CFA Level 1 (2009) - 1

Study Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

15.C N; 30 x 12; 360; I/Y; 9/12; 0.75; PV; -150,000(1 - 0.2) = -120,000; FV; 0; CPT --t PMT '" $965.55

16.C N", 1 x 365 '" 365; IIY '" 12/365 = 0.0328767; PMT =0; PV = -5,000;

CPT --t FV = $5,637.37

17.A 9/0.JJ;$81.82

18.B 4.5 / 65 = 0.0692 or 6.92%

19.A To get the annual payment, enter PV =-10,000; FV =0; IIY = 10; N = 10;

CPT --t PMT =1,627.45. The first year's interest is $1,000 =10,000 x 0.10, so the principal balance going into year 2 is 10,000 - 627.45 = $9,372.55. Year 2 interest

=$9,372.55 x 0.10 '" $937.26.

20.C EAR = (l + 0.18/12)12 - 1 = 19.56%

ANSWERS - COMPREHENSIVE PROBLEMS

I.(lur su~gested solution method is:

PV of third cruise cost = (1.035)11 x 5,000 = $3,130.78 1.08

PV of a/l three = 3,408.94 + .1,266.90 + 3, J 30.78 = $9,806.62. This is the amount needed in the account today so it's rhe PV of a 1O-paymenr annuity due. Solve for payment ar 8% = $1,353.22.

2.This problem is simpler than it may appear. The dividend grew from $0.88 to $1.26 in

eight years. We know, then. that O.88(l + i)8 = 1.26, where i is the compound growth

[FI rhree rimes, gel' 1.045890 and see thac the answer is 4.589%.

This technique for solving for a compound growth race is a very useful one and you will see ic often.

Calculacor solution: PV =0.88, N = 8, FV =-1.26, PMT =0, CPT --t ItY = 4.589.

Srudy Session 2

Cross-Reference to CFA Institute Assigned Reading #5 - The Time Value of Money

Or in begin mode: PV = 400,000; N = 20; FV = 0; PMT = -40,000:

CPT ~ IIY = 8.9196%

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

DISCOUNTED CASH FLOW

ApPLICATIONS

Study Session 2

EXAM FOCUS

LOS 6.a: Calculate and interpret the net present value (NPV) and the internal rate of return (jRR) of an investment, contrast the NPV rule to the IRR rule, and identify problems associated with the IRR rule.

The net present value (NPV) of an investment project is the present value of expected cash inflows associated with the project less the present value of the project's expected cash outflows, discounted at the appropriate cost of capital. The following procedure may be used to compute NPV:

• Identify all costs (outflows) and benefits (inflows) associated with an investment.

•Determine the appropriate discount rate or opportunity COSt for the investment.

•Using the appropriate discount rate, find the PV of each cash flow. Inflows are positive and increase NPY. Outflows are negative and decrease NPV.

•Compute the NPV, the sum of the DCFs.

Mathematically, NPV is expressed as:

Nthe estimated life of the investment

the discount rate = opportunity cOSt of capital

NPV is the PV of the cash flows less the initial (time = 0) outlay.

Study Session 2

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

. Example: Computing·NPV

CaJcula~e.theNPVofaninvestmentprojectwithan-initiaL~:ostof$5 million and

positive'cash flows of$I.6 million at the end of year :1 ,.$2.4~illion at the. end of year 2, and $2.8 million at the end of year 3. Use 12% as thedisc6unrrate.

AnsWer:

Th~NPY'f~r~thisprojectjs the sum of the PVs oftheprojedsindividual cash flows

Figure 2: Calculating NPV with the HP12C@

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

On the TI BAIl Plus calculator, the sequence of [1] [l] scrolls past the variables FO 1, F02, et cetera. The F here stands for frequency, and this is set to 1 by default. We did not enter anything because each cash flow amount occurred only once. If the period 2 cash flow, C02, were repeated three times (at t = 2, 3, and 4) we could input F02 = 3 to account for all three of these. The next input, C03, would then refer to the cash flow at t = 5.

On the HP 12C, you can also account for a repeated cash flow amount. Referring to the above example for the TI BAIl Plus, the fact that the cash flow at period 2 is repeated for three periods is indicated by the sequence 3 [g] Nj immediately after the CF. keystroke to input the amount of the period 2 cash flow. J

Professor's Note: The NPV function can also be used to find the present value of any series ofcash flows (positive or negative) over future periods. Just set CFo = O oand input tht' cash flows CFI through CFN as outlined abovt'. The NPV is the

presellt value o(tht.'sc Cil.i!J flows since there is nou' no initi,71l1egativ{' cash flou' (illitial (au J.

