Financial Econometrics

Chapter 1 Asset Return Calculations

Eric Zivot

Department of Economics, University of Washington

December 31, 1998

Updated: January 7, 2002

1 The Time Value of Money

Consider an amount $V invested for n years at a simple interest rate of R

per annum (where R is expressed as a decimal). If compounding takes place

only at the end of the year the future value after n years is

F Vn = $V Ā· (1 + R)n .

Example 1 Consider putting $1000 in an interest checking account that

pays a simple annual percentage rate of 3%. The future value after n = 1, 5

and 10 years is, respectively,

F V1 = $1000 Ā· (1.03)1 = $1030

F V5 = $1000 Ā· (1.03)5 = $1159.27

F V10 = $1000 Ā· (1.03)10 = $1343.92.

If interest is paid m time per year then the future value after n years is

Āµ Ā¶mĀ·n

R

m

= $V Ā· 1 +

F Vn .

m

1

R

is often referred to as the periodic interest rate. As m, the frequency of

m

compounding, increases the rate becomes continuously compounded and it

can be shown that future value becomes

Āµ Ā¶mĀ·n

R

c

= $V Ā· eRĀ·n ,

= mā’ā $V Ā· 1 +

F Vn lim

m

where e(Ā·) is the exponential function and e1 = 2.71828.

Example 2 If the simple annual percentage rate is 10% then the value of

$1000 at the end of one year (n = 1) for diļ¬erent values of m is given in the

table below.

Compounding Frequency Value of $1000 at end of 1 year (R = 10%)

Annually (m = 1) 1100

Quarterly (m = 4) 1103.8

Weekly (m = 52) 1105.1

Daily (m = 365) 1105.515

Continuously (m = ā) 1105.517

We now consider the relationship between simple interest rates, periodic

rates, eļ¬ective annual rates and continuously compounded rates. Suppose

an investment pays a periodic interest rate of 2% each quarter. This gives

rise to a simple annual rate of 8% (2% Ć—4 quarters). At the end of the year,

$1000 invested accrues to

Āµ Ā¶4Ā·1

0.08

$1000 Ā· 1 + = $1082.40.

4

The eļ¬ective annual rate, RA , on the investment is determined by the rela-

tionship

$1000 Ā· (1 + RA ) = $1082.40,

which gives RA = 8.24%. The eļ¬ective annual rate is greater than the simple

annual rate due to the payment of interest on interest.

The general relationship between the simple annual rate R with payments

m time per year and the eļ¬ective annual rate, RA , is

Āµ Ā¶mĀ·1

R

(1 + RA ) = 1 + .

m

2

Example 3 To determine the simple annual rate with quarterly payments

that produces an eļ¬ective annual rate of 12%, we solve

Āµ Ā¶

R4

1.12 = 1+ =ā’

4

Ā³ Ā“

1/4

(1.12) ā’ 1 Ā· 4

R=

0.0287 Ā· 4

=

= 0.1148

Suppose we wish to calculate a value for a continuously compounded rate,

Rc , when we know the mā’period simple rate R. The relationship between

such rates is given by Āµ Ā¶

Rm

Rc

(1)

e = 1+ .

m

Solving (1) for Rc gives

Āµ Ā¶

R

(2)

Rc = m ln 1 + ,

m

and solving (1) for R gives

Ā³ Ā“

Rc /m

ā’1 . (3)

R=m e

Example 4 Suppose an investment pays a periodic interest rate of 5% every

six months (m = 2, R/2 = 0.05). In the market this would be quoted as

having an annual percentage rate of 10%. An investment of $100 yields

$100 Ā· (1.05)2 = $110.25 after one year. The eļ¬ective annual rate is then

10.25%. Suppose we wish to convert the simple annual rate of R = 10% to

an equivalent continuously compounded rate. Using (2) with m = 2 gives

Rc = 2 Ā· ln(1.05) = 0.09758.

