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Interest Rate Risk Finance 129 Drake Fin 129 DRAKE UNIVERSITY Review of Key Factors Drake Impacting Drake University Fin 129 Interest Rate Volatility Federal Reserve and Monetary Policy Discount Window Reserve Requirements Open Market Operations Review of Key Factors Impacting Drake Drake University Interest Rate Volatility Fin 129 Fisher model of the Savings Market Two main participants: Households and Business Households supply excess funds to Businesses who are short of funds The Saving or supply of funds is upward sloping The investment or demand for funds is downward sloping Saving and Investment Drake Drake University Decisions Fin 129 Saving Decision Marginal Rate of Time Preference Trading current consumption for future consumption Expected Inflation Income and wealth effects Generally higher income – save more Federal Government Money supply decisions Business Short term temporary excess cash. Foreign Investment Drake Borrowing Decisions Drake University Fin 129 Borrowing Decision Marginal Productivity of Capital Expected Inflation Other Drake Equilibrium in the Market Drake University Fin 129 Original Equilibrium Decrease in Income S S D D Increase in Marg. Prod Cap Increase in Inflation Exp. S S D D Drake Loanable Funds Theory Drake University Fin 129 Expands suppliers and borrowers of funds to include business, government, foreign participants and households. Interest rates are determined by the demand for funds (borrowing) and the supply of funds (savings). Very similar to Fisher in the determination of interest rates, Drake Loanable Funds Drake University Fin 129 Now equilibrium extends through all markets – money markets, bonds markets and investment market. Inflation expectations can also influence the supply of funds. Drake Liquidity Preference Theory Drake University Fin 129 Liquidity Preference Two assets, money and financial assets Equilibrium in one implies equilibrium in other Supply of Money is controlled by Central Bank and is not related to level of interest rates Drake The Yield Curve Drake University Fin 129 Three things are observed empirically concerning the yield curve: Rates across different maturities move together More likely to slope upwards when short term rates are historically low, sometimes slope downward when short term rates are historically high The yield curve usually slope upward Three Explanations of the Yield Drake Drake University Curve Fin 129 The Expectations Theories Segmented Markets Theory Preferred Habitat Theory Drake Pure Expectations Theory Drake University Fin 129 Long term rates are a representation of the short term interest rates investors expect to receive in the future. In other words the forward rates reflect the future expected rate. Assumes that bonds of different maturities are In other words, the expected return from holding a one year bond today and a one year bond next year is the same as buying a two year bond today. (the same process that is used to calculate forward rates) Pure Expectations Theory: Drake Drake University A Simplified Illustration Fin 129 Let Rt = today’s time t interest rate on a one period bond Ret+1 = expected interest rate on a one period bond in the next period R2t = today’s (time t) yearly interest rate on a two period bond. Investing in successive one Drake Drake University period bonds Fin 129 If the strategy of buying the one period bond in two consecutive years is followed the return is: (1+Rt)(1+Ret+1) – 1 which equals Rt+Ret+1+ (Rt)(Ret+1) Since (Rt)(Ret+1) will be very small we will ignore it Drake The 2 Period Return Drake University Fin 129 If the strategy of investing in the two period bond is followed the return is: (1+R2t)(1+R2t) - 1 = 1+2R2t+(R2t)2 - 1 (R2t)2 is small enough it can be dropped which leaves Drake Set the two equal to each other Drake University Fin 129 2R2t = Rt+Ret+1 R2t = (Rt+Ret+1)/2 In other words, the two period interest rate is the average of the two one period rates Drake Expectations Hypothesis Drake University R2t = (Rt+Ret+1)/2 Fin 129 When the yield curve is upward sloping (R2t>R1t) it is expected that short term rates will be increasing (the average future short term rate is above the current short term rate). Likewise when the yield curve is downward sloping the average of the future short term rates is below the current rate. (Fact 2) As short term rates increase the long term rate will also increase and a decrease in short term rates will decrease long term rates. (Fact 1) This however does not explain Fact 3 that the