Chapter 4: Mean-Variance Analysis

Transcription

1 Chapter 4: Mean-Variance Analyi Modern portfolio theory identifie two apect of the invetment problem. Firt, an invetor will want to maximize the expected rate of return on the portfolio. Second, an invetor will want to minimize the rik of the portfolio. The two apect amount to the objective of maximizing the expected rate of return for any given, acceptable, level of rik. Alternatively the objective can be tated a: minimize the rik for any given, acceptable, level of expected return. For the purpoe here, rik i aociated with the variance - or more commonly, the tandard deviation - of the portfolio. The goal of ection 4.1 i to contruct the efficient frontier. Every point on the frontier will contitute a poible portfolio which meet the objective of maximum return for a given rik (or, minimum rik for a given return). Once the frontier i identified, then ection 4. addree the quetion of how an individual invetor will chooe among the variou efficient portfolio. The invetor preference decribe how they are willing to trade-off higher return for lower rik, while the efficient frontier decribe how they are able to make the trade-off. Hence, without knowing the invetor preference, we cannot determine which efficient portfolio will be choen. Our job i only to contruct the efficient frontier, then let the invetor decide where they would like to be on it. 1 Section 4.3 introduce a rik-free aet, which turn out to have important implication. The introduction of a rik-free aet allow the invetor to eparate the quetion of identifying the optimum riky portfolio from hi/her own preference a reult known a the Separation Theorem. Section 4.4 recount the context and original development of the eparation theorem. A word of caution i in order before beginning. In order to keep the calculation in the example down to a manageable number, we will be contructing a portfolio from only two riky aet (and adding one rik-free aet in ection 4.3). The reult obtained will generalize to any number of aet. The reult and their interpretation will be tated in the generalized form. In doing o, the explanation may appear to train the actual numerical reult. 4.1 Contruction of the Efficient Frontier reviouly, we had been concerned with variou way in which to think about the determination of aet price (e.g., treat aet a tock or flow). Since aet price are forward looking, the quetion of how expectation are formed kept ariing. A high aet price implied holding other thing uch a rik of default and liquidity contant a high expected payoff. The focu wa on buying an aet with a high expected payoff and a other did thi, then the price would be driven upward. Harry Markowtiz wa the firt to ytematically elevate the iue of rik to a poition on par with expected return. The efficient frontier illutrate the trade-off that exit between expected return and rik. In the contruction of the frontier, we will dicover the 1 The connection to conumer theory of tandard microeconomic hould become abundantly clear. The conumer attempt to maximize utility a decribed by hi/her indifference curve. The indifference curve illutrate how the conumer i willing to trade one good for another. The budget contraint decribe how the conumer i able to trade one good for another given relative price of the good. Aet price are forward looking in the ene that they depend upon what the future i expected to be i.e., what tate of the world actually come to pa. 1

2 benefit of diverification. A imple example will be ued to demontrate the procedure of contructing the efficient frontier. The efficient frontier hould be contructed from a large number of poible aet. However, in order to illutrate the baic procedure, it will be ueful to limit the cope of the aet to two. Suppoe an invetor ha a total of $100,000 to invet in a tock and/or bond it might be preferable to think of thee a index fund. The invetor mut decide how much of the $100,000 to invet in the tock and how much to invet in the bond. Thi deciion will determine the invetor portfolio (i.e., collection of aet). The characteritic of each aet are given in table 5.1. Table 5.1 Stock ond E(r) 7.50% 5.00% σ 1.50% 5.00% ρ -1 Suppoe the invetor held only the bond. The expected return would be 5% with a tandard deviation of 5%. Diatified with thi return and willing to take on more rik if neceary, the invetor ell ome bond and invet in the tock. Suppoe the invetor new portfolio contained 30% tock and 70% bond. What i the expected rate of return on the invetor portfolio? Recall that the expected rate of return on a portfolio i merely the weighted average of the individual rate of return where the weight are the percentage of the aet in the portfolio. (4.1) E r ) = W E( r ) + W E( r ) ( p The W repreent the weight (or, percentage of the aet in the portfolio) and ubcript S and refer to tock and bond repectively. In our example, the expected rate of return on the portfolio i calculated a follow. (4.) E ( r ) W E( r ) + W E( r ) = (.3)(7.5) + (.7)(5) = 5.75% p = Hence, the invetor ha been able to increae the expected rate of return on the portfolio by holding ome tock. What wa the cot of obtaining the higher expected rate of return? The invetor may have believed that he would have to take on more rik (i.e., higher tandard deviation) to obtain a higher expected return. However, did rik increae? In order to calculate the tandard deviation of the portfolio we begin by calculating the variance of a portfolio. 3 (4.3) σ = ( W σ ) + ( W σ ) + W W σ σ ρ 3 The derivation of thi equation i given in the mathematical and tatitical appendix.

