The Repricing Gap Model：重新定价缺口模型.pdf

1 the repricing gap model 1.1introduction among the models for measuring and managing interest rate risk, the repricing gap is certainly the best known and most widely used. it is based on a relatively simple and intuitive consideration: a banks exposure to interest rate risk derives from the fact that interest-earning assets and interest-bearing liabilities show differing sensitivities to changes in market rates. the repricing gap model can be considered an income-based model, in the sense that the target variable used to calculate the effect of possible changes in market rates is, in fact, an income variable: the net interest income (nii the difference between interest income and interest expenses). for this reason this model falls into the category of “earnings approaches” to measuring interest rate risk. income-based models contrast with equity- based methods, the most common of which is the duration gap model (discussed in the following chapter). these latter models adopt the market value of the banks equity as the target variable of possible immunization policies against interest rate risk. after analyzing the concept of gap, this chapter introduces maturity-adjusted gaps, and explores the distinction between marginal and cumulative gaps, highlighting the difference in meaning and various applications of the two risk measurements. the discussion then turns to the main limitations of the repricing gap model along with some possible solutions. particular attention is given to the standardized gap concept and its applications. 1.2the gap concept the gap is a concise measure of interest risk that links changes in market interest rates to changes in nii. interest rate risk is identifi ed by possible unexpected changes in this variable. the gap (g) over a given time period t (gapping period) is defi ned as the difference between the amount of rate-sensitive assets (sa) and rate-sensitive liabilities (sl): gt= sat slt= ? j sat,j ? j slt,j(1.1) the term “sensitive” in this case indicates assets and liabilities that mature (or are subject to repricing) during period t. so, for example, to calculate the 6-month gap, one must take into account all fi xed-rate assets and liabilities that mature in the next 6 months, as well as the variable-rate assets and liabilities to be repriced in the next 6 months. the gap, then, is a quantity expressed in monetary terms. figure 1.1 provides a graphic representation of this concept. by examining its link to the nii, we can fully grasp the usefulness of the gap concept. to do so, consider that nii is the difference between interest income (ii) and interest expenses (ie). these, in turn, can be computed as the product of total fi nancial assets (fa) and the average interest rate on assets (ra ) and total fi nancial liabilities (fl) and average interest rate on liabilities (rl ) respectively. using nsa and nsl as fi nancial assets and copyrighted material 10risk management and shareholders value in banking sensitive liabilities (slt) not sensitive liabilities (nslt) sensitive assets (sat) not sensitive assets (nsat) gapt(0) figure 1the repricing gap concept liabilities which are not sensitive to interest rate fl uctuations, and omitting t (which is considered given) for brevitys sake, we can represent the nii as follows: nii = ii ie = rafa rlfl = ra(sa + nsa) rl(sl + nsl)(1.2) from which: ?nii = ?rasa ?rlsl(1.3) equation (1.3) is based on the simple consideration that changes in market interest rates affect only rate-sensitive assets and liabilities. if, lastly, we assume that the change in rates is the same both for interest income and for interest expenses ?ra= ?rl= ?r(1.4) the result is: ?nii = ?r (sa sl) = ?r ? j saj ? j slj = ?r g (1.5) equation (1.5) shows that the change in nii is a function of the gap and interest rate change. in other words, the gap represents the variable that links changes in nii to changes in market interest rates. more specifi cally, (1.5) shows that a rise in interest rates triggers an increase in the nii if the gap is positive. this is due to the fact that the quantity of rate-sensitive assets which will be renegotiated, resulting in an increase in interest income, exceeds rate-sensitive liabilities. consequently, interest income grows the repricing gap model11 faster than interest expenses, resulting in an increase of nii. vice versa, if the gap is negative, a rise in interest rates