yield curve usually slopes up. Drake Problems with Pure Expectations Drake University Fin 129 The pure expectations theory ignores the fact that there is reinvestment rate risk and different price risk for the two maturities. Consider an investor considering a 5 year horizon with three alternatives: buying a bond with a 5 year maturity buying a bond with a 10 year maturity and holding it 5 years buying a bond with a 20 year maturity and holding it 5 years. Drake Price Risk Drake University Fin 129 The return on the bond with a 5 year maturity is known with certainty the other two are not. Drake Reinvestment rate risk Drake University Fin 129 Now assume the investor is considering a short term investment then reinvesting for the remainder of the five years or investing for five years. Drake Local Expectations Drake University Fin 129 Similarly owning the bond with each of the longer maturities should also produce the same 6 month return of 2%. The key to this is the assumption that the forward rates hold. It has been shown that this interpretation is the only one that can be sustained in equilibrium.* Cox, Ingersoll, and Ross 1981 Journal of Finance Return to maturity expectations Drake Drake University hypothesis Fin 129 This theory claims that the return achieved by buying short term and rolling over to a longer horizon will match the zero coupon return on the longer horizon bond. This eliminates the reinvestment risk. Expectations Theory and Drake Drake University Forward Rates Fin 129 The forward rate represents a “break even” rate since it the rate that would make you indifferent between two different maturities The pure expectations theory and its variations are based on the idea that the forward rate represents the market expectations of the future level of interest rates. However the forward rate does a poor job of predicting the actual future level of interest rates. Drake Segmented Markets Theory Drake University Fin 129 Interest Rates for each maturity are determined by the supply and demand for bonds at each maturity. Different maturity bonds are not perfect substitutes for each other. Implies that investors are not willing to accept a premium to switch from their market to a different maturity. Drake Biased Expectations Theories Drake University Fin 129 Both Liquidity Preference Theory and Preferred Habitat Theory include the belief that there is an expectations component to the yield curve. Both theories also state that there is a risk premium which causes there to be a difference in the short term and long term rates. (in other words a bias that changes the expectations result) Drake Liquidity Preference Theory Drake University Fin 129 This explanation claims that the since there is a price risk and liquidity risk associated with the long term bonds, investor must be offered a premium to invest in long term bonds Therefore, the long term rate reflects both an expectations component and a risk premium. This tends to imply that the yield curve will be upward sloping as long as the premium is large enough to outweigh a possible expected decrease. Drake Preferred Habitat Theory Drake University Fin 129 Like the liquidity theory this idea assumes that there is an expectations component and a risk premium. In other words the bonds are substitutes, but savers might have a preference for one maturity over another (they are not perfect substitutes). However the premium associated with long term rates does not need to be positive. If there are demand and supply imbalances then investors might be willing to switch to a different maturity if the premium produces enough benefit. Preferred Habitat Theory Drake Drake University and The 3 Empirical Observations Fin 129 Thus according to Preferred Habitat theory a rise in short term rates still causes a rise in the average of the future short term rates. This occurs because of the expectations component of the theory. Drake Preferred Habitat Theory Drake University Fin 129 The explanation of Fact 2 from the expectations hypothesis still works. In the case of a downward sloping yield curve, the term premium (interest rate risk) must not be large enough to compensate for the currently high short term rates (Current high inflation with an expectation of a decrease in inflation). Since the demand for the short term bonds will increase, the yield on them should fall in the future. Drake Preferred Habitat Theory Drake University Fin 129 Fact three is explained since it will be unusual for the term premium to be so small or negative, therefore the the yield curve