1). Consider investment/production and consumption of one person in a simple two -period economy without capital markets. Use diagrams to describe investment andconsumption of this rational individual given a subjective rate of time preference. Extendyour analysis of this rational individual by allowing for capital markets to exist.Compare and contrast your results with the ones from the previous discussion.A rational economic agent in a one-person/one-good two-period economy without thepresence of capital marketshas a choice of how he uses his initial endowment income, Yo,and his end of period income, Y1. He can decide how much of the income to consume now,Co, and how much he should invest in production to consume in the future, C1.Theassumptions of this economy are that all actions are undertaken with certainty; forexample, the actions of investment are known. There are also no transaction costs and it isalso assumed that the marginal utility of consumption is positive but decreasing.For the individual to be able to decide whether to consume now or invest to consume in thefuture, he needs to know his subjective trade-offs that must be made between consumptionnow and consumption in the future.This is achieved through analysing his utility functionsand indifference curves.The slope of any point on the indifference curves shows the rate which C0 and C1 can betraded for each other; this is called the marginal rate of substitution (MRS). The MRS can beshown algebraically asThis equation introduces the decision maker’s subjective rate of time preference, . Thesubjective rate of time preference represents the individual’s optimal rate of exchangebetween present and future consumption, it is unique to each person. It shows how much
future consumption must be received, to make investment today worthwhile and still keep thesame total utility.Production opportunities then allow the individual to invest a current unit to be turned intomore than one unit for future consumption. The individual’s schedule of productionopportunities shows the possible combinations of current and future income that theindividual could attain through investment and de-investment. The individual willinvest in allopportunities which yield a higher rate of return than their own subjective rate of timepreference. The Production Possibilities Frontier (PPF) curve aggregates all possibleinvestment opportunities schedules. The slope of a line tangent to the PPF is called themarginal rate of transformation (MRT), which is “the rate at which one dollar of consumptionforgone today is transformed by productive investment into a dollar of consumptiontomorrow.”1Given an individual’s subjective rate of time preference, , the individual’s current andfuture consumption can be derived.1 rd Quoted from page 7, Copeland, T.E. and Weston, J.F., 1992.Financial Theory and Corporate Policy 3 ed.Addison-Wesley.
Z X Fig. 1Figure 1starts with the individual’s initial indifference curve, U1. This initial indifferencecurve crosses the PPF (Curve ZBX) at point A, where the individuals initial and futureincomes are Y0 and Y1 respectively. The individual then compares his rate of subjective timepreference with the marginal rate of return. As the rate of return is greater, the individualstarts to invest as this increases his utility. This process continues until the rate of subjectivetime preference is equal to the rate of return on the last investment, this occurs at point Bwhere the slope is –(1+ri). At point B the individuals indifference curves are tangential to thePFF. The individual has invested Y0-C0 to increase his future expected income by C1-Y1.Themarginal rate of return on his last investment is equal to his subjective rate of timepreference, meaning the individual’s MRS=MRT. The individual’s consumption in thepresent and future is also equal to the production output (P0, P1) as they are the only marketparticipant.
What would happen to this scenario if capital markets were to exist? This would open up theopportunity for intertemporal exchange of consumption bundles between market participants.Individuals can lend and borrow unlimited amounts of bundles at a market determined rate ofinterest, r. Assuming that r is positive, any funds lent in the current period will earn intereston top of the principal paid back at the end of the period. These lending and borrowingopportunities create the capital market line. Any point along the capital market line can beachieved by either lending or borrowing from the individuals initial position. The individualmust repay the principal amount, X0, plus the interest at market rate. This future value of theprinciple and interest combined is denoted, X1, such that the future value can be shownalgebraicially asThe present value of the individuals initial endowment of Y0 and Y1, W0, can be shownalgebraically asAn individuals utility is maximised when their subjective rate of time preference equals themarket rate of interest. The consumption bundle at the point where the individuals subjectiverate of time preference is equal to the market rate of interest is equal to their wealth; and fromthe following equation of the individuals wealth, the equation from the capital market linecan be derivedTherefore, the equation for the capital market line isSubstituting in for W1=W0(1+r) for clarity .
