Investment Analysis & Portfolio Management
Lecture#
07
UNDERSTANDING RISK AND RETURN
INTRODUCTION:
Two key concepts provide the foundation for the field of finance. The first is A dollar
today is worth more than a dollar tomorrow, and is often called the time value of
money. The second is a safe dollar is worth more than a risky dollar. Anyone who
studies finance learns the universal application of these statements and rational
decision making. The tradeoff between risk and return is the principles theme in the
investment decision.
Most people are risk averse, which does not mean, however, they will not take a risk.
It means the only take a risk when they expect to be rewarded for taking it. People
have different degrees of risk aversion; some are more willing to take a chance than
are others.
People invest because they hope to get a return from their investment. Return is the
good stuff that makes people feel better or improves their standard of living. Risk is
the bad stuff of risk averse person seeks to avoid. It is a fact of investment life and is
unavoidable for anyone who seeks more than a trivial rate of return. This chapter
explores the fundamental principles underlying the relationship between risk and
return.
RETURN:
Some return measures are more useful than others. It is important to understand the
calculation and limitations of various measures.
Holding Period Return:
The simplest measure of return is the holding period return. This calculation is
independent of the passage of time and incorporates only a beginning point and an
ending point.
Holding period return = Ending value – Beginning value +
Income Beginning value
Someone might buy 100 shares of stock at $25, receives a 10 cent per share dividend,
and later sell the shares for $30. The holding period return is
$30 – $25 + $0.10 =
20.4% $25
It makes no difference if the holding period return is calculated on the basis of a single
share or 100 shares. The holding period return is exactly the same because every term
is multiplied by 100.
Has this investment done well? The answer depends on how much time transpired
between the purchase and the sale. If the stock was acquired in 1989 and sold in 2000,
the total gain of 20.4% is less than what could have been earned in a bank account. If,
however, the stock was purchased 60 days ago, the return is handsome.
Because we are accustomed to thinking of the rates of return on an annual basis, it is
common to annualize returns. To annualize returns, multiply the holding period return
by
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the fraction 365/days in the holding period. In the previous example, the 20.4% return
came from and holding period of 60 days: 20.4% (365/60) = 124.1%.
Holding period returns must be used with caution. When comparing investment, the
periods should all be the same length. With a collection of stocks, comparing returns
over calendar year 1999 or returns over the past five years is acceptable. One cannot,
however, meaningfully compare Stock A’s 1999 return with Stock B’s 1994-1999
return.
When calculating holding period returns, look out for stock splits or other corporate
actions that would muddy the water. For example, in September 1999, your mother
points out that she purchased 100 shares of Wal-Mart at $25 in September, 1990.
Today’s Wall Street Journal reports a WMT price of $44½, and this modest increase
surprises hard given all the recent news about the companies nationwide growth.
Capital appreciation over this nine-year period seems to be ($44.50 – $25)/ $25 =
78.0%.
Mom is overlooking the fact that the firm split its stock two for one in February 1993
and again in April 1999. A two for one stock split effectively cuts the share price in
half. Someone who owned 100 shares of WMT in early 1990 would have owned 400
shares in September 1999. Anyone unaware of the stock split would calculate an
incorrect holding period return. The split per se would not affect the true return if it is
correctly accounted for in the calculation. The calculation appreciation on Mom’s
stock is actually
4($44.50) – 25 =
612.0% $25
Yield and Appreciation:
A certain amount of ambiguity surrounds the term yield in the investment business. To
many (probable most) people involved with investments, when yield is used by itself,
it usually refers to the dollar amount an investment “throws off” as dividends or
interest. This definition will be followed in this book. The financial pages indicate the
yield of stocks and bonds. Technically, the newspaper shows the current yield, which
is the annual income an investment, is expected to generate divided by its current
market price. For a common stock whose income comes exclusively from dividends,
the current yield is typically referred to as the dividend yield.
A stock might currently sell for $40 and be expected to pay $1 in dividends over the
next 12 months. The newspaper will show its current yield as 2.5% — the $1 dividend
divided by the $40 current share price.
Another stock might excellent prospects for the coming year and be recommended by
many investment advisors. It might, however, pay no dividends. The fact that a stock
pays no dividends does not mean it is a poor investment. Consider Microsoft. Ask a
stockbroker “What is Microsoft’s yield?” and the answer will be “zero”. The
uninitiated person might wonder why anyone would ever buy a stock that was not
expected to yield anything. The explanation lies within another component of return:
appreciation.
Appreciation is the increase in the value of an investment independent of its yield.
