VIEWS: 14 PAGES: 5 POSTED ON: 6/28/2012
This FAQ for Investment, based on Prof. Mehra’s course, is prepared to assist AXP students. It consist of concept framework and analysis of questions. If you find error, omission or ways to improve this FAQ, kindly email me at boonseng.tan@gsb.uchicago.edu. You are encouraged to use this for discussion in your study groups in your country. SYNOPSIS This course is conceptually divided into two sections. For the first week, we cover financial mathematics and use the results for valuation models in a perfect world (i.e. no uncertainty and transaction costs), with special emphasis on valuation and growth. Then, we build the basics for introducing risk by looking at portfolio theory, using the familiar results from the Statistics Course. We extend portfolio theory to CAPM to obtain a measurement for risk. For the second week, we start to look at real world applications involving securities, especially bonds. The final section covers options. This note covers the key concepts and augment with the analysis of some of the questions that you have the solution. 1. Preliminary Matters Let’s begin with the idea that a dollar today is worth more than a dollar in 3 year’s time. This is because if inflation is positive, we loss purchasing power over time. Now, a dollar today is called Present Value (PV) and that for the future is called Future Value (FV). This means that there are discount rates (r) such that FV=PV(1+r1)(1+r2)(1+r3). Rearranging, we obtain a very clumsy expression PV=FV/(1+r1)(1+r2)(1+r3). To make life easy, we define 1/(1+r1)(1+r2)(1+r3) as DF3 (discount factor) and obtain PV=FV*DF. In the case where r1= r2=…, FV=PV(1+r)n. Also, DF1>DF2>DF3…, as long as r >0. This is the case of discounting single cash flow (denote CF). We can extend this idea to multiple cash flows. Consider the case of similar CF and r for t=1,2,3. This is called an Annuity (denote An). Then, An=CF/(1+r)+CF/(1+r)2+… =CF/r*(1-(1/(1+r)n)). If the annuity continues forever, we get the Perpetuity (denote A∞). Given that 1/(1+r)<1, raising this term to ∞ obtain zero. Hence, A∞=CF/r. We can extend these two ideas to growing annuity and growing perpetuity, only the latter interest us. A Growing Perpetuity, GA∞=CF/(r-g). The mathematically inclined reader may refer to Appendix A for proofs. There are good applications for in the Course Pack (CP) and Homework (HW) as follows: CP-A p.98: Mortgage Payment – This is a straightforward application of annuity. Having calculated the annual payment, the next page shows you how much of each payments goes into interest and principal for each year. CP-A p.100: Savings for Retirement – You start saving $X now (first amount at the end of this year) so that you have a sum of money at retirement. You give this to an investment company in exchange for an annual amount C forever. Applying the annuity formula to X will get you the PV of the annuity AT=X/r*(1-1/(1+r)T), but the amount is for your retirement. So you need to find the FV of this amount by multiplying the PV with (1+r)T. This amount (the FV), must equal the perpetuity (=C/r) the investment company pays you. Hence, C/r=AT*(1+r)T. The rest is algebra. CP-A p.104: This is the same problem with specific values given. Another Side Issue: Effective Discount Rate, Annual Percentage Rate, Continuous Compounding The effective discount rate should give you the same PV and FV whether you count in years, months or weeks. Therefore for a 3-year loan (1+ra)3=(1+rm)36=(1+rw)156. However, your bank may calculate in terms of APR (denote R) where FV=PV(1+R/n)n, i.e. APR of 6% for monthly installment means you pay 0.5% on the remaining balance every month. This rate is actually higher than 6% a year effectively. Mathematically, if n becomes ∞, (1+R/n)n=eR. This FAQ for Investment, based on Prof. Mehra’s course, is prepared to assist AXP students. It consist of concept framework and analysis of questions. If you find error, omission or ways to improve this FAQ, kindly email me at boonseng.tan@gsb.uchicago.edu. You are encouraged to use this for discussion in your study groups in your country. 2. Valuation In the perfect world without risk, transaction cost, tax and with perfect information, the value of a security (share, bond or whatever) is the sum of the discounted CF. Then, the return for holding a share is coming from the capital gains and the sum of discounted dividends. Assume you hold the share long enough that the discounted capital gain is negligible and the dividend is constant. Then the value of the share is an annuity of the dividends. If you hold the share forever, the value is perpetuity. This is the dividend discount model (DDM). If we assume (unrealistically) that the dividend can grow at a constant rate forever, then the value of the share is a growing perpetuity. This is the constant growth model (CGM). More realistically, the firm can grow because of investment. If there is no external equity raise (this is assumed throughout the textbook BKM), the amount