎𝑟𝑛𝑖𝑛𝑔 𝑎𝑠𝑠𝑒𝑡𝑠 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ × 6 − $200 𝑚𝑖𝑙𝑙𝑖𝑜𝑛 𝑖𝑛 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠 𝑝𝑒𝑟 𝑚𝑜𝑛𝑡ℎ × 6 = −$600 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

Using the cumulative gap concept, the approximate effect on net interest income with respect to market interest rate changes can be calculated. The key relationship is –

𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑛𝑒𝑡 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑖𝑛𝑐𝑜𝑚𝑒 = 𝑂𝑣𝑒𝑟𝑎𝑙𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒 𝑖𝑛𝑐𝑜𝑚𝑒 (𝑖𝑛 𝑝𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑝𝑜𝑖𝑛𝑡𝑠) × 𝑆𝑖𝑧𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑐𝑢𝑚𝑢𝑙𝑎𝑡𝑖𝑣𝑒 𝑔𝑎𝑝 (𝑖𝑛 𝑑𝑜𝑙𝑙𝑎𝑟𝑠)

For example, suppose market interest rates suddenly rise by 1 full percentage point. Then the bank in the example given above will suffer a net interest income loss of approximately

+0.01 × (−$600 𝑚𝑖𝑙𝑙𝑖𝑜𝑛) = − $ 6 𝑚𝑖𝑙𝑙𝑖𝑜𝑛

• Some firms shade their interest-sensitive gaps toward either asset sensitivity or liability sensitivity, depending on their degree of confidence in their own interest rate forecasts. This  is often referred to as aggressive GAP management.
  • If management firmly believes interest rates are going to fall over the current planning horizon, it will probably allow interest-sensitive liabilities to climb above interest-sensitive assets. If interest rates do fall as predicted, liability costs will drop by more than revenues and the institution’s will grow.
  • Similarly, a confident forecast of higher interest rates will trigger many financial firms to become asset sensitive, knowing that if rates do rise, interest revenues will rise by more than interest expenses.
• Of course, such an aggressive strategy creates greater risk. Consistently correct interest rate forecasting is impossible; most financial managers have learned to rely on hedging against, not forecasting, changes in market interest rates. Interest rates that move in the wrong direction can magnify losses.

• The following table highlights the risks and the possible responses by management of a firm having a positive or negative interest rate gap.

Many financial-service managers have chosen to adopt a purely defensive GAP management strategy where they try and set interest-sensitive GAP as close to zero as possible to reduce the expected volatility of net interest income.

Issues And Weighted Interest Sensitive Gap Management

• Problems with Interest-Sensitive GAP Management – While interest-sensitive gap management works beautifully in theory, practical problems in its implementation always leave financial institutions with at least some interest-rate risk exposure. For example, interest rates paid on liabilities (which often are predominantly short term) tend to move faster than interest rates earned on assets (many of which are long term). Also, changes in interest rates attached to assets and liabilities do not necessarily move at the same speed as do interest rates in the open market. In the case of a bank, for example, deposit interest rates typically lag behind loan interest rates.
• Some financial institutions have developed a weighted interest-sensitive gap approach that takes into account the tendency of interest rates to vary in speed and magnitude relative to each other and with the up and down cycle of business activity. The interest rates attached to assets often change by different amounts and at different speeds than the interest rates attached to liabilities – a phenomenon called basis risk. For example, suppose a bank has the current amount and distribution of interest-sensitive assets and liabilities shown in the table in the next page with rate-sensitive assets totaling $200 million and rate-sensitive liabilities amounting to $223 million, yielding an IS GAP of -$23 million on its present balance sheet. Its federal funds loans generally carry interest rates set in the open market, so these loans have an interest rate sensitivity weight of 1.0, which means that, it is assumed that the bank’s fed funds rate tracks market rates one for one. In the portfolio, there are some riskier, somewhat more rate-volatile investments whose average security yield moves up and down by somewhat more than the interest rate on federal funds loans. So, their interest-rate sensitivity weight is estimated to be 1.3. Loans and leases are the most rate volatile of all with an interest rate sensitivity weight estimated to be 1.5. On the liability side, deposit interest rates and some money market borrowings (such as borrowing from the central bank) may change more slowly than market interest rates and  it is assumed that deposits have a rate-sensitive weight of 0.86 and money market borrowings are slightly more volatile at 0.91
• After multiplying each of the rate-sensitive balance-sheet items by its appropriate interest rate sensitivity weight, the new weighted balance sheet has rate-sensitive assets of $270 and rate- sensitive liabilities of $195. Instead of a negative (liability-sensitive) interest rate gap of – $23, now there is a positive (asset-sensitive) rate gap of +$75.
• Thus, this institution’s interest-sensitive gap has changed direction and, instead of being hurt by rising market interest rates, for example, this financial firm would benefit from higher market interest rates. Suppose the federal funds interest rate rose by 2 percentage points (+0.02). Instead of declining by -$0.46, this bank’s net interest income increases by $1.50 as shown in the bottom of the table in the previous page. Indeed, when it comes to assessing interest rate risk, things are not always as they appear !