When people speak of a stock going up, they are talking about its appreciation.
Suppose an investor buys MSFT at $95, and it rises to $97½. It appreciated by $2½, or
$2.50/$95 = 2.6 %. If it is paid no dividend, its yield was zero. Contrast the MSFT
investment with an interest earning savings account in which a saver deposits $95 to
accumulate interest. One
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year later the account contains $97.50. Even though these situations might seem
identical, technically the savings account showed a yield of 2.6% and appreciation of
zero. The increase in value comes from the interest earned that is left in the account.
Accrued interest does not count as depreciation, nor does an increase in account and
value due to additional deposits. Suppose an investor opens a brokerage account and
buys stocks for $25,000. Five years later the account is worth $45,000. It is
meaningless to say the account appreciated by $20,000 if the investor has been
depositing $150 per month into the account. A good part of increases is because of the
additional investment, not because of investment performance.
This point is especially important when reviewing portfolios managed by an outside
agency. Don’t assume that because the YWCA endowment fund is worth $200,000
more now than it was last year that the fund management was good. The fund
managers should not get credit for bequests or other deposits into the fund over the
past year.
The Time Value of Money:
The notion that one dollar received today is more valuable than one dollar received
tomorrow is usually called the time value of money. It is one of the two key concepts
in finance that form the basis for most valuation equations and pricing and
relationships. Financial theory states that the current price of any financial asset should
be the present value of its expected future cash flows. You have to understand the time
value of money to properly calculate present values.
Time value of money problems involves the relationship among present values, future
values, interest rates and time periods. Most problems involve solving for one of these
values when the other three are known. The simplest time value of money problem is
the single sum problem and can easily be illustrated in the corporate bond market.
PepsiCo Capital Resources has a bond issue coming due in the year 2004. Assume the
redemption date is four years from today. At that time, the company must pay $1,000
for each of its bonds when presented for redemption. Unlike most bond issues, these
bonds pay no periodic interest. Because of the time value of money, investors are
unwilling to pay $1,000 today for a security that will be worth $ 1,000 in four years by
providing no interest income. This bond must sell at a discount in the marketplace.
How much should the discount be? The answer depends on the rate of return available
on other investments of comparable risk in the marketplace. Suppose the Wall Street
Journal shows the price of this bond as $730. Barring default, this bond will gradually
rise in value to be worth $1,000 on its redemption date. The $270 increase in value is
the bond holder’s return. From these values we can calculate the investors anticipated
holding period return:
$1,000 – $730 =
36.99% $730
The holding period of return is not particularly useful in this context because it ignores
the time value of money. What we really want to know is the annual interest rate that
would cause a $730 investment to appreciate to $1,000 in four years. That is, we want
to know the
value of R in equation:
P (1 + R) n = F
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Where P = present value (i.e., price today)
F = future value
R = interest rate per period
n = number of periods
Substituting our numbers, $730 (1 + R) 4 = $1,000. We find R = 8.19%.
Support economic conditions change. Investors become pessimistic about the future
and government’s ability to keep inflation under control. As a consequence, market
interest rates rise by one point. Investors are no longer willing to accept an 8.19% rate
of return on a bond of this risk; they won’t settle for less than 9.19%. What is the most
an investor could pay for the bond to achieve this rate of return? In other words, what
is P, the present value (price of the bond) in the following equation?
P (1 + .0919) 4 = $1,000
Rearranging and doing the math,
($1,000
P=
1
)
(1.0919) 4
= 0.7035 ($1,000)
= $703.50
If the investor pays $703.50 and receives$1,000 in four years, the compound annual
return would be 9.19%.the factor 0.7035 is called the discount factor for four years at
9.19%. Financial calculators are preprogrammed to compute these factors for time
value of money problems. Factors are also routinely presented in tabular form in the
back of accounting and finance textbooks.
Many securities pay more than one cash flow over their life. Adelphia
Communications, a cable TV company, also has a bond maturing in the year 2004, but
this bond pays $95 per year in interest. Its value logically should be influenced by
these additional cash flows. An investor in this bond receives a single sum of $1,000
in four years, but also receives an annuity of $95 per year for the four years. An
Annuity is a series of evenly spaced, equal dollar payments.