available for dividend distribution = earnings – investment. This is the stream of earnings approach. This is an important idea for some of the BKM questions. The stream of earnings idea is more fully developed in the chapter for growth and valuation. In general, value of the share = E1/r + PVGO after the long proof in CP1 p.132-6. This means that investors are paying for the fixed portion of the business (E1, the earnings in period 1, is also the constant dividend the firm would have paid if it has no better investment with higher returns then the cost of capital) and PVGO (Present Value of Growth Opportunities). If we assume that the firm invest a fixed proportion of its earnings b = investment/earnings and raises no external capital, then the firm will grow at a constant rate g=br* (growth=investment* rate of return). The value of the firm is a growing perpetuity, V0=E1(1-b)/(r-br*). In the more complicated case where firm can raise external capital, investment=plough back + external capital, i.e. b=f+k. Take note of two things: (1) the cost of capital / market capitalization rate is the return investor for your share requires, denote as k in BKM and r in the CP. The return from investing in growth project is ROE in BKM and r* in CP. I follow the CP notation. (2) The investment is net investment after deducting depreciation. Therefore, you can get b<0 if you don’t invest but have depreciation. The following are analysis of some questions applying these concepts: BKM Chp. 8 Q15: ChipTech Inc. This is a typical story: The firm initially has r*>r, so it is best to invest most earnings. Then r*=r due to change, the firm becomes indifferent to investing or paying dividend. Find V 0. One good approach is to make a table of how earnings change over time, and then calculate the dividends for the corresponding periods. In all likelihood, these will stabilize after certain period. In (a), the situation for dividend is a straightforward constant growth. In (b), dividend fluctuates and then stabilizes after t3 to become a growing perpetuity. The formula will give you the value of GA ∞ at t2, you have to bring it back to t0. V0 also include CF at t1 and t2. Note that since the firm just goes ex-dividend, you do not need to account for CF0. 3. Market Efficiency The bottom line is that price incorporates all necessary information in an efficient market, so you cannot make abnormal profit knowing the information. The necessary information is past price (weak form), public information (semi strong) and private information (strong). Technical analysis uses past price, fundamental analysis uses public information, insider trading uses private information. Market anomalies arise in inefficient markets. In summary: This FAQ for Investment, based on Prof. Mehra’s course, is prepared to assist AXP students. It consist of concept framework and analysis of questions. If you find error, omission or ways to improve this FAQ, kindly email me at boonseng.tan@gsb.uchicago.edu. You are encouraged to use this for discussion in your study groups in your country. IF Efficiency is: Weak Semi Strong Strong Price incorporate Past Price Public Info Private Info You cannot make $ with Technical analysis Fundamental Analysis Insider Trading 4. From Portfolio Theory to CAPM Let’s start by using 2 risky assets to form a portfolio. From Statistics class, we know how to find the expected portfolio return E(rp) and variance p2 given the weights x1 and x2=(1-x1). Specifically, p2 =x112+x222+2x1x212221,2. When 1,2=1, we obtain the perfect square solution p2=(x11+x22)2 [Try expanding this out to see why] and when 1,2= -1, we obtain another perfect square p2 = (x11 –x22)2= (x22 -x11)2. For the first case, we see that we can obtain a linear relation between E(rp) and p. For the second case, we will have two linear relations. These solutions provided a triangular bound for the possible values of E(rp) and p for any weights x1 and x2 as shown in CP1. p.257. When you consider the case where –1<1,2<1, you get a curve bounded by this triangle as in CP1. p.258. Note especially point A (minimum variance portfolio). Any point above A is on the efficient frontier because it has the highest return given a risk level or the lowest risk given a return. Where on the efficient frontier will an investor choose depends on how risk averse he is. Can an investor choose a risk-less portfolio? The answer is NO unless his assets has 1,2= -1. If you recall, p2=sum of weighted variance + sum of weighted covariance. Variance is always positive, but covariance may be either way depending on 1,2. So, there is a limit to risk reduction in diversification. This is the bottom line in portfolio theory. Now, let’s make a quantum leap adding a risk free asset. If you are investor A, your capital allocation line (CAL) is a combination of rF and the risky portfolio A. This is the lowest line. If you can only tolerate 10% risk, you are better off buying the risky portfolio M and rF than stay on your CAL. Similarly, Investor B