CFA Level 1 (2009) - 1

Study Session 2

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

Example: Time-weighted rate of return

Ashare of stock is purchased at t ::: 0 for $100, .andatthe end of the next year,

t ::: 1, another share is purchased for $120. Auheend ofyear 2, both shares are sold for $130 each.. At the end of both years 1 and 2, the stock paid a $2.00 per share dividend. What is the annual time-weighted rate of return for this investment? (This is the same investment as the preceding example.) . .

Answer:

. Step 1: . Break the. evaluation period inrotwosubperiodSbas~don timing of cash flo~s.

Step 2: Calculate the HPR for each holding period.

HPR j ::: [($120 + 2) / $100] - 1 = 22%

HPRz ::: [($260 + 4) / $240J - 1 = 10%

Step 3: Find the compound annual rate that would have produced a total return equal to the return on the account over the 2-year period.

(l + time-weighted rate of return)Z = (1.22)(1.10)

time-weighted rate of return ::: [(1.22)(1.10)]°·5_1 = 15.84%

This is the geometric meall return, which we examine in more detail later. This allows us to express the time-weighted return as an annual compound rate, even though we have cwo years of data. In the investment management industry, the time-weighted rate ofreturn is the preferred method ofpClformance measurement, because it is not affected bJ! the timing of cash inflows and outflows.

In the preceding examples, the time-weighted rate of return for the portfolio was 15.84%, while the money-weighted rate of return for the same portfolio was 13.86%. The results are different because the money-weighted rate of return gave a larger weight to the year-2 HPR, which was 10%, versus the 22% HPR for year 1. This is because there was more money in the account at the beginning of the second period.

If funds are contributed to an investment portfolio just before a period of relatively poor portfolio performance, the money-weighted rate of return will tend to be lower than the time-weighted rate of rerum. On the other hand, if funds are contributed to a portfolio at a favorable time (just prior to a period of relatively high returns), the money-weighted rate of return will be higher than the time-weighted rate of rerum. The use of the time-

StUdy Session 2

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

weighted return removes these distortions and thus provides a better measure of a manager's ability to select investments over the period. If the manager has complete control over money flows into and out of an account, the money-weighted rate of return would be the more appropriate performance measure.

LOS 6.d: Calculate and interpret the bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill; and convert among holding period yields, money market yields, effective annual yields, and bond equivalent yields.

Pure discount insuuments such as U.S. T-bills are quoted differently from U.S. government bonds. T-bills are quoted on a bank discount basis, which is based on the face vaLue of the instrument instead of the purchase price. The bank discount yield (BDY) is computed using the following formula:

the bill and the purchase price

Fthe face \'al ue (par value) of the bill number of days remaining until maturity

360 bank convention of number of days in a year

The key distinction of the bank discount yield is that it expresses the dollar discount from the face (par) value as a fraction of the face value, not the market price of the instrument. Another notable feature of the bank discount yield is that it is annualized by multiplying the discount-to-par by 3601t, where the market convention is to use

a 360-day year versus a 365-day year. This type of annualizing method assumes no compounding (i.e., simple interest).

Example: Bank discount yield

Calculate the bank discount yield for a T-bill priced at $98,500 with a face value of $100,000 and 120 days until maturity.

Answer:

Substituting the relevant values into the bank discount yield equation in our example, we get:

It is important for candidates to realize that a yield quoted on a bank discount basis is not representative of the return earned by an investor for the following reasons:

•Bank discount yield annualizes using simple interest and ignores the effects of compound interest.

Study Session 2

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

•Bank discount yield is based on the face value of the bond, not its purchase priceinvestment returns should be evaluated relative to the amount invested.

•Bank discount yield is annualized based on a 360-day year rather than a 365-day year.

Holding period yield (HPY) or holding period return, is the total return an investor earns between the purchase date and the sale or maturity date. HPY is calculated using the following formula:

HPY = p) - Po + 0) = PI + °1 _1

Example: HPY

\X'hat is the HPY for a T-bill priced at $98,500 with a face value of $100,000 and 120 days remaining until maturity?

Answer:

Using the HPY equation stated above, we have:

HPY = ($100,000 - $98,500) / $98,500

=$1,500/ $98,500

=1.5228%

0)= 0 here because T-bills are a pure discount instrument (i.e., they make no interest

payments).