usually slopes up. Drake Yield Curves Previous Month Drake University Fin 129 0.053 0.048 0.043 Yield 0.038 8/8/2007 8/15/2007 8/22/2007 0.033 8/29/2007 9/5/2007 9/12/2007 Maturity (Years) 0.028 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Drake Yield Curves Previous 6 Months Drake University Fin 129 0.052 0.047 Yield 0.042 0.037 6/15/2007 5/15/2007 6/15/2007 7/16/2007 8/15/2007 9/12/2007 Maturity (Years) 0.032 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Drake Yield Curves Previous 6 quarters Drake University Fin 129 0.055 0.05 0.045 Yield 0.04 0.035 6/15/2006 9/15/2006 12/15/2006 3/15/2007 6/15/2007 9/12/2007 Maturity (Years) 0.03 0.00 5.00 10.00 15.00 20.00 US Treasury Yields Drake Drake University Jan1989 -June 2006 Fin 129 0.1 1-mo 3-mo 6-mo 1-yr 0.09 2-yr 3-yr 5-yr 7-yr 10-yr 20-yr 30-yr 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 0 12/8/1989 9/3/1992 5/31/1995 2/24/1998 11/20/2000 8/17/2003 5/13/2006 US Treas Rates Drake Drake University May 1990 – Sept 2007 Fin 129 0.1 0.09 0.08 0.07 0.06 0.05 0.04 0.03 0.02 0.01 3-mo 6-mo 10-yr 20-yr 0 5/7/1990 1/31/1993 10/28/1995 7/24/1998 4/19/2001 1/14/2004 10/10/2006 Impact of Interest Rate Volatility Drake Drake University on Financial Institutions Fin 129 The market value of assets and liabilities is tied to the level of interest rates Interest income and expense are both tied to the level of interest rates Static GAP Analysis Drake Drake University (The repricing model) Fin 129 Repricing GAP The difference between the value of interest sensitive assets and interest sensitive liabilities of a given maturity. Measures the amount of rate sensitive (asset or liability will be repriced to reflect changes in interest rates) assets and liabilities for a given time frame. Drake Commercial Banks & GAP Drake University Fin 129 Commercial banks are required to report quarterly the repricing Gaps for the following time frames One day More than one day less than 3 months More than 3 months, less than 6 months More than 6 months, less than 12 months More than 12 months, less than 5 years More than five years Drake GAP Analysis Drake University Fin 129 Static GAP-- Goal is to manage interest rate income in the short run (over a given period of time) Measuring Interest rate risk – calculating GAP over a broad range of time intervals provides a better measure of long term interest rate risk. Drake Interest Sensitive GAP Drake University Fin 129 Given the Gap it is easy to investigate the change in the net interest income of the financial institution. Drake Example Drake University Fin 129 Over next 6 Months: Rate Sensitive Liabilities = $120 million Rate Sensitive Assets = $100 Million If rate are expected to decline by 1% Change in net interest income Drake GAP Analysis Drake University Fin 129 Asset sensitive GAP (Positive GAP) RSA – RSL > 0 If interest rates h NII will If interest rates i NII will Liability sensitive GAP (Negative GAP) RSA – RSL < 0 If interest rates h NII will If interest rates i NII will Would you expect a commercial bank to be asset or liability sensitive for 6 mos? 5 years? Drake Important things to note: Drake University Fin 129 Assuming book value accounting is used -- only the income statement is impacted, the book value on the balance sheet remains the same. The GAP varies based on the bucket or time frame calculated. It assumes that all rates move together. Drake Steps in Calculating GAP Drake University Fin 129 1) Select time Interval 2) Develop Interest Rate Forecast 3) Group Assets and Liabilities by the time interval (according to first repricing) 4) Forecast the change in net interest income. Drake Alternative measures of GAP Drake University Fin 129 Cumulative GAP Totals the GAP over a range of of possible maturities (all maturities less than one year for example). Total GAP including all maturities Other useful measures using Drake Drake University GAP Fin 129 Relative Interest sensitivity GAP (GAP ratio) GAP / Bank Size The higher the number the higher the risk that is present Interest Sensitivity Ratio Drake What is “Rate Sensitive” Drake University Fin 129 Any Asset or Liability that matures during the time frame Any principal payment on a loan is rate sensitive if it is to be recorded during the time period Assets or liabilities linked to an index Interest rates applied to outstanding principal changes during the interval Drake What about Core Deposits? Drake University Fin 129 Against Inclusion Demand deposits pay zero interest NOW accounts etc do pay interest, but the rates paid are sticky For Inclusion