UNDERSTANDING RISK AND RETURN

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Investment Analysis & Portfolio Management (FIN630)

Lesson # 31

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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Investment Analysis & Portfolio Management (FIN630)

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

year later the account contains $97.50. Even though these situations might seem identical,

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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

Where P = present value (i.e., price today)

F = future value

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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)

(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

The present value of the $1,000 return of principle is

$1,000 = $703.51

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(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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