3 Recall, the Greek letter rho (ρ) i the correlation coefficient which when multiplied by the two tandard deviation equal the covariance between the two aet. The important point to notice about the above equation for the variance i that unlike the expected rate of return it i not at leat not alway the imple weighted average of the individual variance. 4 In our example, the variance of the portfolio compoed of 30% tock and 70% bond i the following. (4.4) σ = ( W σ ) + ( W σ ) + W W σ σ ρ = [(.3)(1.5)] + [(.7)(5)] + (.3)(.7)(1.5)(5)( 1) =.065 The tandard deviation of the portfolio (i.e., our meaure of rik) i the quare root of the variance. (4.5) σ σ =.065 =.5% = y adding an aet (i.e., the tock) with a higher rate of return and rik to hi bond-only portfolio our invetor ha been able to increae the expected rate of return not very urpriing and reduce the overall rik of the portfolio thi i very urpriing. Here we ee our firt indication of the power of diverification. In addition, we ee that the bond-only portfolio wa not a very efficient portfolio. That i, there i at leat one portfolio available the one we ued with higher expected return and lower rik. ractice 1. Suppoe the invetor liked what he aw happened o much that he decided to place $70,000 (or, 70%) in the tock and $30,000 (or, 30%) in the bond. What i the expected rate of return and tandard deviation (rik) of thi portfolio? {{{Anwer 1. E ( r ) W E( r ) + W E( r ) = (.7)(7.5) + (.3)(5) = 6.75% p = σ = ( W σ ) + ( W σ ) + W W σ σ ρ = [(.7)(1.5)] + [(.3)(5)] σ σ = 5.56 = 7.5% }}} = + (.7)(.3)(1.5)(5)( 1) = 5.56 Suppoe our invetor having tried three different portfolio (bond-only, 30% tock and 70% bond, 70% tock and 30% bond) realized that the reult were changing in uncertain way. Moving from the bond-only to 30%-70% tock-bond portfolio the rate of return increaed and the rik decreaed. Moving more into tock, the rate of return continued to increae, but o did the rik. Table 5. might help the invetor begin to ee ome pattern in the variou portfolio. 4 The appendix contain further commentary on thi point. 3

4 TALE 5. Stock ond Return (%) Rik (%) 100% 0% % 5% % 10% % 15% % 0% % 5% % 30% % 35% % 40% % 45% % 50% % 55% % 60% 6 35% 65% % 70% % 75% % 80% % 85% % 90% % 95% % 100% 5 5 What are the implication of thi analyi? Firt, think of beginning with the bond-only portfolio. The expected rate of return i 5% with a tandard deviation of 5%. Now, uppoe the invetor decide to allocate 95% of the total invetment in the bond and 5% in the tock. The expected return will increae to 5.13% and the tandard deviation (rik) will actually decreae to 4.13%! Surely, the bond-only portfolio i not efficient. In other word, if we can find another portfolio with a higher expected return and the ame - or, maller - tandard deviation, then the original portfolio hould not be choen regardle of the invetor preference. Second, notice that a we move up the table from the bottom row (where only the bond i held) the rik initially decline, then reache a minimum, then begin to increae. All along thi upward movement in the table, the expected rate of return i increaing. The implication can be clearly een in Figure 5.1. The efficient frontier conit of only the upper portion of the line. Thi upper portion conit of portfolio in which the expected rate of return can be increaed only at the cot of an increae in the tandard deviation (rik). We can graphically pick the efficient portfolio, chooe a given level of rik (i.e., pick a point on the horizontal axi), then move upward until the portfolio with the highet expected rate of return (i.e., move up from the horizontal axi to the highet point on the curved line) i reached. Alternatively, for a given expected rate of return (i.e., pick a point above the minimum rik, on 4