Implicit costs If rates increase, demand deposits decrease as individuals move funds to higher paying accounts (high opportunity cost of holding funds) Drake Expectations of Rate changes Drake University Fin 129 If you expect rates to increase would you want GAP to be positive or negative? Unequal changes in interest Drake Drake University rates Fin 129 So far we have assumed that the change the level of interest rates will be the same for both assets and liabilities. If it isn’t you need to calculate GAP using the respective change. Spread effect – The spread between assets and liabilities may change as rates rise or decrease Drake Strengths of GAP Drake University Fin 129 Easy to understand and calculate Allows you to identify specific balance sheet items that are responsible for risk Provides analysis based on different time frames. Drake Weaknesses of Static GAP Drake University Fin 129 Market Value Effects Basic repricing model the changes in market value. The PV of the future cash flows should change as the level of interest rates change. (ignores TVM) Over aggregation Repricing may occur at different times within the bucket (assets may be early and liabilities late within the time frame) Many large banks look at daily buckets. Drake Weaknesses of Static GAP Drake University Fin 129 Runoffs Periodic payment of principal and interest that can be reinvested and is itself rate sensitive. You can include runoff in your measure of rate sensitive assets and rate sensitive liabilities. Note: the amount of runoffs may be sensitive to rate changes also (prepayments on mortgages for example) Drake Weaknesses of GAP Drake University Fin 129 Off Balance Sheet Activities Basic GAP ignores changes in off balance sheet activities that may also be sensitive to changes in the level of interest rates. Ignores changes in the level of demand deposits Drake Other Factors Impacting NII Drake University Fin 129 Changes in Portfolio Composition An aggressive position is to change the portfolio in an attempt to take advantage of expected changes in the level of interest rates. (if rates are h have positive GAP, if rates are i have negative GAP) Problem: Drake Other Factors Impacting NII Drake University Fin 129 Changes in Volume Bank may change in size so can GAP along with it. Changes in the relationship between ST and LT We have assumes parallel shifts in the yield curve. The relationship between ST and LT may change (especially important for cumulative GAP) Drake Extending Basic GAP Drake University Fin 129 You can repeat the basic GAP analysis and account for some of the problems Include Forecasts of when embedded options will be exercised and include them Drake The Maturity Model Drake University Fin 129 In this model the impact of a change in interest rates on the market value of the asset or liability is taken into account. The securities are marked to market Keep in Mind the following: The longer the maturity of a security the larger the impact of a change in interest rates An increase in rates generally leads to a fall in the value of the security The decrease in value of long term securities increases at a diminishing rate for a given increase in rates Drake Weighted Average Maturity Drake University Fin 129 You can calculate the weighted average maturity of a portfolio. The same three principles of the change in the value of the portfolio (from last slide) will apply M i Wi1M i1 Wi 2 M i 2 Win M in Drake Maturity GAP Drake University Fin 129 Given the weighted average maturity of the assets and liabilities you can calculate the maturity GAP Drake Maturity Gap Analysis Drake University Fin 129 If Mgap is + the maturity of the FI assets is longer than the maturity of its liabilities. (generally the case with depository institutions due to their long term fixed assets such as mortgages). This also implies that its assets are more rate sensitive than its liabilities since the longer maturity indicates a larger price change. Drake The Balance Sheet and MGap Drake University Fin 129 The basic balance sheet identity state that: Asset = Liabilities + Owners Equity or Owners Equity = Assets - Liabilities Technically if Liab >Assets the institution is insolvent If MGAP is positive and interest rate decrease then the market value of assets increases more than liabilities. Likewise, if MGAP is negative an increase in interest rates would cause Drake Matching Maturity Drake University Fin 129 By matching maturity of assets and liabilities owners can be immunized form the impact of interest rate changes. However this does not