Yield Curve

Yield Curve DEFINITION of 'Yield Curve' A line that plots the interest rates, at a set point in time, of bonds having equal credit quality, but differing maturity dates. The most frequently reported yield curve compares the three-month, two-year, five-year and 30-year U.S. Treasury debt. This yield curve is used as a benchmark for other debt in the market, such as mortgage rates or bank lending rates. The curve is also used to predict changes in economic output and growth.

Next Up 1. 2. 3. 4.

SPOT RATE EQUITY CURVE YIELD ELBOW INVERTED YIELD CURVE

5.

BREAKING DOWN 'Yield Curve' The shape of the yield curve is closely scrutinized because it helps to give an idea of future interest rate change and economic activity. There are three main types of yield curve shapes: normal, inverted and flat (or humped). A normal yield curve(pictured here) is one in which longer maturity bonds have a higher yield compared to shorter-term bonds due to the risks associated with time.

An inverted yield curve is one in which the shorter-term yields are higher than the longer-term yields, which can be a sign of upcoming recession. A flat (or humped) yield curve is one in which the shorter- and longer-term yields are very close to each other, which is also a predictor of an economic transition. The slope of the yield curve is also seen as important: the greater the slope, the greater the gap between short- and long-term rates.

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Quantitative Methods - Introduction The Quantitative Methods section of the CFA curriculum has traditionally been placed second in the sequence of study topics, following the Ethics and Professional Standards review. It's an interesting progression: the ethics discussions and case studies will invariably make most candidates feel very positive, very high-minded about the road on which they have embarked. Then, without warning, they are smacked with a smorgasbord of formulas, graphs, Greek letters, and challenging terminology. We know - it's easy to become overwhelmed. At the same time, the topics covered in this section - time value of money, performance measurement, statistics and probability basics, sampling and hypothesis testing, correlation and linear regression analysis - provide the candidate with a variety of highly essential analytical tools and are a crucial prerequisite for the subsequent material on fixed income, equities, and portfolio management. In short, mastering the material in this section will make the CFA's entire Body of Knowledge that much easier to handle. The list of topics within Quantitative Methods may appear intimidating at first, but rest assured that one does not need a PhD in mathematics or require exceptional numerical aptitude to understand and relate to the quantitative approaches at CFA Level 1. Still, some people will tend to absorb quantitative material better than others do. What we've tried to do in this study guide is present the full list of topics in a manner that summarizes and attempts to tone down the degree of technical detail that is characteristic of academic textbooks. At the same time, we want our presentation to be sufficiently deep that the guide can be effectively utilized as a candidate's primary study resource. For those who have already purchased and read the textbook, and for those who already clearly understand the material, this guide should allow for a relatively speedy refresher in those hectic days and weeks prior to exam day. Along the way, we'll provide tips (primarily drawn from personal experience) on how to approach the CFA Level 1 exam and help give you the best chance of earning a passing grade. Expect 12% of the questions on your CFA Level 1 exam to cover topics under Quantitative Methods.

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Quantitative Methods - What Is The Time Value Of Money? The principle of time value of money - the notion that a given sum of money is more valuable the sooner it is received, due to its capacity to earn interest - is the foundation for numerous applications in investment finance. Central to the time value principle is the concept of interest rates. A borrower who receives money today for consumption must pay back the principal plus an interest rate that compensates the lender. Interest rates are set in the marketplace and allow for equivalent relationships to be determined by forces of supply and demand. In other words, in an environment where the market-determined rate is 10%, we would say that borrowing (or lending) $1,000 today is equivalent to paying back (or receiving) $1,100 a year from now. Here it is stated another way: enough borrowers are out there who demand $1,000 today and are willing to pay back $1,100 in a year, and enough investors are out there willing to supply $1,000 now and who will require $1,100 in a year, so that market equivalence on rates is reached.

test knowledge of FV, PV and annuity cash flow streams within questions on mortgage loans or planning for college tuitio he annuity factor formula, and require that the present value of each cash flow be calculated individually, and the resultin