The Limitations Of Interest Sensitive Gap Management

• While interest sensitive gap management is useful, it has limitations as discussed earlier. These limitations are once again summarized as follows –
  • Interest rates paid on liabilities (which are mostly short term) tend to move faster than interest rates earned on assets (which are mostly long term).
  • The point at which certain assets and liabilities can be repriced is not always easy to identify.
  • Interest-sensitive gap management does not consider the impact of changing interest rates on the owners’ (stockholders’) position in the financial firm as represented by the institution’s net worth.
  • There is basis risk since the interest rates attached to assets often change by different amounts and at different speeds than the interest rates attached to liabilities.

Duration Gap Management

• Interest-sensitive gap only looks at the effects of changes in the interest rates on the bank’s net income (or net interest margin). Changing interest rates can also do serious damage to another aspect of a financial firm’s performance-its net worth, the value of the stockholders’ investment in the institution. Just because the net interest margin is protected against interest rate risk doesn’t mean an institution’s net worth is also sheltered from loss, and for most companies net worth is more important than their net interest margin. This requires the application of yet another managerial tool – duration gap management.
• Duration is a value- and time-weighted measure of maturity that considers the timing of all cash inflows from earning assets and all cash outflows associated with liabilities. It measures the average maturity of a promised stream of future cash payments (such as the payment streams that a financial firm expects to receive from its loans and security investments or the stream of interest payments it must pay out to its depositors). In effect, duration measures the average time needed to recover the funds committed to an investment.
• The standard formula for calculating the duration (D) of an individual financial instrument, such as a loan, security, deposit, or nondeposit borrowing, is –

where

𝐷 stands for the instrument’s duration in years and fractions of a year

𝑡 represents the period of time in which each flow of cash is to be received

𝐶𝐹 indicates the volume of each expected flow of cash in each time period (t)

𝑌𝑇𝑀 is the instrument’s current yield to maturity.

𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝐶𝐹 𝑖𝑛 𝑃𝑒𝑟𝑖𝑜𝑑 𝑡

𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑚𝑎𝑟𝑘𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑟 𝑃𝑟𝑖𝑐𝑒 = (1 + 𝑌𝑇𝑀)^t

The calculation of duration has already been discussed in detail in FRM Part 1. This table can be used to recollect those calculations. In this example, t = 5 years, YTM = 10 percent, and Face value of loan = $1000.
The duration of the loan is \( \frac{4,169.87}{1,000.00} \approx 4.17 \, \text{years} \)

Period of Expected Cash Flows	Expected Cash Flow from Loan	Present Value of Expected Cash Flow (at 10% YTM in this Case)	Time Period Cash Is to Be Received (t)	Present Value of Expected Cash Flows × t
1	$100	$90.91	1	$90.91
2	100	82.64	2	165.29
3	100	75.13	3	225.39
4	100	68.30	4	273.21
5	100	62.09	5	318.46
5	1,000	620.92	5	3,104.61
Price or Denominator of Formula = $1,000.00       PV of Cash Flows × t = $4,169.87

• The net worth (NW) of any business or household is equal to the value of its assets less the value o f its liabilities . As market interest rates change, the value of both a financial institution’s assets and its liabilities will change, resulting in a change in its net worth (the owner’s investment in the institution) given by

• According to portfolio theory –

1. A rise in market rates of interest will cause the market value (price) of both fixed-rate assets and liabilities to decline.
2. The longer the maturity of a financial firm’s assets and liabilities, the more they will tend to decline in market value (price) when market interest rates rise.

