CFA Level 1 (2009) - 1

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #6 - Discounted Cash Flow Applications

11.Which of the following statements least likely represents a characteristic of the time-weighted rate of return? It is:

A.not affected by the timing of cash flows.

B.used to measure the compound rate of growth of $1 over a stated measurement period.

C.defined as the internal rate of return on an investment portfolio, taking into account all inflows and outflows.

Use the foUowing data to answer Questions 12 and 13.

Assume an investor purchases a share of stock for $50 at time t = 0, and another share at $65 at time t = 1, and at the end of year 1 and year 2, the stock paid a $2.00 dividend. Also, at the end of year 2, the investor sold both shares for $70 each.

12.The money-weighted rate of return on the investment is:

A.15.45%.

B.16.73%. C. 18.02%.

13.The time-weighted rate of return on the investment is:

A.18.27%.

B.20.13%.

C.21.83%.

14.What is the bank discoum yield for a T-bill that is selling for $99.000, with a face value of $1 00,000, and 95 days remaining until maturity?

A.1.51%.

B.3.79%.

C.6.00%.

15.What is the holding period yield for a T-bill that is selling for $99,000 if it has a face value of $100,000 and 95 days remaining until maturity?

A.1.01%.

B.2.03%.

C.3.79%.

16. What is the effective annual yield for a T-bill that is selling for $99,000 if it has a face value of $100,000 and 95 days remaining until maturity?

A.3.79%.

B.3.94%.

C.4.50%.

17.What is the money market yield for a T-bill that is selling for $99,000 if it has a face value of $1 00,000 and 95 days remaining until maturity?

A.3.79%.

B.3.83%.

C.3.90%.

18.Which of the following is least accurate regarding a bank discount yield?

A.It ignores the opportunity to earn compound interest.

B. It is based on the face value of the bond, not its purchase price.

C.It reflects the nonannualized return an investor will earn over a security's life.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #6 - Discounted Cash Flow Applications

19. A 175-day T-bill has an effective annual yield of 3.80%. Its money-market yield

COMPREHENSIVE PROBLEMS

1.Allison Rogers, CPA, makes the following statement: "The problems with

bank discount yields quoted for T-bills is that they aren't yields, they ignore com pounding, and they are based on a shon year."

A.Is she correct in all regards?

B.Which of these problems is/are remedied by using the holding period yield?

C.Which of these problems is/arc remedied by using a money market yield? D. \'\!hich of rhest problems is/are remedied by using effecrive annual yields?

2.L. Adams buys 1,000 shares of Morris Tool stock for oS35 per share. One year later the srock is $38 per share and has paid a dividend of $1.50 per share. Adams reinvests the dividends in additional shares ar the time. At the end of the second year, the shares are trading for $37 and have paid $2 dividends over the period.

L. Burns buys 500 shares of Morris Tool stock for $35 per share. One year later the stock is $38 per share and has paid a dividend of $1.50 per share. Burns reinvests the dividends in additional shares ar that time and "llso buys 500 additional shares. Ar the end of the second year, the shares arc trading for $37 and have paid $2 in dividends over the period.

A.Compare the annual time-weighted returns for the accounts of the rwo investors (no calculation required).

B.Compare the annual money-weighted returns for the accounts of the rwo investors (no calculation required).

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #6 - Discounted Cash Flow Applications

ANS\X1ERS - CONCEPT CHECKERS

I.B If the NPY and IRR methods give conflicting decisions when selecting among mutually exclusive projects, always seleCt the project with the greatest positive NPY.

") C A projecr will have a negative NPY if irs IRR is less rhan the firm's cost of capital.

3.C When selecting between two mutually exclusive projecrs, neither project should be accepted if they both have a negative NPY.

4.A NPY = 4/1.10 + 31 (1.10)2 + 41 (1.10)3 - $10 = -$0.879038 million, or -$879,038.

Calculator approach: CFO = -10; CF I = 4; CF2 = 3; CF3 = 4; I = 10 -> NPY =

-$0.879038 (million).