leads to a lower nii. table 1.1 reports the possible combinations of the effects of interest rate changes on a banks nii, depending on whether the gap is positive or negative and the direction of the interest rate change. table 1.1 gaps, rate changes, and effects on nii r gap g 0 positive net reinvestment g 0 0 higher rates nii 0 the table also helps us understand the guidelines that may be inferred from gap analysis. when market rates are expected to increase, it is in the banks best interest to reduce the value of a possible negative gap or increase the size of a possible positive gap and vice versa. assuming that one-year rate-sensitive assets and liabilities are 50 and 70 million euros respectively, and that the bank expects a rise in interest rates over the coming year of 50 basis points (0.5%),1the expected change in the nii would then be: e(?nii) = g e(?r) = (20,000,000) (+0.5%) = 100,000(1.6) in a similar situation, the bank would be well-advised to cut back on its rate-sensitive assets, or as an alternative, add to its rate-sensitive liabilities. on the other hand, where there are no expectations about the future evolution of market rates, an immunization policy for safeguarding nii should be based on zero gap. some very common indicators in interest rate risk management can be derived from the gap concept. the fi rst is obtained by comparing the gap to the banks net worth. this allows one to ascertain the impact that a change in market interest rates would have on 1expectations on the evolution of interest rates must be mapped out by bank management, which has various tools at its disposal in order to do so. the simplest one is the forward yield curve presented in appendix 1b. 12risk management and shareholders value in banking the nii/net worth ratio. this frequently-used ratio is an indicator of return on asset and liability management (alm) that is, traditional credit intermediation: ? ? nii nw ? = g nw ?r(1.7) applying (1.7) to a bank with a positive gap of 800 million euros and net worth of 400 million euros, for example, would give the following: ? ? nii nw ? = 800 400 ?r = 2 ?r if market interest rates drop by 50 basis points (0.5%), the bank would suffer a reduction in its earnings from alm of 1%. in the same way, drawing a comparison between the gap and the total interest-earning assets (iea), we come up with a measure of rate sensitivity of another profi t ratio com- monly used in bank management: the ratio of nii to interest-earning assets. in analytical terms: ? ? nii iea ? = g iea ?r(1.8) a third indicator often used to make comparisons over time (evolution of a banks exposure to interest rate risk) and in space (with respect to other banks) is the ratio of rate-sensitive assets to rate-sensitive liabilities, which is also called the gap ratio. analytically: gapratio = sa sl (1.9) unlike the absolute gap, which is expressed in currency units, the gap ratio has the advantage of being unaffected by the size of the bank. this makes it particularly suitable as an indicator to compare different sized banks. 1.3the maturity-adjusted gap the discussion above is based on the simple assumption that any changes in market rates translate into changes in interest on rate-sensitive assets and liabilities instantaneously, that is, affecting the entire gapping period. in fact, only in this way does the change in the annual nii correspond exactly to the product of the gap and the change in market rates. in the case of the bank summarized in table 1.2, for example, the “basic” gap computed as in (1.1), relative to a t of one year, appears to be zero (the sum of rate-sensitive assets, 500 million euros, looks identical to the total of rate-sensitive liabilities). however, over the following 12 months rate-sensitive assets will mature or be repriced at intervals which are not identical to rate-sensitive liabilities. this can give rise to interest rate risk that a rudimentary version of the repricing gap may not be able to identify. one way of considering the problem (another way is described in the next section) hinges on the maturity-adjusted gap. this concept is based on the observation that when there is a change in the interest rate associated with rate-sensitive assets and liabilities, the repricing gap model13 table 1.2 a simplifi ed balance sheet assetsmliabilitiesm 1-month interest-earning interbank deposits 200 1-month interest-bearing interbank deposits 60 3m govt securities30 