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Quantitative Methods - The Five Components Of Interest Rates CFA Institute's LOS 5.a requires an understanding of the components of interest rates from an economic (i.e. non-quantitative) perspective. In this exercise, think of the total interest rate as a sum of five smaller parts, with each part determined by its own set of factors. 1. Real Risk-Free Rate - This assumes no risk or uncertainty, simply reflecting differences in timing: the preference to spend now/pay back later versus lend now/collect later. 2. Expected Inflation - The market expects aggregate prices to rise, and the currency's purchasing power is reduced by a rate known as the inflation rate. Inflation makes real dollars less valuable in the future and is factored into determining the nominal interest rate (from the economics material: nominal rate = real rate + inflation rate). 3. Default-Risk Premium - What is the chance that the borrower won't make payments on time, or will be unable to pay what is owed? This component will be high or low depending on the creditworthiness of the person or entity involved. 4. Liquidity Premium- Some investments are highly liquid, meaning they are easily exchanged for cash (U.S. Treasury debt, for example). Other securities are less liquid, and there may be a certain loss expected if it's an issue that trades infrequently. Holding other factors equal, a less liquid security must compensate the holder by offering a higher interest rate. 5. Maturity Premium - All else being equal, a bond obligation will be more sensitive to interest rate fluctuations the longer to maturity it is.

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Quantitative Methods - Time Value Of Money Calculations Here we will discuss the effective annual rate, time value of money problems, PV of a perpetuity, an ordinary annuity, annuity due, a single cash flow and a series of uneven cash flows. For each, you should know how to both interpret the problem and solve the problems on your approved calculator. These concepts will cover LOS' 5.b and 5.c. The Effective Annual Rate CFA Institute's LOS 5.b is explained within this section. We'll start by defining the terms, and then presenting the formula. The stated annual rate, or quoted rate, is the interest rate on an investment if an institution were to pay interest only once a year. In practice, institutions compound interest more frequently, either quarterly, monthly, daily and even continuously. However, stating a rate for those small periods would involve quoting in small fractions and wouldn't be meaningful or allow easy comparisons to other investment vehicles; as a result, there is a need for a standard convention for quoting rates on an annual basis. The effective annual yield represents the actual rate of return, reflecting all of the compounding periods during the year. The effective annual yield (or EAR) can be computed given the stated rate and the frequency of compounding. We'll discuss how to make this computation next.

interest rate)m - 1 n one year, and m Example: Effective Annual Rate Suppose we are given a stated interest rate of 9%, compounded monthly, here is what we get for EAR: EAR = (1 + (0.09/12))12 - 1 = (1.0075) 12 - 1 = (1.093807) - 1 = 0.093807 or 9.38%

Keep in mind that the effective annual rate will always be higher than the stated rate if there is more than one compounding period (m > 1 in our formula), and the more frequent the compounding, the higher the EAR. Solving Time Value of Money Problems Approach these problems by first converting both the rate r and the time period N to the same units as the compounding frequency. In other words, if the problem specifies quarterly compounding (i.e. four compounding periods in a year), with time given in years and interest rate is an annual figure, start by dividing the rate by 4, and multiplying the time N by 4. Then, use the resulting r and N in the standard PV and FV formulas. Example: Compounding Periods Assume that the future value of $10,000 five years from now is at 8%, but assuming quarterly compounding, we have quarterly r = 8%/4 = 0.02, and periods N = 4*5 = 20 quarters. FV = PV * (1 + r)N = ($10,000)*(1.02)20 = ($10,000)*(1.485947) = $14,859.47 Assuming monthly compounding, where r = 8%/12 = 0.0066667, and N = 12*5 = 60. FV = PV * (1 + r)N = ($10,000)*(1.0066667)60 = ($10,000)*(1.489846) = $14,898.46 Compare these results to the figure we calculated earlier with annual compounding ($14,693.28) to see the benefits of additional compounding periods.