Thus, net worth will vary depending upon the interest rates, and also upon the durations of a financial institution’s assets and liabilities. By equating asset and liability durations, management can balance the average maturity of expected cash inflows from assets with the average maturity of expected cash outflows associated with liabilities. Thus, duration analysis can be used to stabilize, or immunize, the market value of a financial institution’s .

• Price Sensitivity – Duration measures the sensitivity of the market value of financial instruments to changes in interest rates. The percentage change in the market price of an asset or a liability is equal to its duration times the relative change in interest rates attached to that particular asset or liability –

\( \frac{\Delta P}{P} \approx -D \times \frac{\Delta i}{1 + i} \)

where

( \frac{\Delta P}{P} ) represents the percentage change in market price, and

( \frac{\Delta i}{1+i} ) is the relative change in interest rates associated with the asset or liability.

Convexity refers to the presence of a nonlinear relationship between changes in an asset’s price and changes in market interest rates. It is a number designed to aid portfolio managers in measuring and controlling the market risk in a portfolio of assets. An asset or portfolio bearing both a low duration and low convexity normally displays relatively small market risk. Convexity increases with the duration (maturity) of an asset.

Using Duration To Hedge Against Interest Rate Risk

• A financial-service provider interested in fully hedging against interest rate fluctuations wants to choose assets and liabilities such that the duration gap is as close to zero as possible, i.e

𝐷𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑝 =

𝐷𝑜𝑙𝑙𝑎𝑟-𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎𝑠𝑠𝑒𝑡 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 −

𝐷𝑜𝑙𝑙𝑎𝑟-𝑤𝑒𝑖𝑔ℎ𝑡𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦 𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 ≈ 0

Because the dollar volume of assets usually exceeds the dollar volume of liabilities (otherwise the financial firm would be insolvent), a financial institution seeking to minimize the effects of interest rate fluctuations would need to adjust for leverage:

This equation implies that the value of liabilities must change by slightly more than the value of assets to eliminate a financial firm’s overall interest-rate risk exposure. The larger the leverage-adjusted duration gap, the more sensitive will be the net worth (equity capital) of a financial institution to a change in interest rates. If the 𝑙𝑒𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑝 > 0, then change in liabilities will be less than change in assets due to a parallel shift in all interest rates. In this case, a rise in interest rates will tend to lower the market value of net worth as asset values fall further than the value of liabilities.

On the other hand, if 𝑙𝑒𝑣𝑒𝑟𝑎𝑔𝑒 𝑎𝑑𝑗𝑢𝑠𝑡𝑒𝑑 𝑑𝑢𝑟𝑎𝑡𝑖𝑜𝑛 𝑔𝑎𝑝 < 0, then a parallel change in all interest rates will generate a larger change in liability values than asset values. If interest rates fall, liabilities will increase more in value than assets and net worth will decline. Should interest rates rise, however, liability values will decrease faster than asset values and the net worth position will increase in value.

• The change in the market value of a financial institution’s net worth can be calculated as

Δ𝑁𝑊 = ΔA − Δ𝐿

\( \Rightarrow \Delta NW = \left[ -D_A \times \frac{\Delta i}{(1 + i)} \times A \right] – \left[ -D_L \times \frac{\Delta i}{(1 + i)} \times L \right] \)

D_A is the average duration of assets

D_L is the average duration of liabilities
In words,

For example, suppose that a financial firm has an average duration in its assets of three years, an average liability duration of two years, total liabilities of million, and total assets of

million. Interest rates were originally percent, but suddenly they rise to percent.

Change in the value of net worth can be calculated as

\( \Delta NW = \left[ -3 \times \frac{+0.02}{(1 + 0.10)} \times 120 \, \text{million} \right] – \left[ -2 \times \frac{+0.02}{(1 + 0.10)} \times 100 \, \text{million} \right] = -2.91 \, \text{million} \)

Clearly, this institution faces a substantial decline in the value of its net worth unless it can hedge itself against the projected loss due to rising interest rates.