From this equation, it is clear that the slope of the capital market line is –(1+r) and itsintercept is W1.An individual’s consumption and investment strategy with capital markets present is shownin figure 2. Fig. 2Starting with an initial endowment income of Y0 and Y1 at point A, the individual can investto raise their utility. At point D, their indifference curve is tangential to the PPF,meaning theywould have maximised their utility if there was no capital markets present as MRS=MRT.However, the presence of capital markets offers the individual with the opportunity toimprove their utility more. Currently the slope at point D, the rate of return on investment, isgreater than the borrowing rate, the slope of the capital market line, which means that futherinvestment returns more than it costs to borrow the funds. In this situation, the individual willcontinue to invest until the capital market line is tangential to the PPF, which occurs at point
B. Now the individual can reach any position along the capital market line which is tangentialto their own indifference curve. The individual decides to borrow more in the current periodand so his indifference curve shifts out to be tangential to the capital market line at point C,which is where his utility is maximised.In this scenario, production output is (P0, P1). The individuals wealth has increased from W0without capital markets, to W0* with capital markets. His utility has also increased because ofthe existence of capital markets as they started with a utility of U1 which was improved byinvesting to achieve U2, and then they borrowed through the capital markets to achieve autility of U3.The presence of capital markets will always allow an individual to increase their wealth andattain a higher utility than without the markets, except in the occasion where ; which means that the PPF, capital market line and the individual’s indifferencecurve are all tangential at the same point.
2). Graphically demonstrate the Fisher separation theoremfor the case where anindividual ends up lending in financial markets. Explain in detail and label the followingpoints on the graph: initial wealth, W 0; optimal production/investment (P 0; P1); optimalconsumption (C 0;C1); present value of final wealth, W 0*This Fisher separation theorem is the occurrence of a two-step decision process when theindividual is faced with production opportunities and capital market exchange opportunities.The first step is to choose the optimal production level, which is achieved by undertakinginvestments until the objective market interest rate (slope of the capital market line) is equalto the marginal rate of return on investment (slope of the PPF).The second step is to choose the optimal consumption bundle by moving along the capitalmarket line. The individual either lends or borrows until their subjective rate of timepreference (slope of the individual’s indifference curves) is equal to the market rate ofreturn.The Fisher separation theorem is illustrated graphically in figure 3.Fig. 3
The initial endowment of corresponds with utility curve U1, with initial wealth W0,which intersects the PPF at point A. In accordance with the Fisher separation theorem; thecapital market line will shift up and to the right because the rate of return on investment isgreater than the borrowing rate, which means that futher investment returns more than it coststo borrow the funds. Therefore, the capital market line will shift from its initial position of W0to W0*, which is tangential to the PPF at point B. At point B, the , andthisis the optimal production point with P0 and P1 being the current and future productionvalues.The second step of the Fisher separation theorem is choosing the optimal consumptionbundle.The individual begins investing their current consumption, which raises theirindifference curve from U1 to U2. At U2, the indifference curve is tangential to the PPF atpoint C, and the individual has invested of their income. The individual then utilisesthe capital markets and begin to lend more of their current income in the markets, whichshifts their indifference curve from U2to U3, which is tangential to the capital market line atpoint D. This means that they have lent on the capital markets.As a result of these transactions, the present value of their wealth increased from W0 to W0*,which is the present value of their final wealth. Their current consumption has decreasedfrom Y0 to C0*, and their future consumption has increased from Y1 to C1*.
3. If an individual has logarithmic utility and (Markowitz and cost of gamble) riskpremiafor the following gambles: (1) No initial wealth and gamble .8 prob. to obtain 10and .2 prob. to obtain 40; (2) Current wealth 20 and gamble .1 prob. to obtain additional20 and .9 prob. to obtain additional 90; (3) Current wealth 20000 and gamble .8 prob . tolose 4000 and .2 prob. to lose 10000. Compare the two risk premia for each example.i).Current wealth=080% chance to gain £1020% chance to gain £40E(W)= 0.8*10+0.2*40= 16E[U(W)]=0.8*ln(10)+0.2*ln(40)= 1.84+0.74=2.58Wc= exp(2.58)= 13.20W=E(W) because the expected change in wealth is zero.Markowitz risk premium= E(W)-Wc=16-13.2=£2.80Cost of gamble= W-Wc= 16-13.2=£2.80Therefore there is no difference between the risk premium and the cost of gamble. As bothoutcomes of the gamble are positive, this individual would be willing to pay up to £2.80 inorder to take this gamble.ii).Current wealth= £2010% chance to gain +£2090% chance to gain +£90E(W)=0.1*20+0.9*90= 2+81= 83E[U(W)]=0.1*[ln(20+20)]+0.9*[ln(20+90)]=4.60Markowitz risk premium= 83-99.48= -£16.48This individual would need to be paid £16.48 for him to not take the gamble.Cost of gamble= 20 – 99.48= -£79.48iii).Current wealth= £20,00080% chance to lose £4,00020% chance to lose £10,000E(W)= 0.8*16000+0.2*10000=14800.E[U(W)]= 0.8*(ln16000)+0.2*(ln10000)=7.74+1.84=9.58Markowitz risk premium= 14,800-14,472.42= £327.58.