That is, if interest is compounded continuously at an annual rate of 9.758%

then $100 invested today would grow to $100 Ā· e0.09758 = $110.25.

2 Asset Return Calculations

3

2.1 Simple Returns

Let Pt denote the price in month t of an asset that pays no dividends and

let Ptā’1 denote the price in month t ā’ 11 . Then the one month simple net

return on an investment in the asset between months t ā’ 1 and t is deļ¬ned

as

Pt ā’ Ptā’1

(4)

Rt = = %āPt .

Ptā’1

Pt ā’Ptā’1 Pt

Writing ā’ 1, we can deļ¬ne the simple gross return as

=

Ptā’1 Ptā’1

Pt

(5)

1 + Rt =

.

Ptā’1

Notice that the one month gross return has the interpretation of the future

value of $1 invested in the asset for one month. Unless otherwise stated,

when we refer to returns we mean net returns.

(mention that simple returns cannot be less than 1 (100%) since prices

cannot be negative)

Example 5 Consider a one month investment in Microsoft stock. Suppose

you buy the stock in month t ā’ 1 at Ptā’1 = $85 and sell the stock the next

month for Pt = $90. Further assume that Microsoft does not pay a dividend

between months t ā’ 1 and t. The one month simple net and gross returns are

then

$90 ā’ $85 $90

ā’ 1 = 1.0588 ā’ 1 = 0.0588,

Rt = =

$85 $85

1 + Rt = 1.0588.

The one month investment in Microsoft yielded a 5.88% per month return.

Alternatively, $1 invested in Microsoft stock in month t ā’ 1 grew to $1.0588

in month t.

2.2 Multi-period returns

The simple two-month return on an investment in an asset between months

t ā’ 2 and t is deļ¬ned as

Pt ā’ Ptā’2 Pt

ā’ 1.

Rt (2) = =

Ptā’2 Ptā’2

1

We make the convention that the default investment horizon is one month and that

the price is the closing price at the end of the month. This is completely arbitrary and is

used only to simplify calculations.

4

Ptā’1

Pt Pt

Since Ā· the two-month return can be rewritten as

=

Ptā’2 Ptā’1 Ptā’2

Pt Ptā’1

Ā· ā’1

Rt (2) =

Ptā’1 Ptā’2

= (1 + Rt )(1 + Rtā’1 ) ā’ 1.

Then the simple two-month gross return becomes

1 + Rt (2) = (1 + Rt )(1 + Rtā’1 ) = 1 + Rtā’1 + Rt + Rtā’1 Rt ,

which is a geometric (multiplicative) sum of the two simple one-month gross

returns and not the simple sum of the one month returns. If, however, Rtā’1

and Rt are small then Rtā’1 Rt ā 0 and 1 + Rt (2) ā 1 + Rtā’1 + Rt so that

Rt (2) ā Rtā’1 + Rt .

In general, the k-month gross return is deļ¬ned as the geometric average

of k one month gross returns

1 + Rt (k) = (1 + Rt )(1 + Rtā’1 ) Ā· Ā· Ā· (1 + Rtā’k+1 )

kā’1

Y

= (1 + Rtā’j ).

j=0

Example 6 Continuing with the previous example, suppose that the price of

Microsoft stock in month t ā’ 2 is $80 and no dividend is paid between months

t ā’ 2 and t. The two month net return is

$90 ā’ $80 $90

ā’ 1 = 1.1250 ā’ 1 = 0.1250,

Rt (2) = =

$80 $80

or 12.50% per two months. The two one month returns are

$85 ā’ $80

= 1.0625 ā’ 1 = 0.0625

Rtā’1 =

$80

$90 ā’ 85

= 1.0588 ā’ 1 = 0.0588,

Rt =

$85

and the geometric average of the two one month gross returns is

1 + Rt (2) = 1.0625 Ć— 1.0588 = 1.1250.

5

2.3 Annualizing returns

Very often returns over diļ¬erent horizons are annualized, i.e. converted to

an annual return, to facilitate comparisons with other investments. The an-

nualization process depends on the holding period of the investment and an

implicit assumption about compounding. We illustrate with several exam-

ples.