always completely eliminate interest rate risk. Think about duration and funding sources (does the timing of the cash flows match?). Drake Drake University Duration Fin 129 Duration: Weighted maturity of the cash flows (either liability or asset) Weight is a combination of timing and magnitude of the cash flows The higher the duration the more sensitive a cash flow stream is to a change in the interest rate. Drake Duration Mathematics Drake University Fin 129 Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield. Drake Duration Mathematics Drake University Fin 129 CP CP CP CP MV P (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n P (-1)CP (-2)CP (-3)CP (-n)CP (-n)MV n 1 r (1 r) 2 (1 r) 3 (1 r) 4 (1 r) (1 r) n 1 P 1 1CP 2CP 3CP nCP nMV r 1 r (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n The approximate price change for a small change in r Drake Duration Mathematics Drake University Fin 129 P 1 1CP 2CP 3CP nCP nMV (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n r 1 r To find the % price change divide both sides by the original Price P 1 1 1CP 2CP 3CP nCP nMV 1 (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n P r P 1 r The RHS is referred to as the Modified Duration Which is the % change in price for a small change in yield Duration Mathematics Drake Drake University Macaulay Duration Fin 129 Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield). Formally this would be: change in price original price Change in Price Original yield P (1 r) D MAC Original price r P change in yield Change in Yield original yield Duration Mathematics Drake Drake University Macaulay Duration Fin 129 change in price original price Change in Price Original yield P (1 r) D MAC Original price r P change in yield Change in Yield original yield substitute P 1 1CP 2CP 3CP nCP nMV r 1 r (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n 1 1CP 2CP 3CP nCP nMV (1 r) DMAC (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n P 1 r Drake Drake University Macaulay Duration of a bond Fin 129 1CP 2CP 3CP nCP nMV 1 (1 r) n P DMAC (1 r) (1 r) (1 r) 2 3 (1 r) n N t(CP) N(MV) (1 r) t (1 r) N DMAC t 1 N CP MV (1 r) t (1 r) N t 1 Drake Duration Example Drake University Fin 129 10% 30 year coupon bond, current rates =12%, semi annual payments 60 t ($50) 60($1000) (1 .06)t (1 .06)60 DMAC 60 t 1 17.3895 periods 50 $1000 (1 .06)t (1 .06)60 t 1 Drake Example continued Drake University Fin 129 Since the bond makes semi annual coupon payments, the duration of 17.3895 periods must be divided by 2 to find the number of years. 17.3895 / 2 = 8.69475 years This interpretation of duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment. Using Duration Drake Drake University to estimate price changes Fin 129 P (1 r) Rearrange P r D MAC D MAC r P P (1 r) % Change in Price Estimate the % price change for a 1 basis point increase in yield P r .0001 D MAC 8.69925 0.000776 P (1 r) 1.12 The estimated price change is then -0.000776(838.8357)=-0.6515 Drake Using Duration Continued Drake University Fin 129 Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12% Original Price of the bond = 838.3857 If YTM = 12.01% the price is 837.6985 This implies a price change of -0.6871 Our duration estimate was -0.6515 Drake Modified Duration Drake University Fin 129 From before, modified duration was defined as P 1 1 1CP 2CP 3CP nCP nMV 1 (1 r) (1 r) 2 (1 r) 3 (1 r) n (1 r) n P r P 1 r Macaulay Duration Modified Macaulay Duration Duration (1 r) Drake Modified Duration Drake University Fin 129 Using Macaulay Duration P r .0001 D MAC 8.69925 0.000776 P (1 r) 1.12 Modified Macaulay Duration Duration (1 r) P r D MAC D MAC r D MODIFIEDr P (1 r) (1 r) 8.69925 (.0001) 0.000776 1.12 Drake Drake University Duration Fin 129 Keeping other factors constant the duration of a bond will: Increase with the maturity of the bond Decrease with the coupon rate of the bond Will decrease if the interest rate is floating making the bond less sensitive to interest rate changes Decrease if the bond is callable, as interest rates decrease (increasing the likelihood of call) duration increases Drake Duration and Convexity Drake University Fin 129 Using duration to estimate the price change implies that the change in price is the same size regardless of whether the price increased or decreased. The price yield relationship shows that this is not true. Drake Duration and Convexity Drake University Fin 129 3000 2500 2000 Bond Value 1500 1000 500 0 0 0.05 0.1 0.15 0.2 Interest Rate Drake Basic Duration Gap Drake University Fin 129 Duration Gap Drake Basic DGAP Conintued