5 the vertical axi), chooe the portfolio which minimize rik (i.e., move to the right until you hit the curved line). Figure 5.1- The Efficient Frontier Return Rik Doing thi for all poibilitie, we would find that the efficient portfolio lie on the upper part of the chedule. 5 Though omewhat mileading, the efficient frontier may refer to the entire chedule. Undertood to mean thi, we find that in a two aet cae, each aet will lie on the frontier. In general, when the number of poible aet exceed two, no individual aet will be on the frontier again, the power of diverification. ractice. Suppoe an invetor contruct a portfolio with 0% in tock and 80% in bond. The invetor ha calculated the expected return on the tock to be 10% with a tandard deviation of 5%. The expected return on the bond i 6% with a tandard deviation of 1%. In addition, the correlation between tock and bond return i zero. Calculate the expected return and tandard deviation (rik) of thi portfolio. {{{Anwer. The expected rate of return on the portfolio will be: E(r ) = (.)(10) + (.8)(6) = 6.8% 5 The appendix provide the formula for the minimum tandard deviation. Given a correlation of -1, the minimum i quite traightforward to calculate it i zero. In addition, the appendix provide the equation for the weight to chooe in order to minimize the tandard deviation. For example, in the cae of a correlation of -1, the minimum tandard deviation (zero) will occur when the portfolio contain a percentage of bond equal to σ σ + σ. In our example, thi would be approximately 1.5/(1.5+5) =.71. Thu, in order to reduce the tandard deviation to zero, the invetor hould hold a portfolio coniting of 9% tock and 71% bond. 5

6 The variance of the portfolio i: σ = ( W σ ) + ( W σ ) + W W σ σ ρ = [(.)(5)] + [(.8)(1)] + (.)(.8)(5)(1)(0) = note, ince the correlation i zero, the lat term become zero a well. The tandard deviation (rik) of the portfolio i the quare root of the variance: σ = 10.8% In thi example, we have been able to reduce the rik below the rik of holding only the bond while increaing the rate of return on the portfolio above the bond rate of return.}}} What i the impact of the correlation coefficient? We will ue the example in ractice in order to tudy the role of the correlation coefficient. 6 However, we change the value of the correlation coefficient from -1 to +1. Notice that a long a the correlation coefficient i le than +1, diverification can lead to more efficient portfolio. In the unlikely event of the correlation coefficient being exactly +1, then the variance of the portfolio i equal to the weighted average of the individual variance jut like the expected rate of return of the portfolio. Thi implie that there will alway be a trade-off between rik and return. The important point, however, i that the benefit of diverification do not hinge on a negative correlation between aet in a portfolio. However, clearly, there are greater benefit to be had with a negative correlation. The only condition for the benefit of diverification i that the aet do not move perfectly together (or, the cae of the correlation coefficient being exactly +1). The table indicate that the cloer the correlation coefficient come to -1, the greater will be the benefit of diverification. Thi can be een by chooing any portfolio containing both tock and bond (i.e., chooe a row in the table). Now, move acro the row from a correlation of +1 to -1. The expected rate of return of each portfolio in the row remain the ame (thi hould be clear from the calculation for expected rate of return of a portfolio). However, the tandard deviation (rik) continue to decreae! The far right hand ide of the table aume that tock and bond have a perfect negative correlation. In the cae of perfect negative correlation (i.e., -1) between two riky aet, it i alway poible to contruct a portfolio with zero tandard deviation (rik). In thi particular cae, if the portfolio contained approximately 3.4% tock and 67.6% bond, then the tandard deviation (rik) would be zero. The figure preent a graphical repreentation of thee idea. Moving from the figure with a correlation coefficient of +1 6 Recall that the correlation coefficient meaure the relationhip between two variable in our cae, the variable are the expected rate of return of the two aet. The correlation coefficient i cloely related to the covariance between two variable. However, the correlation coefficient i eaier to interpret. The cloer a correlation coefficient i to -1, the tronger the invere relationhip between the two variable. In the cae of exactly -1, we ay that the two variable are perfectly inverely related meaning that they alway move in oppoite direction. The cloer the correlation coefficient i to +1, the tronger i the poitive relationhip (i.e., they tend to move in the ame direction). A correlation coefficient cloe to zero implie a lack of relationhip between the two variable. 6

