Economics

Economics
General Instructions: This report is more like a problem set. You do not need to type your answers, but it
would be helpful to complete 1.B. on a spreadsheet and copy it to your answers. Please write neatly!
1. The Gordon Divide nd Growth ( GDG ) formula from page 147 of Siegel is given by =

−
. Define
as the risk free interest rate plus the equity premium and g as the growth in dividends. ( For now
assume no inflation. We discuss inflation in the next problem.)
A. Write out the infinite geometric series from which this formula is derived , and then use the following
notation to solve the series : S, the sum of the series, α, the first term of the series , and ρ, the commo n
ratio . (For a quick review on geometric series see https://en.wikipedia.org/wiki/Geometric_series .)
A key reason why dividend growth may be greater than zero is that firms reinvest retained earnings,
which in turn contribute to future dividends by increasing the stock of ear ning asse ts . The given table
assumes that the share of earnings paid as dividends, , is 1, and that = 0 and = 0. 1. T otal shares
outstanding are 100. Be sure you understand how all the other values were derived before proceeding!

B . Continue to assume = 0. 1 and the firm has 100 shares outstanding. Present two versions of this
table: one when = 0 .8 and = 0.5 , and derive the value of from the growth in dividends shown in
your table.
C. Now apply GDG to determine the value of one stock in each of the two scenarios described in B.
Summarize in words what your answer shows.
2. In an inflationary environment dividends grow with inflation. Assume the most recently observed
dividend payment is
0
and the required return on equity = + so it now includes inflation . Let
= ( the only growth in dividends is from inflation) . Now
0
=

0
( 1+)
( 1+ )
+

0
( 1+)
2
( 1+ )
2
+

0
(1+ )
3
( 1+ )
3
…
A. Solve this infinite series to find
0
assuming
0
= 10, = 0. 1 = 0. 03 or 3%.
B. Now move 5 years ahead. The market is evaluating the stock with the same assumptions except that

0
= 10( 1 + )
5
since the nominal dividend is now higher. If you sell the stock what i s your nominal
capital gain and your real capital gain assuming no taxes ? How did you gain from holding the stock?
C. Now suppose that nominal capital gains are taxed at the rate of 0.5. What is your real after tax gain in
this tax environment from the tra nsactions described in part B? Why is this tax considered unfair?
time Earnings Dividend
Retained
Earnings g
Total Earning
Assets
1 10 10 0 10,000.00
2 10 10 0 0 10,000.00
3 10 10 0 0 10,000.00
4 10 10 0 0 10,000.00
5 10 10 0 0 10,000.00
Per Share
Marshall Report 3 Econ 135

3. A crucial ingredient in the GDG formula is the discount rate, that is the value of R , in the
denominator of the present discount formula. The discount rate will differ for different stocks, and the
concept of a stock’s beta will help us pin it down, although there is considerable debate on this topic. A
stock’s beta is a measure of how it s return varies with the overall market rate of return. If the covariance
is high, the risk of the stock is not diversifiable and so the stock should earn a higher return, or in other
words, its future earnings should be discounted at a higher rate.
A. Using the standard deviation formula for a portfolio from Report 2,

= � ∑ ∑

=1

=1
�
1
2
�
,
show that if you have lots of stocks with no covariance (so 0
ij ji
σσ= = ), you can reduce the
portfolio standard deviation towards 0 as the number of stocks increases. . (You may assume
all the stocks have the same variance and you hold equal shares of each stock so
1
ij ww
N
= = .)
B. The capital asset pricing model (CAPM) argues that a particular stock’s expected rate of
return ,
i
R , is given by
() i f M fi
RR R Rβ =+−
where
M
R is the market return,
f
R is the risk – free interest rate, and
i
β is stock i ‘s beta. (If you have
had some econometrics, note that
i
β can be found by running a regression on time series of the stock’s
return on the other variables, usually with monthly data over a five year horizon .)
Let’s assume that
M
R =0.05 and 0.02
f
R = . Now suppose we have 3 stocks with the following betas:
1 23
0.5 1 1.5 β ββ = = =
Each stock pays a $1 annual dividend and there is no inflation or expected growth in the dividend.
Determine the price of each stock using both GDG and CAPM.
C. According to Siegel, for which type of stock is the CAPM model the worst predictor ? Briefly summarize
his evidence. Please define the terms used in your answer in your own words.