Drake University Fin 129 $ Weighted Duration N DA w i Da i of Asset Portfolio i 1 Asset i where w i Market Value of All Assets Da i Macaulay Duration of asset i $ Weighted Duration N DL w jDl j of Liability Portfolio j1 Asset j where w j Market Value of All Liabilitie s Dl j Macaulay Duration of Liability j Drake Basic DGAP Drake University Fin 129 If the Basic DGAP is + If Rates h in the value of assets > in value of liab If Rate i in the value of assets > in value of liab Drake Basic DGAP Drake University Fin 129 If the Basic DGAP is (-) If Rates h in the value of assets < in value of liab If Rate i in the value of assets < in value of liab Drake Basic DGAP Drake University Fin 129 Does that imply that if DA = DL the financial institution has hedged its interest rte risk? No, because the $ amount of assets > $ amount of liabilities otherwise the institution would be insolvent. Drake DGAP Drake University Fin 129 Let MVL = market value of liabilities and MVA = market value of assets Then to immunize the balance sheet we can use the following identity: Drake DGAP and equity Drake University Fin 129 Let DMVE = DMVA – DMVL We can find DMVA & DMVL using duration From our definition of duration: Δi ΔP D P Applying the formula (1 i) Drake Drake University Fin 129 ΔMVE ΔMVA - ΔMVL Δy Δy -DA MVA - - DL MVL 1 y 1 y Δy -(DA)MVA - (DL)MVL 1 y MVL Δy - (DA) - (DL) 1 y MVA MVA Δy ΔMVE -DGAP MVA 1 y Drake DGAP Analysis Drake University Fin 129 If DGAP is (+) An in rates will cause MVE to An in rates will cause MVE to If DGAP is (-) An in rates will cause MVE to An in rates will cause MVE to The closer DGAP is to zero the smaller the potential change in the market value of equity. Drake Weaknesses of DGAP Drake University Fin 129 It is difficult to calculate duration accurately (especially accounting for options) Each CF needs to be discounted at a distinct rate can use the forward rates from treasury spot curve Must continually monitor and adjust duration It is difficult to measure duration for non interest earning assets. Drake More General Problems Drake University Fin 129 Interest rate forecasts are often wrong To be effective management must beat the ability of the market to forecast rates Varying GAP and DGAP can come at the expense of yield Offer a range of products, customers may not prefer the ones that help GAP or DGAP – Drake Duration in Practice Drake University Fin 129 Impact of convexity Shape of the yield curve Default Risk Floating Rate Instruments Demand Deposits Mortgages Off Balance Sheet items Drake Convexity Revisited Drake University Fin 129 The more convexity the asset or portfolio has, the more protection against rate increases and the greater the possible gain for interest rate falls. The greater the convexity the greater the error possible if simple duration is calculated. All fixed income securities have convexity The larger the change in rates, the larger the impact of convexity Drake Flat Term Structure Drake University Fin 129 Our definition of duration assumes a flat term structure and that the all shirts in the yield curve are parallel. Discounting using the spot yield curve will provide a slightly different measure of inflation. Drake Default Risk Drake University Fin 129 Our measures assume that the risk of default is zero. Duration can be recalculated by replacing each cash flow by the expected cash flow which includes the probability that the cash flow will be received. Drake Floating Rates Drake University Fin 129 If an asset or liability carries a floating interest rate it readjusts its payments so the future cash flows are not known. Duration is generally viewed as being the time until the next resetting of the interest rate. Drake Demand Deposits Drake University Fin 129 Deposits have an open ended maturity. You need to define the maturity to define duration. Method 1 Look at turnover of deposits (or run). If deposits turn over 5 times a year then they have an average maturity of 73 days (365/5). Method 2 Think of them as a puttable bond with a duration of 0 Method 3 Look at the % change in demand deposits for a given level of interest rate changes. Simulation Drake Mortgages Drake University Fin 129 Mortgages and mortgage backed securities have prepayment risk associated with them. Therefore we need to model the prepayment behavior of the mortgage to understand the cash flow. Drake Off Balance Sheet Items Drake University Fin 129 The value of derivative products also are impacted by duration changes. They should be included in any portfolio duration estimate or GAP analysis.