variable-rate cds (next repricing in 3 months) 200 5yr variable-rate securities (next repricing in 6 months) 120 variable-rate bonds (next repricing in 6 months) 80 5m consumer credit 80 1yr fi xed-rate cds160 20yr variable-rate mortgages (next repricing in 1 year) 70 5yr fi xed-rate bonds180 5yr treasury bonds 170 10yr fi xed-rate bonds120 10yr fi xed-rate mortgages200 20yr subordinated securities80 30yr treasury bonds130 equity120 total1000 totals1000 this change is only felt from the date of maturity/repricing of each instrument to the end of the gapping period (usually a year). for example, in the case of the fi rst item in table 1.2, (interbank deposits with one-month maturity), the new rate would become effective only after 30 days (that is, at the point in time indicated by p in figure 1.2) and would continue to impact the banks profi t and loss account for only 11 months of the following year. today1 yearp =1/12 fixed rate new rate conditions 11 months gapping period: 12 months time figure 1.2an example of repricing without immediate effect more generally, in the case of any rate-sensitive asset j that yields an interest rate rj, the www. interest income accrued in the following year would be: iij= sajrjpj+ saj(rj+ ?rj) (1 pj)(1.10) where pjindicates the period, expressed as a fraction of the year, from today until the maturity or repricing date of the jthasset. the interest income associated with a generic 14risk management and shareholders value in banking rate-sensitive asset is therefore divided into two components: (i) a known component, represented by the fi rst addendum of (1.10), and (ii) an unknown component, linked to future conditions of interest rates, represented by the second addendum of (1.10). thus, the change in interest income is determined exclusively by the second component: ?iij= saj(1 pj) ?rj(1.11) if we wish to express the overall change of interest income associated with all the n rate-sensitive assets of the bank, we get: ?ii = n ? j=1 saj?rj(1 pj)(1.12) similarly, the change in interest expenses generated by the kthrate-sensitive liability can be expressed as follows: ?iek= slk?rk(1 pk)(1.13) furthermore, the overall change of interest expenses associated with all the m rate- sensitive liabilities of the bank comes out as: ?ie = m ? k=1 slk?rk(1 rk)(1.14) assuming a uniform change in the interest rates of assets and liabilities (?rj=?rk=?r j, k), the estimated change in the banks nii simplifi es to: ?nii = ?ii ?ie = ? j saj(1 pj) ? j slj(1 pj) ?r gma?i (1.15) where gmastands for the maturity-adjusted gap, i.e. the difference between rate-sensitive assets and liabilities, each weighted for the time period from the date of maturity or repricing to the end of the gapping period, here set at one year.2 using data from table 1.2, and keeping the gapping period at one-year, we have: ?ii = ? j irj= ? j saj(1 pj) ?r = 312.5 ?r ?ie = ? k ipk= ? k slk(1 pk) ?r = 245 ?r and fi nally ?nii = gma?r = (312.5 245) ?r = 67.5 ?r 2as saita (2007) points out, by using (1.15), on the basis of the maximum possible interest rate variation (?iwc, worst case) it is also possible to calculate a measure of “nii at risk”, i.e. the maximum possible decrease of the nii: imar = gma?iwc. this is somewhat similar to “earnings at risk” (see chapter 23), although the latter refers to overall profi ts, not just to net interest income the repricing gap model15 therefore, where the “basic” gap is seemingly zero, the maturity-adjusted gap is nearly 70 million euros. a drop in the market rates of 1% would therefore cause the bank to earn 675,000 euros less. the reason for this is that in the following 12 months more assets are repriced earlier than liabilities. 1.4marginal and cumulative gaps to take into account the actual maturity profi le of assets and liabilities within the gapping period, an alternative to the maturity-adjusted gap is the one based on marginal and cumulative gaps. it is important to note that there is no such thing as an “absolute” gap. instead, different gaps exist for different gapping periods. in this sense, then, we can refer to a 1-month gap, a 3-month gap, a 6-month gap, a 1-year gap and so on.3 an accurate interpretation of a banks exposure to market rate changes therefore requires us to analyze several gaps relative to various maturities. in doing so, a distinction must be drawn between: cumulative gaps (gt1,gt2,gt3 ), defi ned as the difference between