'i. A Use your resr-taking skills here. You kno\\" from lhe previous question rhar rhe NI'\' is negarive a[ 10'}". Thus, the ]RI\ musr bL'less than 10<\". This leaves only choice:\

rhe answt:r. Calculawf solurion; IRIZ ~

(I. C r~orh projects should be accepted since increase shareholder wealth.

BWhen rhe NPY and IRR rankings conflict. alwavs selecr the project wirh rhe highest posirive NP\, in order to maximize shareholder wealrh.

8.A NPV = PY(cash inflows) - CFo = ($1.8 million 1 0.105) - $] 5 million = $2,142,857. Accept the project.

9.C As a perpetuiry, the following relationship applies: $1.8 million / IRR = $15 million. Thus, IRR = 1.8/ 15 = 12%. Since IRR > cost of capital (hurdle rare), accept the project.

10.B The project should be accepted on the basis of irs positive NPY and its IRR, which exceeds the cost of capitaL

J I. C The money-weighted rate of return is the IRR of an investment's net cash flows.

12. C One way to do this problem is to set up the cash flows so that the PY of inflows = PYof outflows and plug in each of the multiple choices. 50 + 65 1 (l + r) = 21 (l + r) +

1441 (l + 1')2 -> r = 18.02%. Or on your financial calculator, solve for IRR:

-50- 65-2 + 2(70+2) =0

I+IRR (1+IRR)2

Calculating MoneyWeighted Retum With the TJ Business Analyst JJ Plus®

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #6 - Discounted Cash Flow Applications

13.C HPR1 = (65 + 2) / 50 - 1= 34%, HPR2 = (140 + 4) / 130 - 1 = 10.77% Time-weighted return = [(1.34)(1.1077)]°·5 - 1 = 21.83%

14.B (1,000 / 100,000) x (360 / 95) = 3.79%

15.A (100,000 - 99,000) / 99,000 = 1.01 %

16.B (I + 0.0101)365/95 - 1 = 3.94%

17.B (360 x 0.0379) / (360 - (95 x 0.0379}) = 3.83%, or (1,000/99,000)(360/ 95) = 3.83%

18.C This is actually the defini cion of the holding period yield. The other answers are true statements regarding the bank discount yield.

365

19, C Since the effective yield is 3.8%, we know [l,~oo]m = 1.038 and

AJternatively, we can get the HPY from the EAY of 3.8!!-!; as (1.038) 36' -1 = 1.804%.

ANSWERS - COMPREHENSIVE PROBLEMS

1.A. She is correct in all regards. Bank discount yields are not true yields because they are based on a percentage of face (maturity) value instead of on the original amount

invested. They are annualized without compounding since the actual discount from face value is simply multiplied by the number of periods in a "year." The "year" used is 360 days, so that is a shortcoming as well.

B.The holding period yield uses the increase in value divided by the amount invested (purchase price), so it solves the problem that the BOY is not a true yield.

C.The money market yield is also a true yield (a percentage of the initial investment), but does not solve the other two problems since it does not involve compounding and is based on a 360-day year.

D, The effective annual yield solves all three shortcomings. It is based on the holding period yield (so it is a true yield), is a compound annual rate, and is based on a 365-day year.

2.A. Both investors have held the same single srock for both periods, so the time-weighted returns must be identical for both accounts.

B.The performance of the stOck (annual total return) was better in the first year than in the second. Since Burns increased his holdings for the second period by more than Adams, the Burns account has a greater weight on the poorer returns in a money-weighted returns calculation and will have a lower annual money-weighted rate of return over the two-year period than Adams.

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The following is a review of the Quantitative Methods: Baliic Concepts principles designed to address the learning outcome statements set forth by CFA lnstitute®. This topic is also covered in:

STATISTICAL CONCEPTS AND MARKET RETURNS

Study Session 2

EXAM Focus

LOS 7.a: Differentiate between descriptive statistics and inferential statistics, between a population and a sample, and among the types of measurement scales.