To start, if our investment horizon is one year then the annual gross and

net returns are just

Pt

= (1 + Rt )(1 + Rtā’1 ) Ā· Ā· Ā· (1 + Rtā’11 ),

1 + RA = 1 + Rt (12) =

Ptā’12

Pt

ā’ 1 = (1 + Rt )(1 + Rtā’1 ) Ā· Ā· Ā· (1 + Rtā’11 ) ā’ 1.

RA =

Ptā’12

In this case, no compounding is required to create an annual return.

Next, consider a one month investment in an asset with return Rt . What

is the annualized return on this investment? If we assume that we receive

the same return R = Rt every month for the year then the gross 12 month

or gross annual return is

1 + RA = 1 + Rt (12) = (1 + R)12 .

Notice that the annual gross return is deļ¬ned as the monthly return com-

pounded for 12 months. The net annual return is then

RA = (1 + R)12 ā’ 1.

Example 7 In the ļ¬rst example, the one month return, Rt , on Microsoft

stock was 5.88%. If we assume that we can get this return for 12 months then

the annualized return is

RA = (1.0588)12 ā’ 1 = 1.9850 ā’ 1 = 0.9850

or 98.50% per year. Pretty good!

Now, consider a two month investment with return Rt (2). If we assume

that we receive the same two month return R(2) = Rt (2) for the next 6 two

month periods then the gross and net annual returns are

1 + RA = (1 + R(2))6 ,

RA = (1 + R(2))6 ā’ 1.

6

Here the annual gross return is deļ¬ned as the two month return compounded

for 6 months.

Example 8 In the second example, the two month return, Rt (2), on Mi-

crosoft stock was 12.5%. If we assume that we can get this two month return

for the next 6 two month periods then the annualized return is

RA = (1.1250)6 ā’ 1 = 2.0273 ā’ 1 = 1.0273

or 102.73% per year.

To complicate matters, now suppose that our investment horizon is two

years. That is we start our investment at time t ā’ 24 and cash out at time

Pt

t. The two year gross return is then 1 + Rt (24) = Ptā’24 . What is the annual

return on this two year investment? To determine the annual return we solve

the following relationship for RA :

(1 + RA )2 = 1 + Rt (24) =ā’

RA = (1 + Rt (24))1/2 ā’ 1.

In this case, the annual return is compounded twice to get the two year

return and the relationship is then solved for the annual return.

Example 9 Suppose that the price of Microsoft stock 24 months ago is

Ptā’24 = $50 and the price today is Pt = $90. The two year gross return is

1+Rt (24) = $90 = 1.8000 which yields a two year net return of Rt (24) = 80%.

$50

The annual return for this investment is deļ¬ned as

RA = (1.800)1/2 ā’ 1 = 1.3416 ā’ 1 = 0.3416

or 34.16% per year.

2.4 Adjusting for dividends

If an asset pays a dividend, Dt , sometime between months t ā’ 1 and t, the

return calculation becomes

Pt + Dt ā’ Ptā’1 Pt ā’ Ptā’1 Dt

Rt = = +

Ptā’1 Ptā’1 Ptā’1

where PtPtā’1 is referred as the capital gain and

ā’Ptā’1 Dt

is referred to as the

Ptā’1

dividend yield.

7

3 Continuously Compounded Returns

3.1 One Period Returns

Let Rt denote the simple monthly return on an investment. The continuously

compounded monthly return, rt , is deļ¬ned as

Ć !

Pt

(6)

rt = ln(1 + Rt ) = ln

Ptā’1

where ln(Ā·) is the natural log function2 . To see why rt is called the con-

tinuously compounded return, take the exponential of both sides of (6) to

give

Pt

ert = 1 + Rt = .