7 where the efficient frontier i a traight line indicating that the portfolio rate of return and tandard deviation are imply weighted average to the figure baed on a -1, the efficient frontier get pulled toward the vertical axi illutrating that rik i falling. 7

8 Table 5.3 Correlation = +1 Correlation = +0.5 Correlation = 0 Correlation = -0.5 Correlation = -1 Allocation ortfolio ortfolio ortfolio ortfolio ortfolio ortfolio ortfolio ortfolio ortfolio ortfolio Stock ond Return Rik Return Rik Return Rik Return Rik Return Rik

9 Efficient Frontier: Correlation = Return Rik Efficient Frontier: Correlation = Return Rik 9

10 Efficient Frontier: Correlation = Return Rik Efficient Frontier: Correlation = Return Rik 10

11 Efficient Frontier: Correlation = Return Rik The efficient frontier allow u to find the portfolio with maximum return for given rik. The frontier act much like a conumer budget contraint in indicating how the invetor i able to trade-off rik for return. Jut a the cae of the conumer budget contraint, chooing to be inide the efficient frontier lead to an inefficient outcome. The quetion now i to begin to addre the quetion of how the invetor hould chooe between the efficient portfolio. 4. Illutrating reference for Rik and Return Exactly which efficient portfolio hould an invetor chooe? The anwer will depend upon the invetor particular preference. I the invetor willing to take on more rik in order to gain a higher expected rate of return? The invetor preference can be illutrated with a et of indifference curve. Along any particular indifference curve, the invetor ha the ame amount of utility (or, atifaction). The indifference curve will lope upward. Thi indicate that in order to leave the invetor with the ame utility, the invetor mut be compenated with higher expected rate of return for greater level of rik. A higher indifference curve i alway better. Thi imply demontrate that the invetor will achieve a higher level of utility ince hi/her expected rate of return can be higher for any given level of rik. Alternatively, you may read the indifference curve horizontally a tating a lower rik for any given level of expected return. 11

12 Return Return Increaing utility Young Executive Rik Little Old Lady Rik The figure above preent two type of invetor preference. The indifference curve for the young executive demontrate that he require little additional expected return for taking on more rik. The little old lady on the other hand require a large increae in her expected rate of return for taking on additional rik. We can think of reaon why thee type of invetor view the rik-return trade-off in their particular way. The young executive ha a teady income in the form of a alary and a long invetment time horizon, which allow him to view the cot of additional return omewhat mildly. The little old lady on the other hand may not have another ource of income and ha a hort invetment horizon (note, not necearily a hort time left to live, but rather need to be cahing out of ome of her invetment oon). The important general point i that for both type of invetor, they till hope to get on the highet indifference curve poible. A we move up indifference curve, the invetor achieve higher expected rate of return for the ame or, le rik. Thu, the higher indifference curve are uperior regardle of preference. 7 We can now turn to the quetion of which efficient portfolio the invetor hould chooe. The invetor problem i to maximize the expected rate of return and minimize the rik of the portfolio ubject to the available efficient portfolio. Graphically, the invetor i attempting to reach the highet indifference curve poible, given the contraint of the efficient frontier. The optimum riky portfolio for an individual invetor will be given by the point at which the indifference curve (illutrating the invetor preference for the rik/return trade-off) i jut tangent to the efficiency frontier (illutrating all poible efficient portfolio) - thi i point O on the graph. 7 Throughout we aume invetor are rik-avere. It i perfectly poible to treat invetor a rik-neutral or even rik-loving, but the cot of the complication that arie from thoe aumption would eem to far outweight the benefit for u at thi point. 1