is better off with M and rF instead of B and rF. If everyone buys M, then M must be the market portfolio! Everyone shifts to the highest CAL call capital market line (CML), assuming we agree fully on every stock return, risk and covariance in our portfolios. Two further points. (1) Everyone will be on the CML with the most risk averse at the left most. To be on the right of M, the weight on rF is negative, i.e. short/borrow rF. If rF borrowing/lending rates differ, the CML kink at M. (2) The slope of the CAL is the risk to variability ratio (Shrape’s ratio). In particular, CML slope is (rM-rF)/M. The CML is therefore E(rp)=rF+(rM-rF)/M *p. We extend the idea to single stock. Following the proofs in CP1 p.303-306, we obtain the Capital Asset Pricing Model (CAPM): E(rj)=rF+(rM-rF)/M2*jM= rF+(rM-rF). Note: =jM/M2 =covariance/variance (as in regression slope). The CAPM equation is the security market line (SML). Re-writing the SML as E(rj)=rF+jM, is the market price of (nondiversifiable/ systematic) risk that comes from the covariance term. The BIG idea is that CAPM enable us to price (systematic) risk. CP2 p66-81 discussed testing of CAPM. Here are some applications: 6. Capital Budgeting under Uncertainty This FAQ for Investment, based on Prof. Mehra’s course, is prepared to assist AXP students. It consist of concept framework and analysis of questions. If you find error, omission or ways to improve this FAQ, kindly email me at boonseng.tan@gsb.uchicago.edu. You are encouraged to use this for discussion in your study groups in your country. This is the first application in CP2. The firm is deciding on a capital project with one uncertain CF. In a certain world, PV=CF/(1+r). In an uncertain world, you either adjust r to obtain PV=E[CF]/(1+rF+cov(rj,rm)), or adjust CF to obtain PV=(E[CF]- cov(CFj,rm))/(1+rF). The terms cov(rj,rm) and cov(CFj,rm) are adjustment factors for r and CF derived from CAPM. Introducing debt, we get USU= LSL where S is the share price. Rearranging, L= u[1+BL/SL]. In the presence of tax, we obtain L=u[1+BL(1-T)/SL]. Finally, E[rL]=rF+(E[rM]-rF)U+( E[rM]-rF) u(BL/SL)= rF+ operation risk + financial risk These formulae are also useful for Cases in Financial Strategy later! 7. Bond Valuation, Duration and Immunization The section on bond valuation is relatively straightforward. The big idea in this section is Immunization against interest rate risk in bonds. You buy a bond to protect against future liability like what pension fund managers do. [You saw the idea with zero-coupon when you saw the arbitrage opportunity earlier]. However, interest rate fluctuation can hit you, so you manage your bond portfolio so that its duration equals the timing of your CF. 8. Option: Preliminary, Binomial Pricing, Black-Scholes. If you hold a Call option, you hope stock price goes Up so that you can Buy at below market price, vice versa for Put. If you hold American Call, you NEVER exercise before expiry because you could sell the Call for better money (you can only exercise European Call upon expiry). You can replicate the payoff for a Call + risk-free Asset with a Put + Stock (that’s the parity). The Binomial Pricing for (Call) Option requires two state prices in the next period to obtain the option price today. Two ways of doing this: (1) if you don’t know the probability of the state prices, replicate the payoff of the call option with a portfolio of stock + risk free (2) if you know the probability or can calculate it, price of option = probability weighted state prices. This can be used for multi-period. The Black Shole extends this idea using continuous discounting. This FAQ for Investment, based on Prof. Mehra’s course, is prepared to assist AXP students. It consist of concept framework and analysis of questions. If you find error, omission or ways to improve this FAQ, kindly email me at boonseng.tan@gsb.uchicago.edu. You are encouraged to use this for discussion in your study groups in your country. Appendix A: Proofs for Annuity, Perpetuity and Growing Perpetuity Let An=X/(1+r)+X/(1+r)2+…X/(1+r)n ……(1) Multiply (1) with 1/(1+r), An/(1+r)=X/(1+r)2+X/(1+r)3+….X/(1+r)n+1 ….(2) Subtract (2) from (1), An(1-(1/1+r))=X/(1+r)-X/(1+r)n+1 Simplifying, An(r/(1+r))=X/(1+r)*(1-1/(1+r)n) An=X/r*(1-(1/1+r)n) (Proved) Note: The trick is that multiplying by 1/(1+r) will make the first term become the second term and so on, and taking the difference will eliminate all the in between terms. The careful reader also notice that 1/(1+r)n is in the discount factor DFn. Now, 1/(1+r)<1, so when n approaches ∞, 1/(1+r)n becomes zero. Then A∞=X/r (Proved) Let GAn=X/(1+r)+X(1+g)/(1+r)2+…X(1+g)n-1/(1+r)n ……(1) Multiply (1) with (1+g)/(1+r), GAn(1+g)/(1+r)=X(1+g)/(1+r)2+….X(1+g)n/(1+r)n+1 ….(2) Subtract (2) from (1), GAn(1-(1+g)/(1+r))=X/(1+r)-X(1+g)n/(1+r)n+1 Simplifying, GAn((r-g)/(1+r))=X/(1+r)*(1-((1+g)/(1+r))n) GAn=X/(r-g)*(1-((1+g)/(1+r))n) Assuming that r>g, (1+g)/(1+r)<1 and the second term is zero when n becomes ∞ Therefore, GA∞=X/(r-g), provided r>g. (Proved)