Ptā’1

Rearranging we get

Pt = Ptā’1 ert ,

so that rt is the continuously compounded growth rate in prices between

months t ā’ 1 and t. This is to be contrasted with Rt which is the simple

growth rate in prices between months t ā’ 1 and t without any compounding.

Ā³Ā“

Furthermore, since ln x = ln(x) ā’ ln(y) it follows that

y

Ć !

Pt

rt = ln

Ptā’1

= ln(Pt ) ā’ ln(Ptā’1 )

= pt ā’ ptā’1

where pt = ln(Pt ). Hence, the continuously compounded monthly return, rt ,

can be computed simply by taking the ļ¬rst diļ¬erence of the natural loga-

rithms of monthly prices.

Example 10 Using the price and return data from example 1, the continu-

ously compounded monthly return on Microsoft stock can be computed in two

ways:

rt = ln(1.0588) = 0.0571

2

The continuously compounded return is always deļ¬ned since asset prices, Pt , are

always non-negative. Properties of logarithms and exponentials are discussed in the ap-

pendix to this chapter.

8

or

rt = ln(90) ā’ ln(85) = 4.4998 ā’ 4.4427 = 0.0571.

Notice that rt is slightly smaller than Rt . Why?

Given a monthly continuously compounded return rt , is straightforward

to solve back for the corresponding simple net return Rt :

Rt = ert ā’ 1

Hence, nothing is lost by considering continuously compounded returns in-

stead of simple returns.

Example 11 In the previous example, the continuously compounded monthly

return on Microsoft stock is rt = 5.71%. The implied simple net return is then

Rt = e.0571 ā’ 1 = 0.0588.

Continuously compounded returns are very similar to simple returns as

long as the return is relatively small, which it generally will be for monthly or

daily returns. For modeling and statistical purposes, however, it is much more

convenient to use continuously compounded returns due to the additivity

property of multiperiod continuously compounded returns and unless noted

otherwise from here on we will work with continuously compounded returns.

3.2 Multi-Period Returns

The computation of multi-period continuously compounded returns is con-

siderably easier than the computation of multi-period simple returns. To

illustrate, consider the two month continuously compounded return deļ¬ned

as Ć !

Pt

= pt ā’ ptā’2 .

rt (2) = ln(1 + Rt (2)) = ln

Ptā’2

Taking exponentials of both sides shows that

Pt = Ptā’2 ert (2)

9

so that rt (2) is the continuously compounded growth rate of prices between

months t ā’ 2 and t. Using PPt = PPt Ā· Ptā’1 and the fact that ln(x Ā· y) =

Ptā’2

tā’2 tā’1

ln(x) + ln(y) it follows that

Ć !

Pt Ptā’1

Ā·

rt (2) = ln

Ptā’1 Ptā’2

Ć ! Ć !

Pt Ptā’1

= ln + ln

Ptā’1 Ptā’2

= rt + rtā’1 .

Hence the continuously compounded two month return is just the sum of the

two continuously compounded one month returns. Recall that with simple

returns the two month return is of a multiplicative form (geometric average).

Example 12 Using the data from example 2, the continuously compounded

two month return on Microsoft stock can be computed in two equivalent ways.

The ļ¬rst way uses the diļ¬erence in the logs of Pt and Ptā’2 :

rt (2) = ln(90) ā’ ln(80) = 4.4998 ā’ 4.3820 = 0.1178.

The second way uses the sum of the two continuously compounded one month

returns. Here rt = ln(90) ā’ ln(85) = 0.0571 and rtā’1 = ln(85) ā’ ln(80) =

0.0607 so that

rt (2) = 0.0571 + 0.0607 = 0.1178.

Notice that rt (2) = 0.1178 < Rt (2) = 0.1250.

The continuously compounded kā’month return is deļ¬ned by

Ć !

Pt

= pt ā’ ptā’k .

rt (k) = ln(1 + Rt (k)) = ln

Ptā’k

Using similar manipulations to the ones used for the continuously com-

pounded two month return we may express the continuously compounded

kā’month return as the sum of k continuously compounded monthly returns:

kā’1

X

rt (k) = rtā’j .

j=0

The additivitity of continuously compounded returns to form multiperiod

returns is an important property for statistical modeling purposes.