13 Return Rik Although the invetor may like to be on the higher indifference curve, it i imply not poible given the characteritic of the riky aet. On the other ide, any other point on the efficiency frontier reult in a lower indifference curve, thu a lower level of utility. The optimum riky portfolio will be different for invetor with different rik-return preference hence, different hape of their indifference curve uch a the young executive and little old lady. The next ection demontrate that thi reult may not hold in all cae. 4.3 The Separation Theorem A rik-free aet can be introduced into our portfolio. The important implication of thi introduction may appear urpriing. The firt tak, however, will be to deal with the technical apect. Thi can be done in a quick and dirty way. Reconider the tock-bond characteritic of ractice. We can work with the cae of a zero correlation coefficient along with a 50% compoition of tock and bond (the rate of return i 8% and tandard deviation 13.87%, ee the table) - call thi the riky portfolio. Now, uppoe you could buy a rik-free treaury bill paying 5%. A portfolio (call it the complete portfolio ) can be compoed of the rik-free aet (treaury bill) and the riky aet (tock and bond). The deciion concern how much of your total amount (e.g., $100,000) i allocated toward each type of aet. The reulting complete portfolio will have an expected rate of return and tandard deviation. We can illutrate the procedure with a little practice problem. roblem 3. An invetor ha $100,000 to invet. The invetor ha choen to contruct a portfolio containing 5% of a rik-free treaury bill (5% rate of return and zero tandard deviation) and 75% of riky aet. The riky portion of the complete portfolio i compoed of 50% tock (10% expected rate of return and 5% tandard deviation) and 50% bond (6% rate of return and 1% 13

14 tandard deviation). Calculate the expected rate of return and tandard deviation of the complete portfolio. Notice, we have already calculated the expected rate of return and tandard deviation for the riky portfolio. All that thi example require i to ue the previou formula with the two type of aet: rik-free and riky portfolio. {Anwer 3. E ( r ) W E( r ) + W E( r ) = (.75)(8) + (.5)(5) = 7.5% C = f f where ubcript C tand for the Complete portfolio (including the rik-free aet), for the riky portfolio, and f for the rik-free aet. σ C = ( W σ ) + ( W σ ) + W W σ σ ρ = [(.75)(13.87)] And, f + [(.5)(0)] f f f + (.75)(.5)(13.87)(0)(0) = σ C = σ C }}} = = 10.4% The intuition of the analyi can be een with the aid of a graph. We will avoid ome of the technical detail. Eentially, we are forming a linear combination of a rik-free aet and a riky portfolio in order to contruct a complete portfolio. If we began with only the rik-free aet, then - uing our numerical example - the complete portfolio would have a rate of return of 5% and zero tandard deviation (thi i the point on the vertical axi). On the other hand, if the complete portfolio did not contain the rik-free aet, then the expected rate of return would be 8% with a tandard deviation of 13.87% - thi i point Z on the graph (note, thi ha been drawn a the optimum riky portfolio for convenience). y varying the percentage of our total invetment allocated to the rik-free aet and riky portfolio, the expected rate of return and tandard deviation will be given by a traight line between the point on the vertical axi repreenting the rik-free aet and point Z, the riky portfolio. The point X repreent the complete portfolio of ractice 3 where the expected rate of return turn out to be 7.5% with a tandard deviation of 10.4%. 14