An investor in this bond receives income from two sources: the return of the $1,000
principal in 4 years, and the $95 per year annuity. One way to determine the present
value of the annuity is to decompose it into four single sums of $95 each and find the
present value of each, but this method is inefficient. A more convenient expression for
the present value of an annuity is shown below:
P = C [1/R – 1/R (1 + R) n]
Where C = periodic payment
Suppose the risk of this bond is comparable to that of the PepsiCo capital resources
board, and is trading at a price that also implies a 9.19% rate of return. The present
value of the annuity is then
P = $95 [1/.0919 – 1/.0919 (1 + .0919)4] = $306.49
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The present value of the $1,000 return of principle is
$1,000 =
$703.51
(1.0919)4
The bondholder is entitled to both the return of principle and the annuity, so the bond
market price must be the sum of these two values: $306.49 + $703.51 = $1,010.
The holding period return over the remaining 4 years of this bond’s life would be
$1,000 – $ 1,010 + 4($95) =
36.63% $1,010
Compounding:
Compounding refers to the earning of interest on previously earned interest. Its effects
are more pronounced as the frequency with which interest is computed and credited to
the principal balance increases. At a financial institution, interest on a savings account
might be calculated once per year, semiannually, quarterly, monthly, or daily. Each of
these methods constitutes discreet compounding because the number of times per year
the bank calculates the interest can be counted.
Suppose an account earns 8% per year, compounded quarterly. In this scenario, the
account holder does not earn 8% every three months. Rather, the account is credited
with ¼ of 8% four times per year. After three months, an initial deposit of $100 would
earn $2, resulting in an account balance of $102. Three months later, the $102 has
earned 2%, so its value is $102(1.02) = $104.04. Interest is added again three months
later, and once more at the end of the year. At the end of one year account is worth
$100(1.02)4 = $108.24.
If the 8% interest were compounded annually, at the end of one year the account
balance would be $100(1.08), or $108.00. Note that with quarterly compounding the
account earns 24 cents more than with annual compounding. The rule is this: if money
is invested at an annual rate of R for t years an interest is compounded n times per year
and multiply the number of years in the problem by n. mathematically,
F = P (1 + R/n) nt
Where F = future value
n = number of compounding periods per
year t = investment horizon in years
Compounding can also occur hourly, by the minute, by the second, or by any
arbitrarily small time interval. In the limit, compounding occurs continuously, with an
in finite a number of time intervals. This changes the equation to
F = P (1 + R/∞) ∞t
This mathematical result forms the basis for natural logarithms. The quantity (1 + R/n)
nt
approaches eRt as n approaches infinity. The value e is 2.71828. Most financial
calculators have e programmed as an internal function. The equation can be restated as
F = PeRt
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Compound Annual Return:
The compound annual rate is a more useful return measure than the holding period
return. It takes account of the time value of money and the fact that investment
horizons are not always the same. This is also called the effective annual rate.
Supports an investor paid $40 for a non-and dividend paying stock for 4.5 years ago.
Today the stock sells for $78. Assuming no stock splits along the way, the holding
period return is ($78 – $40)/ $40 = 95%.After 4.5 years have passed, the 95% figure
probably lacks a frame of reference for performance comparison purposes. Because
we are accustomed to thinking of interest rates per year, we usually look at annual
returns to provide that frame of reference. The compound annual return is the annual
interest rate that makes the time value of money relationship hold: $40(1 + R)
4.5
=
$78. In this equation, R is 16%, a meaningful number. It tells us that if the $40 had
been invested at 16%, after 4.5 years the investment would be worth $78. The
compound annual returns on competing investments can be directly compared.
A danger with compound annual returns, however, stems from computing them over
short
periods of time. Suppose Wal-Mart closes today at $51, a $1.00 from yesterday’s
close. What is the compound annual return? Solve for R in the equation $50(1 + R)
1/365
= $51.
The answer is 137,641%! Associating this annual rate with your $50 Wal-Mart stock
means that in 12 months, a share would be worth $68,870 — not a likely scenario.
A recent new story provides a useful example of the importance of associating time
with returns. In January 1928, Julia Ford Bundy Blue, a widow of one of the founders
of international business machines, bequeathed 100 shares of IBM trust on behalf of a
retirement home Altadena, California. At that time and the stock sold for $123 per
share, making the bequest worth $12,300. 66 years later the trust dissolved and paid
$4.5 million to the retirement home. The Associated Press reported this story with the
headline, “66 year old IBM stock yields 36,600 percent.” The headline creates two
problems here. First is the incorrect use of the terms yield. Increase in principle value
is not part of yield. Second, the appreciation occurred over 66 years, so the 36,600%
figure needs to be translated to frame of reference terms. A fund that began with a
value of $12,300 and 66 years later was worth $4.5 million showed a compound
annual return of 9.35% per year over the same period. The latter figure, however, does
not make headlines.
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