10

3.3 Annualizing Continuously Compounded Returns

Just as we annualized simple monthly returns, we can also annualize contin-

uously compounded monthly returns.

To start, if our investment horizon is one year then the annual continu-

ously compounded return is simply the sum of the twelve monthly continu-

ously compounded returns

rA = rt (12) = rt + rtā’1 + Ā· Ā· Ā· + rtā’11

11

X

= rtā’j .

j=0

Deļ¬ne the average continuously compounded monthly return to be

11

1X

rm = rtā’j .

12 j=0

Notice that

11

X

12 Ā· rm = rtā’j

j=0

so that we may alternatively express rA as

rA = 12 Ā· rm .

That is, the continuously compounded annual return is 12 times the average

of the continuously compounded monthly returns.

Next, consider a one month investment in an asset with continuously

compounded return rt . What is the continuously compounded annual return

on this investment? If we assume that we receive the same return r = rt

every month for the year then rA = rt (12) = 12 Ā· r .

4 Further Reading

This chapter describes basic asset return calculations with an emphasis on

equity calculations. Campbell, Lo and MacKinlay provide a nice treatment

of continuously compounded returns. A useful summary of a broad range

of return calculations is given in Watsham and Parramore (1998). A com-

prehensive treatment of ļ¬xed income return calculations is given in Stigum

(1981) and the oļ¬cial source of ļ¬xed income calculations is āThe Pink Bookā.

11

5 Appendix: Properties of exponentials and

logarithms

The computation of continuously compounded returns requires the use of

natural logarithms. The natural logarithm function, ln(Ā·), is the inverse of

the exponential function, e(Ā·) = exp(Ā·), where e1 = 2.718. That is, ln(x) is

deļ¬ned such that x = ln(ex ). Figure xxx plots ex and ln(x). Notice that ex

is always positive and increasing in x. ln(x) is monotonically increasing in x

and is only deļ¬ned for x > 0. Also note that ln(1) = 0 and ln(ā’ā) = 0. The

exponential and natural logarithm functions have the following properties

1. ln(x Ā· y) = ln(x) + ln(y), x, y > 0

2. ln(x/y) = ln(x) ā’ ln(y), x, y > 0

3. ln(xy ) = y ln(x), x > 0

d ln(x) 1

4. = x, x > 0

dx

d 1d

5. (chain-rule)

ln(f (x)) = f (x)

ds f (x) dx

6. ex ey = ex+y

7. ex eā’y = exā’y

8. (ex )y = exy

9. eln(x) = x

dx

= ex

10. e

dx

d f (x) d

= ef (x) dx f (x) (chain-rule)

11. e

dx

6 Problems

Exercise 6.1 Excel exercises

Go to http://finance.yahoo.com and download monthly data on Mi-

crosoft (ticker symbol msft) over the period December 1996 to December

2001. See the Project page on the class website for instructions on how to

12

download data from Yahoo. Read the data into Excel and make sure to re-

order the data so that time runs forward. Do your analysis on the monthly

closing price data (which should be adjusted for dividends and stock splits).

Name the spreadsheet tab with the data ādataā.

1. Make a time plot (line plot in Excel) of the monthly price data over the

period (end of December 1996 through (end of) December 2001. Please

put informative titles and labels on the graph. Place this graph in a

separate tab (spreadsheet) from the data. Name this tab āgraphsā.

Comment on what you see (eg. price trends, etc). If you invested

$1,000 at the end of December 1996 what would your investment be

worth at the end of December 2001? What is the annual rate of return

over this ļ¬ve year period assuming annual compounding?

2. Make a time plot of the natural logarithm of monthly price data over

the period December 1986 through December 2000 and place it in the

āgraphā tab. Comment on what you see and compare with the plot of

the raw price data. Why is a plot of the log of prices informative?