The internal rate of return ORR) is defined as the rate of return that equates the PV of an investment's expected benefits (inflows) with the PV of its costs (outflows). Equivalently, the IRR may be defined as the discount rate for which the NPY of an investment is zero.

Calculating IRR requires only that we identify the relevant cash flows for the investment opportunity being evaluated. Market-determined discount rates, or any other external (market-driven) data, are not necessary with the IRR procedure. The general formula for the IRR is:

In the majority of IRR applications to capital budgeting, the initial cash flow, CFo' represents the initial cost of the investment opportunity, and is therefore a negative value. As such, any discount rate less than the IRR will result in a positive NPY, and a discount rate greater than the IRR will result in a negative NPY. This implies that the NPV of an investment is zero when the discount rate used equals the IRR.

Example: Computing IRR

What is the IRR for the investment described in the preceding example?

Answer:

Substituting the investment's cash flows into the previous IRR equation results in the following equation:

Solving this equation yields an IRR = 15.52%.

Study Session 2

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

It is possible to solve IRR problems through a trial and error process. That is,keep guessing IRRs untilyou get the one that provides an NPV equal rozero. Practically speaking, a financial calculator oran electronic spreadsheet canandshould:be employed. The procedures for computing IRR with· the TI BAIl Plus and.HP12C financial calculators are illustrated in the following figures, respectively.

Calculating IRRwiththe TIBusiness Analyst II Plus®

The NPV Decision Rule Versus the IRR Rule

NPV decision rule. The basic idea behind NPY analysis is that if a project has a positive NPY, this amount goes to the firm's shareholders. As such, if a firm undertakes a project with a positive NPY, shareholder wealth is increased.

The NPY decision rules are summarized:

Accept projects with a positive NPY. Positive NPY projects will increase shareholder wealth.

•Reject projects with a negative NPY. Negative NPY projects will decrease shareholder wealth.

•When two projects are mutually exclusive (only one can be accepted), the project with the higher positive NPY should be accepted.

IRR decision rule. Analyzing an investment (project) using the IRR method provides the analyst with a result in terms of a rate of return.

Study Session 2

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

The following are decision rules of IRR analysis:

•Accept projects with an IRR that is greater than the firm's (investor's) required rate of return.

Reject projects with an IRR that is less than the firm's (investor's) required rate of return.

Note that for a single project, the IRR and NPV rules lead to exactly the same accept/ reject decision. If the IRR is greater than the required rate of return, the NPV is positive, and if the IRR is less than the required rate of return, the NPV is negative.

Problems Associated With the IRR Method

When the acceptance or rejection of one project has no effect on the acceptance or rejection of another, the twO projects are considered to be independent projects. When only one of two projects may be accepted, the projects are considered to be mutually exclusive. For mutually exclusive projects. the NPY and IRR methods can

give conflicting project rankings. This (all happell when the projects' initial costs arc or dirfLTClH sii'.es or wht:n the timin~ oj' the clSh Hows is different. Let's look at an n;\ll1plc that illustrates how NPY and IRR call vicki remAieting results.

Example: Conflicting decisions between NPV and IRR

Assume NPV and IRR analysis of two mutually exclusive projects produced the results shown in the following figure. As indicated, the IRR criteria recommends that Project A should be accepted. On the other hand, the NPV criteria indicates acceptance of Project B. Which project should be selected?

Ranking Reversals with NPV and IRR

Answer:

Investing in Project A increases shareholder wealth by $2,272.72, while investing in Project B increases shareholder wealth by $6,363.64. Since the overall goal of the firm is to maximize shareholder wealth, Project B should be selected because it adds the most val ue to the firm.

Mathematically speaking, the NPV method assumes the reinvestment of a project's cash flows at the opportunity cost of capital, while the IRR method assumes that the reinvestment rate is the IRR. The discount rate used with the NPV approach represents the market-based opportunity cost of capital and is the required rate of rerum for the shareholders of the firm.

Study Session 2

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

Given that shareholder wealth maximization is the ultimate goal of the firm, always select the project with the greatest NPV when the IRR and NPV rules provide conflicting decisions.