Exam Tips and Tricks units - either by calling for quarterly or monthly compounding or by expressing time in months and the interest rate in years - is to go too fast. Remember to make sure the units agree for r and N, and are consistent with the frequency of compounding, prior Present Value of a Perpetuity A perpetuity starts as an ordinary annuity (first cash flow is one period from today) but has no end and continues indefinitely with level, sequential payments. Perpetuities are more a product of the CFA world than the real world - what entity would obligate itself making to payments that will never end? However, some securities (such as preferred stocks) do come close to satisfying the assumptions of a perpetuity, and the formula for PV of a perpetuity is used as a starting point to value these types of securities. The formula for the PV of a perpetuity is derived from the PV of an ordinary annuity, which at N = infinity, and assuming interest rates are positive, simplifies to:

Therefore, a perpetuity paying $1,000 annually at an interest rate of 8% would be worth: PV = A/r = ($1000)/0.08 = $12,500 FV and PV of a SINGLE SUM OF MONEY If we assume an annual compounding of interest, these problems can be solved with the following formulas:

oney, R = annual interest rate,

Example: Present Value At an interest rate of 8%, we calculate that $10,000 five years from now will be: FV = PV * (1 + r)N = ($10,000)*(1.08)5 = ($10,000)*(1.469328) FV = $14,693.28 At an interest rate of 8%, we calculate today's value that will grow to $10,000 in five years: PV = FV * (1/(1 + r)N) = ($10,000)*(1/(1.08)5) = ($10,000)*(1/(1.469328)) PV = ($10,000)*(0.680583) = $6805.83 Example: Future Value An investor wants to have $1 million when she retires in 20 years. If she can earn a 10% annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to:

A. $100,000 B. $117,459 C. $148,644 D. $161,506 Answer: The problem asks for a value today (PV). It provides the future sum of money (FV) = $1,000,000; an interest rate (r) = 10% or 0.1; yearly time periods (N) = 20, and it indicates annual compounding. Using the PV formula listed above, we get the following: PV = FV *[1/(1 + r) N] = [($1,000,000)* (1/(1.10)20)] = $1,000,000 * (1/6.7275) = $1,000,000*0.148644 = $148,644 Using a calculator with financial functions can save time when solving PV and FV problems. At the same time, the CFA exam is written so that financial calculators aren't required. Typical PV and FV problems will test the ability to recognize and apply concepts and avoid tricks, not the ability to use a financial calculator. The experience gained by working through more examples and problems increase your efficiency much more than a calculator. FV and PV of an Ordinary Annuity and an Annuity Due To solve annuity problems, you must know the formulas for the future value annuity factor and the present value annuity factor.

r

/r

yments

FV Annuity Factor The FV annuity factor formula gives the future total dollar amount of a series of $1 payments, but in problems there will likely be a periodic cash flow amount given (sometimes called the annuity amount and denoted by A). Simply multiply A by the FV annuity factor to find the future value of the annuity. Likewise for PV of an annuity: the formula listed above shows today's value of a series of $1 payments to be received in the future. To calculate the PV of an annuity, multiply the annuity amount A by the present value annuity factor.

The FV and PV annuity factor formulas work with an ordinary annuity, one that assumes the first cash flow is one period from now, or t = 1 if drawing a timeline. The annuity due is distinguished by a first cash flow starting immediately, or t = 0 on a timeline. Since the annuity due is basically an ordinary annuity plus a lump sum (today's cash flow), and since it can be fit to the definition of an ordinary annuity starting one year ago, we can use the ordinary annuity formulas as long as we keep track of the timing of cash flows. The guiding principle: make sure, before using the formula, that the annuity fits the definition of an ordinary annuity with the first cash flow one period away. Example: FV and PV of ordinary annuity and annuity due An individual deposits $10,000 at the beginning of each of the next 10 years, starting today, into an account paying 9% interest compounded annually. The amount of money in the account of the end of 10 years will be closest to: A. $109,000 B. $143.200 C. $151,900 D. $165,600 Answer: The problem gives the annuity amount A = $10,000, the interest rate r = 0.09, and time periods N = 10. Time units are all annual (compounded annually) so there is no need to convert the units on either r or N. However, the starting today introduces a wrinkle. The annuity being described is an annuity due, not an ordinary annuity, so to use the FV annuity factor, we will need to change our perspective to fit the definition of an ordinary annuity. Drawing a timeline should help visualize what needs to be done:

Figure 2.1: Cashflow Timeline The definition of an ordinary annuity is a cash flow stream beginning in one period, so the annuity being described in the problem is an ordinary annuity starting last year, with 10 cash flows from t0 to t9. Using the FV annuity factor formula, we have the following:

FV annuity factor = ((1 + r)N - 1)/r = (1.09)10 - 1)/0.09 = (1.3673636)/0.09 = 15.19293 Multiplying this amount by the annuity amount of $10,000, we have the future value at time period 9. FV = ($10,000)*(15.19293) = $151,929. To finish the problem, we need the value at t10. To calculate, we use the future value of a lump sum, FV = PV*(1 + r) N, with N = 1, PV = the annuity value after 9 periods, r = 9. FV = PV*(1 + r)N = ($151,929)*(1.09) = $165,603. The correct answer is "D". Notice that choice "C" in the problem ($151,900) agrees with the preliminary result of the value of the annuity at t = 9. It's also the result if we were to forget the distinction between ordinary annuity and annuity due, and go forth and solve the problem with the ordinary annuity formula and the given parameters. On the CFA exam, problems like this one will get plenty of takers for choice "C" - mostly the people trying to go too fast!! PV and FV of Uneven Cash Flows The FV and PV annuity formulas assume level and sequential cash flows, but if a problem breaks this assumption, the annuity formulas no longer apply. To solve problems with uneven cash flows, each cash flow must be discounted back to the present (for PV problems) or compounded to a future date (for FV problems); then the sum of the present (or future) values of all cash flows is taken. In practice, particularly if there are many cash flows, this exercise is usually completed by using a spreadsheet. On the CFA exam, the ability to handle this concept may be tested with just a few future cash flows, given the time constraints. It helps to set up this problem as if it were on a spreadsheet, to keep track of the cash flows and to make sure that the proper inputs are used to either discount or compound each cash flow. For example, assume that we are to receive a sequence of uneven cash flows from an annuity and we're asked for the present value of the annuity at a discount rate of 8%. Scratch out a table similar to the one below, with periods in the first column, cash flows in the second, formulas in the third column and computations in the fourth.

Cash Flow

Present Value Formula

$1,000

($1,000)/(1.08)1

$1,500

($1,500)/(1.08)2

$2,000

($2,000)/(1.08)3

$500

($500)/(1.08)4

$3,000

($3,000)/(1.08)5 Taking the sum of the results in column 4, we have a PV = $6,208.86. Suppose we are required to find the future value of this same sequence of cash flows after period 5. Here's the same approach using a table with future value formulas rather than present value, as in the table above:

Cash Flow

Future Value Formula

$1,000

($1,000)*(1.08)4

$1,500

($1,500)*(1.08)3

$2,000

($2,000)*(1.08)2

$500

($500)*(1.08)1

$3,000

($3,000)*(1.08)0

Taking the sum of the results in column 4, we have FV (period 5) = $9,122.86. Check the present value of $9,122.86, discounted at the 8% rate for five years: PV = ($9,122.86)/(1.08)5 = $6,208.86. In other words, the principle of equivalence applies even in examples where the cash flows are unequal.

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Quantitative Methods - Time Value Of Money Applications I. MORTGAGES Most of the problems from the time value material are likely to ask for either PV or FV and will provide the other variables. However, on a test with hundreds of problems, the CFA exam will look for unique and creative methods to test command of the material. A problem might provide both FV and PV and then ask you to solve for an unknown variable, either the interest rate (r), the number of periods (N) or the amount of the annuity (A). In most of these cases, a quick use of freshmen-level algebra is all that's required. We'll cover two real-world applications - each was the subject of an example in the resource textbook, so either one may have a reasonable chance of ending up on an exam problem. Annualized Growth Rates The first application is annualized growth rates. Taking the formula for FV of a single sum of money and solving for r produces a formula that can also be viewed as the growth rate, or the rate at which that sum of money grew from PV to FV in N periods.