15 Return CAL Y 8% X Z 5% 13.87% Rik The capital allocation line (CAL) repreent the complete portfolio for variou allocation between the rik-free aet and riky portfolio. The lope (rie/run) of the CAL i given by the following: E( r ) r f σ 8% 5% = 13.87% =.17 The lope i ometime called the reward-to-variability ratio (or, Sharpe ratio). Thi lope equal the increae in expected return than an invetor can obtain per unit of additional tandard deviation (rik). The portion of the line between the rik-free rate of return and point Z i where the invetor i lending a portion of hi/her total invetment to the default free borrower (i.e., the government). Thi i illutrated by point X. What about the portion of the line beyond point Z. Thi part of the line would reult if the invetor could borrow at the rik-free rate, then purchae more than 100% of hi/her own invetment money into the riky portfolio. Thi i illutrated by point Y. Now of coure, except for the government, an invetor cannot actually borrow at thi rik-free rate. If we wanted more realim, then the CAL will have a kink at point Z indicating the lope of the line get flatter a the interet rate on a loan i greater than the rik-free rate. You can alo think of the portion of the CAL beyond Z a indicating that the invetor i buying on margin till though, the cot of doing o will exceed the rik-free rate. What would have happened if a riky portfolio with a higher expected rate of return and tandard deviation had been choen? For example, uppoe you had choen a riky portfolio compoed of 75% tock and 5% bond. Uing our previou number, the expected rate of return would be 9% with a tandard deviation of 18.99% (thi can be een in the previou table). The lope of the CAL would decline lightly. Thi indicate a lower expected rate of return for an additional unit of rik. You could continue along thi path - chooing variou riky portfolio and drawing the CAL. Which CAL would be bet? The one with the highet lope! Thi will occur when the CAL i jut tangent to the efficient frontier. Thi i the one we have drawn in the 15

16 figure. We have everything needed to tate and apply the eparation theorem. efore doing o, conider the path taken to get to thi point. Imagine that you are an invetment counelor. It i your job to et up a financial portfolio for a client. Your firt tep would be to calculate the expected rate of return and tandard deviation for every poible riky financial aet. The probabilitie aociated with the expected value and variance formula can be determined by (a) looking at the pat price data for each aet, (b) conidering the financial poition (e.g., ue of ome accounting and financial ratio) and propect (e.g., a new CEO, a new product line, competition, etc.), and/or (c) ubjective meaure (e.g., gut feeling). Thi i, of coure, a fairly daunting tak. The econd tep i to form variou combination of the riky aet in order to define the efficient frontier (a fairly eay proce with a good computer). Once thi i done, you could top here with the calculation and attempt to undertand your client peronal preference between expected rate of return (a good thing) and rik (a bad thing). Thi i where you would be attempting to dicover your client particular indifference curve. Having done o, you can advie the client to buy a particular portfolio of financial aet. However, and thi i where the eparation theorem come in, uppoe that you have identified what you conider to be an excellent portfolio an optimum portfolio of riky aet. Would you really want to advie your client to chooe another portfolio imply becaue of their peronal preference for return/rik? Shouldn t there be a way to purchae the excellent portfolio and till meet your client peronal preference? Thi take u to the next tep. The third tep i to find a rik-free aet. The U.S. Treaury ill erve thi role nicely. However, you could chooe omething like a money market mutual account for your client where the rate of return wa lightly higher while the rik remain pretty near zero. Whichever rik-free aet you chooe, you mut now contruct the Capital Allocation Line (CAL). You do thi by combining - in variou amount - the riky portfolio on each point of the efficient frontier with the rik-free aet. Identify the CAL with the highet lope. Thi i the one that give the greatet expected return for each additional unit of rik. It i alo the one that i jut tangent to the efficient frontier. The point - labeled Z - on the efficient frontier that i jut tangent to the CAL i that excellent (or, optimal riky) portfolio. Thi i the riky portfolio that you hould advie all your client to hold regardle of their peronal preference for return/rik. It doen t matter if your client i the little old lady or the young executive. The only difference between the client, reflected in their preference, will be how much of their total invetment to allocate to the rikfree aet and how much to thi optimal riky portfolio. For the little ol lady, you may advie her to have a complete portfolio like point X in the figure. In thi cae, he would be holding ome portion of her wealth in rik-free government T-bill. For the young executive, you might encourage the young invetor - willing to take on even more rik in the hope of higher return - to borrower at the rik-free interet rate in order to purchae more of the optimal riky portfolio than what he/he could buy with their current wealth. Graphically, you are moving the invetor up and to the right along the CAL to a point like Y. ut notice, you are till adviing to buy into only the identified optimal riky portfolio. We have eparated the deciion of which riky 16