The effective annual yield (EAY) is an annualized value, based on a 365-day year, that accounts for compound interest. It is calculated using the following equation:

EAY = (1 + HPy)365 / ( - 1

Example: EAY

Compute the EAY using the l20-day HPY of 1.5228% from the previous example.

Answer:

The HPY is converted to an EAY as follows:

EAY = (1.015228)3651120 - 1 = 1.047042 -1 = 4.7042%

Note that we can convert from an EAY to HPY by using the reciprocal of the

Study Session 2

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

The money market yield (or CD equivalent yield) is equal to the annualized holding period yield, assuming a 360-day year. Using the money market yield makes the quoted yield on a T-bill comparable to yield quotes for interest-bearing money market instruments that pay interest on a 360-day basis. The money marker yield is

360xHPY.

#days

rMM = HPY x (3601r)

Given the bank discount yield, rBD' the money market yield, rMM , may be calculated using the equation:

360 x rED

rMM =

360 - (t X rBD )

~Proftssor's Note: J coultlllf remember thisftrmula with (/ gUll tlll1~)' hMd, bm J

~(and.you) Clill emi!;), co/wert a BDV to an HPY and thr HPj" to tI 1/1(1//('.1' marker

yield using rht' prl'1,jowforJ7lll{a.f.

Example: Money market yield, rMM

What is the money market yield for a 120-clay T-bill that has a bank discount yield of 4.50%?

Answer:

Given the rBD for the T-bill, the first equation for rMM is applied as follows:

360xO.045 IMM= 360-(120xO.045)

== 16.2 I 354.6

.= 4.569%

Alternatively, we could first calculate the HPY for the T-bilJ and then annualize that.

Study Session 2 Cross-Reference to CFA Institute Assigned Reading #6 - Discounted Cash Flow Applications

Converting Among Holding Period Yields, Money Market Yields, and Effective Annual Yields

Once we have established HPY, EAY, or rMM , we can use one as a basis for calculating the other two. Remember:

The HPY is the actual rerurn an investor will receive if the money market instrument is held until maturity.

•The EAY is the annualized HPY on the basis of a 365-day year and incorporates the effects of compounding.

•The rMM is the annualized yield that is based on price and a 360-day year and does not account for the effects of compounding-it assumes simple interest.

.8ciurlple: ,Converting. among EAY, .Hpy aitdr1\1M ...

·Assume you purchased a $1'00,000 T-billthatmaturesin 150daysfora price of' $98,000. The broker who sold you the T-bill quoted the money market yield at 4:898%. Compute the HPY and the EAY.

Answer:

Money market to holding periodyie1d-rMM is an annualized yield based on a 360-day year. To change the rMM in this example inroits HPY, we need to convert it toa 150,.. .

::= 0.02041::= 2.041%

:fIolding period yield to effective amiualyield-=th~EAYis equal to the annuali~ed' .' HPY based on a 365-day year. Now that wehavecompured the HPY, simply annualize using a 365-day year to calculate the EAYasfollows:

EAY ::= (I + 0.02041)365/150 - 1

::= 1.05039 - 1 "" 5.039%

Note that to convert the EAY back into the HPY, apply the reciprocal of the exponent to theEAY. This is the same as taking one plus theEAY to the power (t/365J, For example, we can convert the EAY we just calculated back to the HPY as follows:

HPY::= (1.05039)150/365 - 1 ::= 2.041%

Professor's Note: On the Levell CFA®exam, you may be asked to convert. '. from any one ofthese .three yields into .one ofthe others. You.should note . that the EAYand rMM are merelyt.mnualizedversionsofthe HPYlfyou .. concentrate on converting back andforthbetween the HPYand the other yield figures, youwillbe well-prepared to answer these types ofquestions on the Level 1 exam. .

Study Session 2

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

Bond Equivalent Yield

The bond-equivalent yield refers to 2 x the semiannual discount rate. This convention stems from the fact that yields on U.S. bonds are quoted as twice the semiannual rate because the coupon interest is paid in two semiannual payments.

Example: Bond-eqllivalent yield calculation (1)

,. I·

A·3-monthloan~as.ah?ldingperiod yield of 2%. What is theyield:~~:Cl:'~ond­ equivalentbisis?". ,'':<.

Answer:,

."".

Theflrst step is to~~nvertthe.3-month yield to an effective seniiannuai(6~month) yield: " . ..'