• The interest rate hedging mentioned above can be done by

1. calculating the duration of each loan, deposit, and the like
2. weighting each of these durations by the market values of the instruments involved, and
3. adding all value-weighted durations together to derive the duration of a financial institution’s entire portfolio.

For example, suppose management of a bank finds that it holds a U.S. Treasury $1,000 par bond with 10 years to final maturity, bearing a 10 percent coupon rate with a current price of $900. The duration of the bond can be easily calculated as 7.49 years using the formula given earlier. Suppose this bank holds $90 million of these Treasury bonds, each with a duration of 7.49 years. The bank also holds other assets with durations and market values as given in the table. Weighting each asset duration by its associated dollar volume, the duration of the asset portfolio can be calculated as follows:

Assets Held	Actual or Estimated Market Values of Assets	Asset Durations
Treasury bonds	$90 million	7.49 years
Commercial loans	100 million	0.60 year
Consumer loans	50 million	1.20 years
Real estate loans	40 million	2.25 years
Municipal bonds	20 million	1.50 years

Consider another example as follows –

The positive duration gap of +0.60 year means that the bank’s net worth will decline if interest rates rise and increase if interest rates fall. Management may be anticipating a decrease in the level of interest rates. If there is significant risk of rising market interest rates, however, the asset-liability management committee will want to use hedging tools to reduce the exposure of net worth to interest rate risk.

Suppose interest rates on both assets and liabilities rise from 8 to 10 percent. Then change in value of net worth can be calculated as

Hence, the institution’s net worth would fall by approximately $3.34 million if interest rates increased by 2 percentage points.

On the other hand, it can also be shown that the institution’s net worth would rise by approximately $3.34 million if interest rates increased by 2 percentage points.

• Overall, the impact of interest rate changes on the market value of net worth depends upon three crucial size factors:
• The size of the duration gap (DA – DL), with a larger duration gap indicating greater exposure of a financial firm to interest rate risk.
• The size of a financial institution (A and L), with larger institutions experiencing a greater change in net worth for any given change in interest rates.
• The size of the change in interest rates, with larger rate changes generating greater interest rate risk exposure.

In summary, the impact of changing market interest rates on net worth is indicated by entries in this table.

If the Financial Institution’s Leverage-Adjusted Duration Gap Is: Leverage-Adjusted Duration Gap	And If Interest Rates:	The Financial Institution’s Net Worth Will:
Positive \( \left( D_A > D_L \times \frac{\text{Liabilities}}{\text{Assets}} \right) \)	Rise Fall	Decrease Increase
Negative \( \left( D_A < D_L \times \frac{\text{Liabilities}}{\text{Assets}} \right) \)	Rise Fall	Increase Decrease
Zero \( \left( D_A = D_L \times \frac{\text{Liabilities}}{\text{Assets}} \right) \)	Rise Fall	No Change No Change

The Limitations Of Duration Gap Management

• While duration is simple to interpret, it has limitations –
  • Finding assets and liabilities of the same duration is often a frustrating task. It would be much easier if the maturity of a loan or security equaled its duration; however, for financial instruments paying out gradually over time, duration is always less than calendar maturity.
  • Some accounts held by depository institutions, like checkable deposits and passbook savings accounts, may have a pattern of cash flows that is not well defined, making the calculation of duration difficult.
  • Customer prepayments distort the expected cash flows from loans and so do customer defaults (credit risk) when expected cash flows do not happen.
  • Duration gap models assume that a linear relationship exists between the market values (prices) of assets and liabilities and interest rates, which is not strictly true.
  • If there are major changes in interest rates and different interest rates move at different speeds, the accuracy and effectiveness of duration gap management decreases.
  • yield curves in the real world typically do not change in parallel fashion-short-term interest rates tend to move over a wider range than long-term interest rates, for example-and a big change in market interest rates (say, one or two percentage points) can result in a distorted reading of how much interest rate risk a financial manager is really facing.
  • Duration itself can shift as market interest rates move, and the durations of different financial instruments can change at differing speeds with the passage of time.