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# SYNOPSIS

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```									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 =x112+x222+2x1x212221,2. When 1,2=1, we obtain the perfect square solution
p2=(x11+x22)2 [Try expanding this out to see why] and when 1,2= -1, we obtain another
perfect square p2 = (x11 –x22)2= (x22 -x11)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)

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