17 portfolio to hold from the invetor peronal preference. The preference come in only when deciding how much of the rik-free aet to hold. 4.4 Tobin Development of the Separation Theorem The eparation theorem wa not an attempt to implify the invetor olution to Markowitz Mean-Variance Analyi. Rather, Jame Tobin original paper (1958) wa intended to provide a more coherent foundation for Keyne Liquidity reference Theory of the Interet Rate. It will be argued in thi appendix that Keyne formulation of the liquidity preference theory of interet wa extremely weak both, from the perpective of development of hi earlier work in A Treatie on Money and the later development of portfolio theory. Tobin eparation theorem wa tangential to Mean-Variance Analyi while directly relating to broader iue within macroeconomic theory and policy Keyne Awkward Money Demand Function {{{need to write but the focu will be on: Difficulty of Monetary olicy to lower the long-term interet rate Inelatic expectation Divergence of opinion re normal rate (price) ortfolio i all or nothing deciion}}} 4.4. The Separation Theorem in Relation to Liquidity reference Tobin conidered the cae in which the government iued two type of financial aet: money and bond. Since the government iue both, the rik of default i the ame and zero. In order to implify matter, Tobin aume that the bond iued by the government i a conol. Why i thi aumption a implification? A conol i a pecial type of bond not actually iued by the U.S. government, but ha been iued by other government and can be approximated with a very long term to maturity (e.g., Diney 100 year bond) which make a et yearly payment. The point i that the bond never mature thu, doe not make a final payment (i.e., face-value). The price and interet rate of a conol are extremely eay to compute. The preent value of all the future yearly payment reduce to a nice formula. (A.1) C1 C C3 = L = 3 (1 + r) (1 + r) (1 + r) C r Where i the price of the bond, r i the interet rate (or, more pecifically, the yield to maturity), and C the yearly coupon payment. Conol are ueful to aume when firt introducing bond becaue it become abolutely clear that the price of the bond and interet rate on the bond move inverely. We can olve for the interet rate by cro-multiplying: 17

18 (A.) C r = Thi i imilar to a dividend yield (D/ where D i the yearly dividend payment, in thi cae an expectation mut be formed, and i the price of the hare). The expected rate of return on the bond i compoed of the interet rate on the bond (A.) and the expectation of a capital gain or lo (g). e e (A.3) g = = 1 The expected rate of return on the bond i therefore written a: (A.4) E( r ) = r + g Auming a martingale probability for the expected price implie that thi i equal to the current price. (A.5) E( ) = e = Thi implie that the expected rate of return from holding the bond will be equal to the interet rate hence, the average value of g i zero. (A.6) E( r ) = r Note, however, that the expected price i being treated a a random variable with mean of zero and a contant tandard deviation (e.g., conider it a a random variable with a normal ditribution thu, you need two piece of information to identify it particular normal curve, the mean and tandard deviation). Thi i important becaue one tend to forget about the capital gain/lo ince it drop out of the expected rate of return calculation but, it play an important role in the ret of the analyi. What about the money aet? We aume for implicity that money i defined in uch a way that it pay zero interet and, of coure, ha no capital gain/lo. efore moving on, conider the different definition of money: M1 (currency + checkable depoit), M (M1 + mall aving account), M3 (M + large aving account). Today, bank do in fact pay interet on checking account and have alway paid it on aving account). Furthermore, Keyne had argued that in ome circumtance money hould be defined to include hort-term government Treaury bill. For Keyne, the eential difference between money and bond wa their price fluctuation. The price of a hort-term T-bill will not fluctuate very much with a change in the interet rate. On the other hand, the price of a long-term government bond (uch a a conol) would fluctuate greatly with a change in the interet rate. In modern terminology, the percentage change in the price brought about by a one percent change in the interet rate i called the duration. The difference between money and bond, for Keyne, amount to the notion that 18