assets and liabilities that call for renegotiation of interest rates by a set future date (t1,t2 t1,t3 t2, etc.) period or marginal gaps (g? t1,g ? t2,g ? t3 ), defi ned as the difference between assets and liabilities that renegotiate rates in a specifi c period of time in the future (e.g. from 0 to t1, from t1 to t2, etc.) note that the cumulative gap relating to a given time period t is nothing more than the sum of all marginal gaps at t and previous time periods. consequently, marginal gaps can also be calculated as the difference between adjacent cumulative gaps. for example: gt2= g? t1+ g ? t2 g? t2= gt2 gt1 table 1.3 provides fi gures for marginal and cumulative gaps computed from data in table 1.2. note that, setting the gapping period to the fi nal maturity date of all assets and liabilities in the balance sheet (30 years), the cumulative gap ends up coinciding with the value of the banks equity (that is, the difference between total assets and liabilities). as we have seen in the previous section, the one-year cumulative gap suggests that the bank is fully covered from interest risk, that is, the nii is not sensitive to changes in market rates. we know, however, that this is misleading. in fact, the marginal gaps reported in table 1.3 indicate that the bank holds a long position (assets exceed liabilities) in the fi rst month and in the period from 3 to 6 months, which is set off by a short position from 1 to 3 months and from 6 to 12 months. 3 for a bank whose non-fi nancial assets (e.g. buildings and real estate assets) are covered precisely by its equity (so there is a perfect balance between interest-earning assets and interest-bearing liabilities), the gap calculated as the difference between all rate-sensitive assets and liabilities on an infi nite time period (t = ) would realistically be zero. extending this horizon indefi nitely, in fact, all fi nancial assets and liabilities prove sensitive to interest rate changes. therefore, if fi nancial assets and liabilities coincide, the gap calculated as the difference between the two is zero. 16risk management and shareholders value in banking table 1.3 marginal and cumulative gaps periodrate-sensitive assets rate-sensitive liabilities marginal gap gt cumulative gapgt 01 months20060140140 13 months3020017030 36 months2008012090 612 months70160900 15 years1701801010 510 years2001208070 1030 years1308050120 total1000880 we see now how marginal gaps can be used to estimate the real exposure of the bank to future interest rate changes. to do so, for each time period indicated in table 1.3 we calculate an average maturity (tj*), which is simply the date halfway between the start date (tj 1) and the end date (tj) of the period: t j tj+ tj1 2 for example, for the second marginal gap (running from 1 to 3 months) the value of t*2 equals 2 months, or 2/12. using tjto estimate the repricing date for all rate-sensitive assets and liabilities that fall in the marginal gap gtj , it is now possible to write a simplifi ed version of (1.15) which does not require the knowledge of the actual repricing date of each rate-sensitive asset or liability, but only information on the value of various marginal gaps: ?nii = ?r ? j|tj?1 g? tj(1 t j) = ?r g w 1 (1.16) gw 1 represents the one-year weighted cumulative gap. this is an indicator (also called nii duration) of the sensitivity of the nii to changes in market rates, computed as the sum of marginal gaps, each one weighted by the average time left until the end of the gapping period (one year). for the portfolio in table 1.2, gw 1 is 45 million euros (see table 1.4 for details on the calculation). as we can see, this number is different (and less precise) from the maturity- adjusted gap obtained in the preceding section (67.5). however, its calculation did not require to specify the repricing dates of the banks single assets and liabilities (which can actually be much more numerous than those in the simplifi ed example in table 1.2). what is more, the “signal” this indicator transmits is similar to the one given by the maturity- adjusted gap. when rates fall by one percentage point, the bank risks a reduction in nii of approximately 450,000 euros. the repricing gap model17 table 1.4 example of a weighted cumulative gap calculation periodg?ttjtj1 tjg?t(1 tj) up to 1 month140 1/12 1/2423/24134.2 up to 3 months170 3/12 2/1210/12141.7 up to 6 months120 6/12 9/2415/2475.0 up to 12 months9019/123/1222.5