3. Using the monthly price data over the period December 1996 through

December 2001 in the ādataā tab, compute simple (no compounding)

monthly returns (Microsoft does not pay a dividend). When computing

returns, use the convention that Pt is the end of month closing price.

Make a time plot of the monthly returns, place it in the āgraphsā tab

and comment. Keep in mind that the returns are percent per month

and that the annual return on a US T-bill is about 5%.

4. Using the simple monthly returns in the ādataā tab, compute simple

annual returns for the years 1996 through 2001. Make a time plot of the

annual returns, put them in the āgraphsā tab and comment. Note: You

may compute annual returns using overlapping data or non-overlapping

data. With overlapping data you get a series of annual returns for every

month (sounds weird, I know). That is, the ļ¬rst month annual return

is from the end of December, 1996 to the end of December, 1997. Then

second month annual return is from the end of January, 1997 to the

end of January, 1998 etc. With non-overlapping data you get a series of

5 annual returns for the 5 year period 1996-2001. That is, the annual

return for 1997 is computed from the end of December 1996 through

13

the end of December 1997. The second annual return is computed from

the end of December 1997 through the end of December 1998 etc.

5. Using the monthly price data over the period December 1996 through

December 2001, compute continuously compounded monthly returns

and place then in the ādataā tab. Make a time plot of the monthly

returns, put them in the āgraphsā tab and comment. Brieļ¬‚y compare

the continuously compounded returns to the simple returns.

6. Using the continuously compounded monthly returns, compute contin-

uously compounded annual returns for the years 1997 through 2001.

Make a time plot of the annual returns and comment. Brieļ¬‚y compare

the continuously compounded returns to the simple returns.

Exercise 6.2 Return calculations

Consider the following (actual) monthly closing price data for Microsoft

stock over the period December 1999 through December 2000

End of Month Price Data for Microsoft Stock

December, 1999 $116.751

January, 2000 $97.875

February, 2000 $89.375

March, 2000 $106.25

April, 2000 $69.75

May, 2000 $62.5625

June, 2000 $80

July, 2000 $69.8125

August, 2000 $69.8125

September, 2000 $60.3125

October, 2000 $68.875

November, 2000 $57.375

December, 2000 $43.375

1. Using the data in the table, what is the simple monthly return between

December, 1999 and January 2000? If you invested $10,000 in Microsoft

at the end of December 1999, how much would the investment be worth

at the end of January 2000?

14

2. Using the data in the table, what is the continuously compounded

monthly return between December, 1999 and January 2000? Convert

this continuously compounded return to a simple return (you should

get the same answer as in part a).

3. Assuming that the simple monthly return you computed in part (1)

is the same for 12 months, what is the annual return with monthly

compounding?

4. Assuming that the continuously compounded monthly return you com-

puted in part (2) is the same for 12 months, what is the continuously

compounded annual return?

5. Using the data in the table, compute the actual simple annual return

between December 1999 and December 2000. If you invested $10,000 in

Microsoft at the end of December 1999, how much would the investment

be worth at the end of December 2000? Compare with your result in

part (3).

6. Using the data in the table, compute the actual annual continuously

compounded return between December 1999 and December 2000. Com-

pare with your result in part (4). Convert this continuously com-

pounded return to a simple return (you should get the same answer

as in part 5).

7 References

References

[1] Campbell, J., A. Lo, and C. MacKinlay (1997), The Econometrics of

Financial Markets, Princeton University Press.

[2] Handbook of U.W. Government and Federal Agency Securities and Re-

lated Money Market Instruments, āThe Pink Bookā, 34th ed. (1990), The

First Boston Corporation, Boston, MA.

[3] Stigum, M. (1981), Money Market Calculations: Yields, Break Evens and

Arbitrage, Dow Jones Irwin.

15

[4] Watsham, T.J. and Parramore, K. (1998), Quantitative Methods in Fi-

nance, International Thomson Business Press, London, UK.

16