19 duration i mall for money and large for bond. For u, thi amount to auming that the capital gain/lo on money i negligible hence, the variance of thi i o low that it can be afely ignored. Thu, money ha a zero expected rate of return and variance hould be a afe aumption we can alway define the money aet o that it ha the characteritic, or very cloe. At thi point, we can apply the tool developed to handle the Mean-Variance Analyi. The expected rate of return on the portfolio i merely a weighted average of the expected return for the individual aet. (A.7) E( r ) = W E( r M E( r ) = r E( r ) = W r M ) = 0, E( r M ) + W E( r ) The variance of the portfolio i not the imple weighted average of the individual aet variance. We have een that thi depend upon the covariance (hence, correlation) between the aet. σ = ( W σ ) + ( W σ ) + ( W σ )( W σ ) ρ M M M M Thi implifie greatly once we recall that the rate of return on the money aet ha zero variance. Hence, we get the following. (A.8) σ = ( W σ ) The tandard deviation (i.e., rik) of the portfolio i imply the quare root of the variance. (A.9) σ = W σ All of thi ha been accomplihed with the tool ued in contructing the efficient frontier. The difference i that we have introduced an aet without rik and return what i called a rik-free aet. The introduction of thi type of aet carrie greater ignificance than what one might expect. In the preent context, our goal i to contruct the demand chedule for the rik-free aet. The available trade-off between rik and return can be developed from equation (A.7) and (A.9). Thi trade-off will be given by the lope of the line depicting the relationhip between rik and return on the portfolio. (A.10) r E( r ) = σ σ Notice, thi line i derived for olving (A.7) and (A.9) for the proportion of the portfolio held in bond and equating the reult. Graphically, the line i depicted in Figure A.1. 19

20 E(r) Figure A.1 Rik of ortfolio Notice, the line will tilt upward for an increae in the interet rate on bond or a decreae in the rik of the bond. The pecific place an invetor will chooe will depend upon how they view the rik and return trade-off. Since it depend upon their ubjective preference for rik and return we can depict their willingne in term of indifference curve a done previouly. In order to tranlate the deciion concerning rik and return into the reulting proportion of bond and money held we imply rearrange equation (A.9). (A.11) W σ = σ Recall, the proportion of money held in the portfolio will be given by one minu the proportion of bond held. Thi relationhip i graphed in Figure A.. 0

21 Wb Figure A. Rik of ortfolio We can now put everything together to derive a money demand chedule baed on portfolio choice (of the Mean-Variance variety). Figure A.3 depict the reult. Notice, the demand for money i read upward on the lower quadrant. Thu, a the interet rate increae, the top line will tilt upward normally leading to an increae in the proportion of bond held and decreae in the proportion of money normally i conditional upon how invetor for rik, i.e., their indifference curve. We, therefore, derive the demand for money chedule by allowing the interet rate to vary. If the rik of bond decline, the upper line tilt upward again and the lower line tilt downward! 1

22 E(r) Sigmabond Rik of ortfolio 1 percentage of wealth held in the form of money Wb Figure A.3