The word statistics is used to refer to data (e.g., the average return on XYZ stock was 8% over the last ten years) and to the methods we use to analyze data. Statistical methods fall into one of two categories, descriptive statistics or inferential statistics.

DescriptizJe statistiC.' are used ro summarize the important characteristics of large data sets. The focus of this topic review is on the use of descriptive statistics to consolidate a mass of numerical data inro useful information.

Inferential statistics, which will be discussed in subseguent ropic reviews, pertain to the procedures used to make forecasts, estimates, or judgments about a large set of dara on the basis of the statistical characteristics of a smaller set (a sample).

A population is defined as the set of all possible members of a stated group. A crosssection of the returns of all of the stocks traded on the New York Srock Exchange (NYSE) is an example of a population.

It is freguendy roo costly or time consuming to obtain measurements for every member of a population, ifit is even possible. In this case, a sample may be used. A sample is defined as a subset of the population of interest. Once a population has been defined,

a sample can be drawn from the population, and the sample's characteristics can be used to describe the population as a whole. For example, a sample of 30 stocks may be selected from among all of the stocks listed on the NYSE to represent the population of all NYSE-traded stocks.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

Types of Measurement Scales

Different statistical methods use different levels of measurement, or measurement scales. Measurement scales may be classified into one of four major categories:

•Nominal scales. Nominal scales are the least accurate level of measurement. Observations are classified or counted with no parricular order. An example would be assigning the number 1 to a municipal bond fund, the number 2 to a corporate bond fund, and so on for each fund style.

•Ordinal scales. Ordinal scales represent a higher level of measurement than nominal scales. When working with an ordinal scale, every observation is assigned

to one of several categories. Then these categories are ordered with respect to a specified characteristic. For example, the ranking of 1,000 small cap growth

stocks by performance may be done by assigning the number 1 to the 100 best performing stocks, the number 2 to the next 100 best performing stocks, and so on, a~~igning the number 10 to the 100 worst performing stocks. Based on this type

of measurement, it can be concluded that a stock ranked 3 i~ better than a stock ranked 4. but the ~cale reveals nothing about performance differences or whcther the difference between a 3 and a 4 is the same a~ the difference between a 4 and a 5.

•interval scale. Intetval scale measurements provide relative ranking, like ordinal scales, plus the assurance that differences between scale values are equal. Temperature

mea~uremenrin degrees is a prime example. Cerrainly, 49°e i~ hotter than 32°C, and the temperature difference between 49°(: and 32°e is the same as the difference between Gloe and 50°C. The weakness of the inrerval ~cale is that a measurement of zero does not necessarily indicate the total absence of what we are measuring. This means that interval-seale-based ratios are meaningless. For example, 30°F is nor three times as hot as 100 E

•Ratio scales. Ratio scales represenr the most refined level of measurement. Ratio scales provide ranking and equal differences between scale values, and they also have a true zero point as the origin. Order, intervals, and ratios all make sense with a ratio scale. The measurement of money is a good example. If you have zero dollars, you have no purchasing power, bur if you have $4.00, you have twice as much purchasing power as a person with $2.00.

~Professor's Note: Candidates sometimes use the French word for black, nair, to

~remember the types ofscales in order o.{precision.' Nominal, Ordinal, Interval, Ratio.

LOS 7.b: Explain a parameter, a sample statistic, and a frequency distribution.

A measure used ro describe a characteristic of a population is referred to as a parameter. While many population parameters exist, investment analysis usually utilizes just a few, particularly the mean rerurn and the standard deviation of rerurns.

In the same manner that a parameter may be used to describe a characteristic of a population, a sample statistic is used ro measure a characteristic of a sample.

A frequency distribution is a tabular presentation of statistical data that aids the analysis of large data sets. Frequency distributions summarize statistical data by assigning it ro

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #7 - Sraristical Concepts and Market Returns

specified groups, or intervals. Also, the data employed with a frequency distribution may be measured using any type of measurement scale.

o Professor's Note: Intervals are also known as classes.

