THE IMPACT OF THE CAPITAL MARKET
CHAPTER 7 MODELS OF BANKING BEHAVIOUR
7.5 ASSET MANAGEMENT
The analysis of Section 7.4 also helps us in arriving at some general principles on how a bank manages its assets. We can pose two questions. First, how does a bank allocate its assets between high-risk, high-return loans and low-risk, low-return assets?
Second, how does a bank allocate its assets between a risk-free asset and risky assets?
This question can be answered by appealing to portfolio theory and the Markowitz separation theorem. Portfolio theory tells us that we can separate the asset allocation decision into two stages. The ¢rst stage involves the construction of a composite
ASSET MANAGEMENT 97
BOX 7.3
The conditions for the existence of intermediation Let the expected utility function of the bank be described by:
EfUðÞg ¼EðÞ 122 ð7:3:1Þ
where
EðÞ ¼EðrLÞLþrMMEðrDÞD ð7:3:2Þ andEðrLÞ ¼EðrLþ"LÞ;"LNð0; 2LÞ, EðrDÞ ¼EðrDþ"DÞ,"DNð0; 2DÞand is the coefficient of risk aversion.
Substituting (7.3.2) into (7.3.1) and noting that:
CovðrL;rDÞ ¼rLrDrLrD
EfUðÞg ¼EðrLÞLþrMMEðrDÞD12f2rLL2þ2rDD22rLrDrLrDðLÞðDÞ ð7:3:3Þ whererLrDis the correlation coefficient between the stochastic yield on loans and deposits. Substituting forMfrom the balance sheet of the bank, the first- order conditions for utility maximization are:
@EfUðÞg
@L ¼EðrLÞ rM12½2L2rL2rLrDrLrDD ¼0 ð7:3:4Þ
@EfUðÞg
@D ¼rMEðrDÞ 12½2D2rD2rLrDrLrDL ¼0 ð7:3:5Þ EðrLÞ rM¼rLðrLLrLrDrDDÞ ð7:3:6Þ EðrDÞ rM¼ rDðrDDrLrDrLLÞ ð7:3:7Þ If the yields of loans and deposits were independent EðrL;rDÞ ¼0, then the correlation coefficient between the yields on loans and deposits is zero.
We can see that (7.3.6) is positive and (7.3.7) is negative. You can also see that intermediation is impossible if the correlation is unity – the bracketed part of (7.3.6) and (7.3.7) cannot both be positive. Notice that a negative correlation enables intermediation to take place and creates the condition for a risk-averse bank to conduct asset transformation. However, a zero or negative correlation is too restrictive a condition for the existence of inter- mediation or a risk markup on the risk-free rate. If the correlation between the loan rate and deposit rate was positive (as is likely in reality), by subtracting (7.3.7) from (7.3.6) we can see that the condition for a positive spread (margin of intermediation) of the correlation coefficient can be positive but small:
rLrD< ð2rLþ2rDÞ rLrDðDþLÞ
asset made of an optimal combination of risky assets. The second stage involves a comparison between the composite risky asset and the risk-free asset.
The optimal combination of risky assets is a unique combination that cannot be improved on, either in terms of higher return or lower risk. Consider two assetsA andBwith two di¡erent risk return characteristics as shown in Figure 7.3. On the vertical axis we have expected return and on the horizontal we have a measure of risk. AssetAhas a low-risk, low-return characteristic. AssetBhas a high-risk, high- return characteristic.
It would be di⁄cult to choose betweenAandBwithout knowing something about the risk preferences of the bank manager. However, we know that an asset such asCwould be preferable to bothAorB, because it has a higher return thanA for the same level of risk and the same return asBbut for lower risk. The question is: Can we combineAandBin such a way as to generate a positionCthat would be superior to both? By appealing to portfolio theory we can construct a superior position toAandBbased on the covariance of the relative stochastic returns. Box 7.4 shows how this is done. Figure 7.4 shows the e⁄cient frontier that describes the risk-minimizing combination of assets for varying coordinates of expected return and risk measured by the variance.
The composite asset is made up of the optimal combination of loans that minimizes the risk associated with the stochastic returns. The expected return from the total portfolio will include the allocation of assets between the risk-free asset
ASSET MANAGEMENT 99
FIGURE 7.3
Optimal combination of assets Expected
return
Risk A
C B
BOX 7.4
A primer on portfolio theory
LetRAandRBbe the rates of return on assetsAandB, respectively, and2Aand 2Bbe their respective variances. The return on a portfolio that holds a propor- tionofAandð1ÞofB. The return on the portfolio is given by:
R¼RAþ ð1ÞRB ð7:4:1Þ The variance ofRis:
2R¼VarðREðRÞÞ
) 2VarðRAÞ þ ð1Þ2VarðRBÞ þ2ð1ÞCovðRA;RBÞ
) 22Aþ ð1Þ22Bþ2ð1ÞA;BAB ð7:4:2Þ Recall that:
A;B¼CovðRA;RBÞ AB
Minimizing (7.4.2) with respect to:
@2R
@ ¼22A2ð1Þ2Bþ2A;BAB4A;BAB¼0 ) 2A ð1Þ2BþA;BAB2A;BAB¼0
) ð2Aþ2BÞ 2B2A;BAB¼ A;BAB
) ð2Aþ2B2A;BABÞ ¼2BA;BAB
Collecting terms and solving forgives:
¼ BðBA;BAÞ
½2Aþ2B2A;BAB
The proportioncan be chosen to minimize the risk of the total portfolio. The choice of the optimal value ofwill depend on the correlation between asset Aand assetB. LetA;B¼ 1. Then:
¼ B
AþB
You should be able to confirm that substituting this value into (7.4.2)
ASSET MANAGEMENT 101
FIGURE 7.4 Efficient frontier
produces a value for the variance of zero. The figure below shows the three cases of the correlation coefficient being1, 1 and in-between.