LOS 6.b: Define, calculate, and interpret a holding period return (total return).

A holding period can be any period of time. The holding period of an investment may be a matter of days or as long as several years. The holding period return is simply the percentage change in the value of an investment over the period it is held. If the asset has cash flows, such as dividend or interest payments, we refer to the return, including the value of these interim cash flows, as the total return.

As an example, consider a Treasury bill purchased for $980 and sold three months later for $992. The holding period return can be calculated as:

HPR = ending value - beginning value = ending value _ 1

beginning value beginning value

992

and we have -- -1 = 0.0 122 or 1.22% 980

\Y/e \'!ould say that the investor's 3-month holding period return was 1.22%.

As an example of a security that has interim cash paymems, consider a stock that is purchased for $30 and is sold for $33 six months later, during which time it paid $0.50 in dividends. The holding period return (total return in this case) can be calculated as:

HPR = ending value - beginning value + cash flow received beginning value

= ending value + cash flow received -1 beginning value

which is the investor's total return over the holding period of six months.

LOS 6.c: Calculate, interpret, and distinguish between the moneyweighted and time-weighted rates of return of a portfolio and appraise the performance of portfolios based on these measures.

The money-weighted return applies the concept of IRR to investment portfolios. The money-weighted rate of rerum is defined as the internal rate of rerum on a portfolio, taking into account all cash inflows and outflows. The beginning value of the account is an inflow as are all deposits into the account. All withdrawals from the account are outflows, as is the ending value.

Study Session 2

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

Example: Money-weighted rate of return

Assume an investor buys a share of stock for $100 at t ::: 0 and at the end of the next year (t = 1), she buys an additional share for $120. At the end of year 2, the investor sells

both shares for $130 each. At the end of each year in the holding period, the stock paid a $2.00 per share dividend. What is the money-weighted rate of return?

Step 1: Determine the timing of each cash flow and whether the cash flow is an inflow (+), into the account, or an outflow (-), available from the account.

Step 2: Net the cash flows for each time period and set the PY of cash inflows equal to the present value of cash outflows.

Step 3: Solve for r to find the money-weighted rate of return. This can be done using trial and error or by using the IRR function on a financial calculator or spreadsheet.

The intuition here is that we deposited $100 into the account at t ::: 0, then added $118 to the account at t ::: 1 (which, with the $2 dividend, funded the purchase of one more share at $120), and ended with a toral value of $264.

To compute rhis value with a financial calculator, use these net cash flows and follow the procedure(s) described to calculate the IRR.

Net cash flows: CF o = +100; CF j = +120 - 2 = +118; CF2 := -260 + -4::: -264

Calculating Money-Weighted Return With the TI Business Analyst II PLus®

Note that the values for F01, F02, etc., are all equal to one.

Study Session 2

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

.CalculatmgMoney-WeightedReturnWiththe HP 12C@

.Tf"]TFI.N:J[fl~IREG1>':··.":(:~';:· .. C1~arMemoryRegister~

- -~:- ''- )'.:;

The money-weightedr~te:'ofreturnfor this problem is 13.86%.

Projessorj· Note: In the preceding eXflrnple. we elltered the flows into the tlCCOUIlt as positive and the ending value aj' a mgative (the investor could withdraw this

~amount from the account). Note that there is no difference in the solutioll {f

~we nztn the cash flows into the account as negative values (out ofthe investor:(

poc/let) and the ending value as tl p().iitive value (into the investor's pocket). As long tlS ptlyments into the tlccoullt and payments out ofthe account (including the ending value) Me entered with oppr}(iU' signs, the computed IRR will be correct.

Time-weighted rate of rerum measures compound growth. It is the rate at which $ LOa compounds over a specified performance horizon. Time-weighting is the process of averaging a set of values over time. The annual time-weighted return for an investment may be computed by performing the following steps:

Step 1: Value the portfolio immediately preceding significant additions or withdrawals. Form subperiods over the evaluation period that correspond to the dates of deposits and withdrawals.

Step 2: Compute the holding period return (HPR) of the portfolio for each subperiod.

Step 3: Compute the product of (I + HPR) for each subperiod to obtain a total return for the entire measurement period [i.e., (l + HPR j ) x (l + HPR2) .••

(l + HPRn)J· If the total investment period is greater than one year, you must take the geometric mean of the measurement period return to find the annual time-weighted rate of return.