The following procedure describes how to construct a frequency distribution.

Step 1: Define the intervals. The first step in building a freq uency distribution is to define the intervals to which data measurements (observations) will be assigned. An interval, also referred to as a class, is the set of values that an observation may take on. The range of values for each interval must have a lower and upper limit and be all-inclusive and nonoverlapping. Intervals must be mutuaLLy exclusive in a way that each observation can be placed in only one interval. and the total set of intervals should cover the roral range of values for the entire population.

The number of inrcl"\'als used is an importanr consideration. If roo fe\\ inrcf\'als are used, the dara mav be roo broadly wmmarized, and imporrant characreristics may be lost. On rhe orher hand, if too many inrervals are used, rhe dara may not be summarized enough.

Step 2: Tally the observatlOllS, After the inrervals have been defined, rhe observations must be rallied, or assigned ro rheir appropriate interval.

Step 3: Count the observations. Having tallied the data set, the number of observations that are assigned to each interval must be counted. The absolute frequency, or simply the frequency, is rhe actual number of observations that fall within a given interval.

Useme,data,jnTible A to construct a freqtiengdistdoution for thereturns on Imelco's

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

.

TableB: TalIYaildIJitervil{~~iIDif~rRe~urnsData·

Absolute Frequency·

2

3

7

3

2

20

Tallying and counting the observations generates a frequency distribution that

summarizes the pattern of annual returns on InteIco common stock. Notice tl1at the .. , interval with the greatest (absolute) freq~encyis the (0% S; Rr < 10%)imerval,w-hiCh

includes seven return observations. Par my frequency distribution, the interval with the greatest frequency is referred to as the modal interval.

LOS 7.e: Calculate and interpret relative frequencies and cumulative relative frequencies, given a frequency distribution, and describe the properties of a dataset presented as a histogram or a frequency polygon.

The relative frequency is another useful way to present data. The relative frequency is calculated by dividing the absolute frequency of each return interval by the total number of observations. Simply stated, relative frequency is the percentage of total observations falling within each interval. Continuing with our example, the relative frequencies are presented in Figure 1.

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

Figure I: Absolute and Relative Frequencies of Inte1co Returns

Ir is also possible ro compute the cumulative absolute frequency and cumulative relative frequency by summing the absolure or relarive frequencies srarting at rhe lowesr inrerval and progressing through the highest. The cumulative absolute frequencies and cumulative relative frequencies for the lntdco srock returns example are presented in Figure 2.

Figure 2: Absolute, Relative, and Cumulative Frequencies of Imelco Returns

Notice that the cumulative absolute frequency or cumulative relative frequency for any given interval is the sum of the absolute or relative frequencies up to and including the given interval. For example, the cumulative absolute frequency for the (0% :s Rt < 10%) interval is 13 = 1 + 2 + 3 + 7 and the cumulative relative frequency for this interval is 5%

+ 10% + 15% + 35% = 65%.

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #7 - Statistical Concepts and Market Returns

HISTOGRAMS AND FREQUENCY POLYGONS

A histogram is the graphical presenration of the absolute frequency disrribution. A hisrogram is simply a bar charr of conrinuous data that has been classified inro a frequency disrriburion. The anractive feature of a hisrogram is that it allows us ro quickly see where most of the observations are concenrrated.

To construct a histogram, the inrervals are scaled on the horizonral axis and the absolute frequencies are scaled on the verrical axis. The hisrogram for the Inrelco rerurns data in Table B from the previous example is provided in Figure 3.

Figure 3: Histogram of Intelco Stock Return Data

7

6

5

>.

U

c

u~ -4

Q.)

u: 3

2

Interval

To construct a frequency polygon, the midpoint of each interval is ploned on the horizonral axis, and the absolute frequency for that inrerval is planed on the verrical axis. Each point is then connected with a srraight line. The frequency polygon for the retUrns data used in our example is illusrrated in Figure 4.