Expected return
A
B ρA,B = -1
ρA,B = 1
-1 < ρA,B < 1
Risk
Risk Expected
Return
High-risk assets
Low-risk assets
and the risky composite asset. The expected return of the total portfolio is:
EðRRÞ ¼~ !EðRÞ þ ð1!ÞrT ð7:9Þ where!is the proportion of the bank’s assets that are risky, andrTis the risk-free asset. Given that the returns on the risk-free asset are deterministic, then the variance of the returns on total portfolio is:
2RR~ ¼!22R ð7:10Þ Equation (7.9) can also be expressed as:
EðRRÞ ¼~
EðRÞ rT
!
!2þrT ð7:11Þ Substituting (7.10) into (7.11), we have:
EðRRÞ ¼~
EðRÞ rT
!2R
2RR~ þrT
) 2RR~ þrT
ð7:12Þ Equation (7.12) de¢nes the opportunity locus giving the tradeo¡ between expected return and risk for the portfolio as shown in Figure 7.6. To make the choice of asset allocation between the risk-free asset and the composite risky asset (the proportion
!) we need to know the risk preferences of the bank. This is shown in Figure 7.5 by the utility function that is tangential to the opportunity locus.
FIGURE 7.5 Portfolio allocation
Composite risky asset C
Opportunity locus EðRRÞ~
rT
2RR~ UðRR; ~ 2RR~Þ
Now let us consider how the allocation alters when the composite asset becomes more risky.
Figure 7.6 shows the opportunity locus shifting down fromACtoAC0. We can think of the ray from the originAC andAC0as unit lengths and any point on the line de¢nes a proportion (0< ! <1). The initial allocation to risky assets is given by the ratioAB=ACand the proportion of assets held in the risk-free asset is shown byBC=AC. The increase in the riskiness of the portfolio is shown by a shift in the opportunity set toAC0. PointC0is to the right ofCand shows the same expected return asCbut with a higher level of risk. The new equilibrium is shown asB0. The share of the composite risky asset will have fallen toAB0=AC0and the share of the risk-free asset has increased toB0C0=AC0. We have shown that an increase in the riskiness of the portfolio drives the bank to reduce its exposure in the composite risky asset (loans) and hold a higher share of the risk-free asset. Notice that from equation (7.14) a fall in the yield of loans as a whole has qualitatively the same e¡ect as a rise in the riskiness of loans.
Consider what happens if the risk-free rate rises (Figure 7.7) but there is no change in the overall composite loan rate. The opportunity locus shifts fromACto A0C. The proportion of the portfolio held in the risk-free asset increases from BC=AC toB0C=A0C. This is clearly sensible if it was at all realistic. However, in reality a rise in the risk-free rate of interest will raise interest rates generally and the
ASSET MANAGEMENT 103
FIGURE 7.6
An increase in the riskiness of loans
C C’
B
B’
U0
U1
EðRRÞ~
2RR~ A¼rT
allocation will depend on the relative rate of interest or the margin between the risk- free rate and the composite risky rate. A general rise in interest rates will have the op- portunity locus shift up to the left in parallel toAC.
The portfolio model yields the following conclusions:
1. A bank tries to diversify its loan assets between high-risk, high-return lending (new ventures, SMEs,2 etc.) and low-risk, low-return lending (blue-chip companies, secured lending to households, etc.).
2. The bank holds a proportion of its assets in risk-free liquid assets.
3. An increase in riskiness on the asset portfolio (ceteris paribus) will see banks moving into the risk-free liquid asset.
4. An increase in the return on loans (ceteris paribus) will see banks moving away from the risk-free asset.
5. An increase in the risk-free rate of return (ceteris paribus) will result in banks increasing their holding of the risk-free asset.
While the conclusions of the portfolio model seem like common sense in the practice of banking, there are a number of de¢ciencies in the model that need to be borne in mind. The results are sensitive to the speci¢cation of the utility
FIGURE 7.7
An increase in the risk-free rate of interest
C C’
A
B B’
U0
U1
A’
EðRRÞ~
2RR~
2SME¼Small- to Medium-sized Enterprises.
function. We have implicitly assumed a quadratic or negative exponential utility function in terms of the return on the portfolio. This means that the utility function can be expressed in terms of the ¢rst two moments (mean and variance) of the distribution of the returns on assets. This also implies that the distribution of returns is normal. But, in reality, returns on a loan portfolio are not normally distributed. The standard loan contract calls for the repayment of principal and interest. The interest return is not normally distributed. Borrowers may be delinquent and pay less or even default, but they do not pay more than what is - speci¢ed in the contract. So, there is only downside risk and no equivalent compensating upside risk.
The portfolio model cannot accommodate the customer^loan relationship. The bank will often lend without collateral with the aim of building up a long-term relationship with a customer. A bank may be willing to provide an unsecured loan to an impoverished student on the grounds that it would be useful to gain the loyalty of the student for the future when he or she has graduated and is handling a company account. The customer^loan relationship ensures that lending and overdraft facilities exist in bad times as well as good.
The portfolio model also does not incorporate the intricate bilateral deter- mination of lending terms. The reason being that individual loans have di¡erent characteristics of interest to a bank other than return and risk. Lending characteristics would include maturity, collateral, and credit rating. Such characteristics are di⁄cult to capture in the simple portfolio model.