Finance 462 Solutions to Problem Set #9. First, to simplify, set the unemployment rate to 5% (.05)

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1 Finance 46 Soutions to Probem Set #9 1) With no fees, we have the foowing demand fooans: Q = UR First, to simpify, set the unempoyment rate to 5% (.05) Q = (.05) = To cacuate the easticity, we first need the derivative with respect to the interest rate. dq d = 64 Next, divide by the quantity of oans and mutipy by the interest rate to get the easticity. dq dr r Q = 64 Q To get tota revenues as a function of L, first sove the demand curve for the interest rate = Q = Q Monthy revenues equa the interest rate charged (divided by tweve) times the quantity of oans issued times $100,000. = 1608Q 13Q 1 Margina revenue is the derivative with respect to L MR = 1608 (13) Q = Q Now, take the cost function

2 .05 TC = 10, ( 100,000) Q = 10,000 Q Margina cost is the derivative with respect to Q MC = 416 Now, to get the optima amount of oans, set MR=MC and sove for Q Q = 416 Q = 46 Now, given L, the interest rate can be found using the demand curve. = Q = (46) =.1194 = 11.94% Therefore,.1194 TR = $ 100, = $45,770 TC = 10, (46) = $9,136 Profits = $16,634 At the profit maximizing point, easticity of demand is dq dr Q.1194 = 64 = If we add the fees, the procedure is the same, by the demand curve becomes: Q = (100) 90.4(.05) = Soving for the interest rate, we get 89 1 = Q = Q Tota Revenues now incude interest income and fee income: $100 + Q = 19Q 13Q 1

3 Tota Costs are unchanged at.05 TC = 10, ( 100,000) Q = 10,000 Q Set margina revenue equa to margina cost as in part (a) and the optimum is 33 oans and an interest rate of 9%. Profits are equa to$43. In part (d), things get interesting. The demand curve doesn t change, but now, Tota revenues become: + Q 1 ( $100) Q = 39Q 13 Because a the fees are being paid up front. The optima interest rate becomes.5%, 74 oans are created, and profits are $63,000! ) Perfecty competitive firms are restricted to charge a price (or, more accuratey, a spread) equa to margina cost. Therefore, in a perfecty competitive market, the entire increase in margina cost wi be refected in interest rates. A monopoist, however, charges a markup above margina cost. This markup is chosen optimay given that the bank wi ose some demand if it raises its price. Therefore, to save some saes, the bank wi increase price (here, the oan rate) by an amount ess than the margina cost increase. 3) Optima decisions are made at the margin for both competitive firms and monopoists. Therefore, a change in a fixed cost (a cost that is independent of saes) wi have no effect on price ony on profits. 4) Given the information on Assets/Liabiities, your baance sheet is as foows (interest paid/received in parentheses): Assets Liabiities Cash: $15,000 Checking: $00,000 Short Term Loans: $160,000 (7%) Savings: $50,000 (%) Government Securities: $100,000 (3%) Equity Capita: $5,000 a) The reserve requirement states that the bank must hod 5% of tota deposits (5% of $50,000 is $1,500) in the form of either cash or deposits at the Federa Reserve. Therefore, your current cash hodings of $15,000 in cash can be broken up into $1,500 in required reserves and $,500 in excess

4 reserves. Your equity is defined as tota assets minus tota iabiities. In this case, $75,000 (Assets) - $50,000 (Liabiities) resuts in $5,000 in equity. This as a percentage of non-cash assets is 9.6% - we above the requirement of 4%. b) Your profit is defined as your revenues from interest on oans/securities, minus your interest paid on deposits. In this case, your profits are (.07)*(160,000) + (.03)*(100,000) (.0)*(50,000) = $11,00+$3,000 - $1,000 = $13,00. Return on Assets expresses this profit as a percentage of tota assets, or (13,00/$75,000)*100 = 4.8%. Return on equity expresses profits as a percentage of equity, or (13,00/5,000)*100 = 5.8%. c) A $10,000 withdrawa woud resut is a deduction of $10,000 from the cash category of assets and an equa deduction of $10,000 deduction from checking undeiabiities (baance sheets must aways baance!). Note that this withdrawa reduces our cash position to $5,000 or % of deposits. Therefore, we must raise cash to get back above the reserve requirement. This can be done by iquidating some assets (seing securities or recaing oans) or by taking out a short-term oan in the fed funds market. d) A $30,000 defaut is a much bigger probem. This deducts $30,000 in oans from the asset side and $30,000 from equity capita on the iabiity side. Now, our equity is -$5,000 (obviousy beow the requirement). This coud ony be soved by bringing in addition capita (i.e., investing more capita in the bank). 5) Given the information on Assets/Liabiities, your baance sheets shoud ook ike the foowing (duration of assets/iabiities in parentheses): Assets Liabiities Cash: $18,000 (0) Checking: $100,000 (0) 1 yr. Loans: $90,000 (1) 1yr. CDs: $50,000 (1) yr. Loans: $80,000 () yr. CDs: $0,000 () Equity Capita: $18,000 a) A $10,000 withdrawa woud resut is a deduction of $10,000 from Cash and Checking with no impact on equity. This woud, however, ower cash reserves beow the requirement and wi need to be rectified) most ikey through a fed funds oan or a repo). b) A $0,000 oad defaut woud deduct $0,000 from yr. Loans and equity capita. With equity capita at -$,000, more woud need to invest in the bank for it to stay in business. c) The duration of iabiities (the weighted average of the duration of each individua iabiity where the weights are equa to each individua iabiity s percentage of tota iabiities). In this case (I dropped the zeroes for

5 simpicity): (18/188)*0 + (90/188)*1 + (80/188)* = Simiay, the duration of assets is (100/170)*0 + (50/170)*1 + (0/170)* =.53. Lasty the duration gap is the difference between the duration of assets and the iabiity to asset ratio weighted duration of iabiities. In this case, 1.34 (170/188)*.53 =.86. Duration represents the sensitivity to interest rates. In this exampe, a 3% rise in the interest rate wi cause a 3*(.86) =.58% drop is equity as a percentage of assets (in this exampe, the duration of assets is bigger than the duration of iabiities. Therefore, when interest rates rise, the vaue of assets fas by more than the vaue of iabiities, resuting in a decrease in net worth.