This means that the individual would be willing to pay insurance against the gamble up to thevalue of £327.58.Cost of gamble= 20,000-14,472.42= £5,527.58.
4. Show three alternative types of behaviour towards risk, by considering the shape ofthe utility function as a function of wealth. What is a risk premium? How can beapproximated? (Please give no proofs.) Use indifference curves map of an individual inmean (expected return) - standard deviation (risk) space to show risk aversion and riskloving.Assuming that marginal utility of wealth is positive, i.e. more wealth is better; there are threetypes of behaviour a person can have towards taking risks.The first type of behaviour is of risk lovers. This is shown by a convexed utility function, asthe individual’s utility of expected wealth will be lower than their expected utility of wealth.These types of people love to take risks.The second type of behaviour is of risk neutrality. This is characterised by a linear utilityfunction.This type of people are indifferent to taking risks.
The third type of behaviour is of a risk averter. This is shown by a concaved utility function,as the individual’s utility of expected wealth will be greater than their expected utility ofwealth.These types of people are against risk taking behaviour.A risk premium is the maximum amount of wealth that an individual would be willing to giveup in order to avoid a gamble. It is calculated by taking the difference between anindividual’s expected wealth when they take the gamble and the certainty equivalent wealth,which is the level of wealth that an individual would accept with certainty if the gamble wasto be removed. Shown in mathematical form this looks likeThe graph below shows an individual’s indifference curves in risk-return space when theyhave the behaviour of a risk averter.
U4 U3 U2 U1The indifference curves show all an individual’s combinations of risk and return that yield thesame expected utility of wealth.U1<U2<U3<U4 as the person is receiving the same amount ofreturn for less risk when comparing the indifference curves by expected return, i.e. increasingutility.The following graph now shows the indifference curves of an individual in risk-return spacewhen they are a risk lover.
U1<U2<U3<U4 as the person is receiving the same amount of return for more risk whencomparing the indifference curves by expected return, i.e. increasing utility.5. Given the exponential utility function U(W) = -e-W (i) Sketch a graph of the function. (ii) Does the function exhibit positive marginal utility and risk aversion? (iii) Does the function have decreasing absolute risk aversion?(iv) Does the function have constant relative risk aversion?i).
ii). Marginal utility can be derived by taking the first derivative of the utility function withrespect to Wealth.Therefore, the function exhibits positive marginal utility. This is confirmed by the graphabove as the slope of any line which is tangent to the utility function at any point is alwayspositive.To calculate whether the graph shows risk aversion, marginal utility must be differentiated tofind the rate of change in marginal utility.The utility function therefore exhibits risk aversion, which is supported by analysing thegraph above which is concaved.iii). The definition of Absolute Risk Aversion (ARA) is given by the equationARA measures risk aversion for a given level of wealth. Substituting in the values from theprevious part of the question can produce the ARA.Therefore, this function does not have a decreasing ARA; instead it has a constant ARA.iv). The equation for Relative Risk Aversion (RRA) is given by:Then substituting in the values from the previous question the RRA can be calculatedTherefore, this utility function does not exhibit a constant relative risk aversion; insteadchanges depending on wealth.
6. A businessman faces a 10 % chance of having a fire that will reduce his netwealth to £ 1, a 10 % chance that fire will reduce it to £ 100,000, and an 80 % chancethat nothing detrimental will happen, so that his business will retain its worth (£200,000). What is the maximum amount he will pay for insurance if he has logarithmicutility function U(W) = ln(W)? Explain your answer in detail.Current wealth= £200,000Gamble:10% chance to be worth £110% chance to be worth £100,00080% chance to be worth £200,000E(W)= (0.1*1)+(0.1*100,000)+(0.8*200,000)= £170,000.10E[U(W)]= (0.1*ln)+(0.1*ln[100,000])+(0.8*ln[200,000])=10.91615066Wc = e10.91615066 = 55,058.45The maximum risk premium that the business man is willing to pay is:The Markowitz risk premium has been used to calculate the business man’s maximum risk premiumbecause by definition the Markowitz risk premium is the maximum amount of money that he wouldpay to insure against the gamble. So this businessman is willing to pay an insurance premium up to£114,941.55 to protect against fires.