1.022 - 1 = 4.04%

The second step it to double it (2 x 4.04 = 8.08%)

Example: Bond-equivalent yield calculation (2)

The effective annual yield on an investment is 8%. What is the yield on a bondequivalent basis?

Answer:

The first step is to convert the effective annual yield to arieffectiveseniiannual yield.

1.08°·5 - 1 =3.923%

The second step IS to double it: 2 x 3.923 = 7.846%.

Study Session 2

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

KEY CONCEPTS

LOS 6.a

The NPV is the present value of a project's future cash flows, discounted at the firm's cost of capital, less the project's cost. IRR is the discount rate that makes the NPV = 0 (equates the PV of the expected future cash flows to the project's initial cost).

The NPV rule is to accept a project if NPV > 0; the IRR rule is to accept a project if IRR> required rate of return. For an independent (single) project, these rules produce the exact same decision.

For mutually exclusive projects, IRR rankings and NPV rankings may differ due to differences in project size or in the riming of rhe cash Rows. Choose rhe project with the higher NPV as long as it is posirive.

A projecr may have mulriple IRRs or JlO m.H..

LOS 6.b

The holding period rerum (or yield) is calculated as:

LOS 6.c

The money-weighted rate of return is the IRR calculated using periodic cash flows into and out of an account and is the discount rare that makes the PV of cash inflows equal to the PV of cash outflows.

The time-weighted rate of return is calculated from the account'S periodic holding period returns and is the preferred performance measure.

LOS 6.d

The bank discount yield is the percentage discount from face value, annualized by multiplying by 360/days to maturity:

To convert a bank discount yield (RBO) to a holding period yield:

R (days)

BD 360

days)

- RBO [360

Study Session 2

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

CONCEPT CHECKERS

1.Which of the following statements least accurately describes the IRR and NPV methods?

A.The discount rate that gives an investment an NPV of zero is the investment's IRR.

B.If the NPV and IRR methods give conflicting decisions for mutually exclusive projects, the IRR decision should be used to select the project.

C.The NPV method assumes that a project's cash flows will be reinvested at the cost of capital, while the IRR method assumes they will be reinvested at the IRR.

2.Which of the following statements least accurately describes the IRR and NPV methods?

A.A project's IRR can be positive even if the NPV is negative.

3, Which of the following statements least accurate~l' describes the IRR and NPV methods?

A. The NPV tells how much the value of the firm has increased if you accept the project.

B.'V;Then evaluating independent projects, the IRR and NPV methods always yield the same acceptlreject decisions.

C.When selecting between mutually exclusive projects, the project with the highest NPV should be accepted regardless of the sign of the NPV calculation.

4.A company is considering entering into a joint venture that will require an investment of $10 million. The project is expected to generate cash flows of $4 million, $3 million, and $4 million in each of the next three years, respectively. Assuming a discount rate of 10%, what is the project's NPV?

A.-5879,000.

B.-$309,000.

C.+$243,000.

5.A company is considering entering into a joint venture that will require an investment of $10 million. The project is expected to generate cash flows of $4 million, $3 million, and $4 million in each of the next three years, respectively. Assuming a discount rate of 10%, what is the project's approximate IRR?

A.5%.

B.10%.

C.15%.

Study Session 2

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

6.What should an analyst recommend based on the following information for two independent projects?

A.Accept A and reject B.

B.Reject A and accept B. C. Accept A and accept B.

7.What should an analyst recommend based on the rollowing inrormation ror two mutually exclusive projects~

A.Accept A and reject B.

B.Reject A and accept B.

C.Accept A and accept B.

8.Goodeal Inc. is considering the purchase 0f a new material handling system for a cost of $15 million. This system is expected to generate a positive cash flow of $1.8 million per year in perpetuity. What is the NPV of the proposed investment if the appropriate discount rate is 1O. 5%~

A.$2,142,857.

B.$13,200,000.

C.$17,142,857.

$1.8 million per year in perpetui[y Wha[ is the IRR of the proposed inves[menr if the appropriate hurdle rate is 10.5%~

A.8.3%.

B.10.5%.

C.12.0%.

10.Should a company accept a project that has an IRR of 14% and an NPV of $2.8 million if the cost of capital is 12%~

A.Yes, based only on the NPY.

B.Yes, based on the NPV and the IRR.

C.No, based on both the NPV and IRR.