For example, if a company's earnings were $100 million five years ago, and are $200 million today, the annualized five-year growth rate could be found by: growth rate (g) = (FV/PV)1/N - 1 = (200,000,000/100,000,000) 1/5 - 1 = (2) 1/5 - 1 = (1.1486984) - 1 = 14.87% Monthly Mortgage Payments The second application involves calculating monthly mortgage payments. Periodic mortgage payments fit the definition of an annuity payment (A), where PV of the annuity is equal to amount borrowed. (Note that if the loan is needed for a $300,000 home and they tell you that the down payment is $50,000, make sure to reduce the amount borrowed, or PV, to $250,000! Plenty of folks will just grab the $300,000 number and plug it into the financial calculator.) Because mortgage payments are typically made monthly with interest compounded monthly, expect to adjust the annual interest rate (r) by dividing by 12, and to multiply the time periods by 12 if the mortgage loan period is expressed in years.

Since PV of an annuity = (annuity payment)*(PV annuity factor), we solve for annuity payment (A), which will be the monthly payment:

he loan)/(PV annuity factor) Example: Monthly Mortgage Payments Assuming a 30-year loan with monthly compounding (so N = 30*12 = 360 months), and a rate of 6% (so r = .06/12 = 0.005), we first calculate the PV annuity factor: PV annuity factor = (1 - (1/(1 + r)N)/r = (1 - (1/(1.005)360)/0.005 = 166.7916 With a loan of $250,000, the monthly payment in this example would be $250,000/166.7916, or $1,498.88 a month.

up on the test, partly because they give an unfair advantage to those with higher-function calculators and because questions mus wn with understanding natural logs or transcendental numbers. II. RETIREMENT SAVINGS Savings and retirement planning are sometimes more complicated, as there are various lifecycles stages that result in assumptions for uneven cash inflows and outflows. Problems of this nature often involve more than one computation of the basic time value formulas; thus the emphasis on drawing a timeline is sound advice, and a worthwhile habit to adopt even when solving problems that appear to be relatively simple. Example: Retirement Savings To illustrate, we take a hypothetical example of a client, 35 years old, who would like to retire at age 65 (30 years from today). Her goal is to have enough in her retirement account to provide an income of $75,000 a year, starting a year after retirement or year 31, for 25 years thereafter. She had a late start on saving for retirement, with a current balance of $10,000. To catch up, she is now committed to saving $5,000 a year, with the first contribution a year from now. A single parent with two children, both of which will be attending college starting in five years, she won't be able to increase the annual $5,000 commitment until after the kids have graduated. Once the children are finished with college, she will have extra disposable income, but is worried about just how much of an increase it will take to meet her

ultimate retirement goals. To help her meet this goal, estimate how much she will need to save every year, starting 10 years from now, when the kids are out of college. Assume an average annual 8% return in the retirement account. Answer: To organize and summarize this information, we will need her three cash inflows to be the equivalent of her one cash outflow. 1.The money already in the account is the first inflow. 2. The money to be saved during the next 10 years is the second inflow. 3. The money to be saved between years 11 and 30 is the third inflow. 4.The money to be taken as income from years 31 to 50 is the one outflow. All amounts are given to calculate inflows 1 and 2 and the outflow. The third inflow has an unknown annuity amount that will need to be determined using the other amounts. We start by drawing a timeline and specifying that all amounts be indexed at t = 30, or her retirement day.

Next, calculate the three amounts for which we have all the necessary information, and index to t = 30. (inflow 1) FV (single sum) = PV *(1 + r)N = ($10,000)*(1.08)30 = $100,627 (inflow 2) FV annuity factor = ((1 + r)N - 1)/r = ((1.08)10 - 1)/.08 = 14.48656 With a $5000 payment, FV (annuity) = ($5000)*(14.48656) = $72,433 This amount is what is accumulated at t = 10; we need to index it to t = 30. FV (single sum) = PV *(1 + r)N = ($72,433)*(1.08)20 = $337,606 (cash PV annuity factor = (1 - (1/(1 + r)N)/r = (1 - (1/(1.08)25/0.08 = 10.674776.outflow)

With payment of $75,000, PV (annuity) = ($75,000)*(10.674776) = $800,608. Since the three cash inflows = cash outflow, we have ($100,627) + ($337,606) + X = $800,608, or X = $362,375 at t = 30. In other words, the money she saves from years 11 through 30 will need to be equal to $362,375 in order for her to meet retirement goals. FV annuity factor = ((1 + r)N - 1)/r = ((1.08)20 - 1)/.08 = 45.76196 A = FV/FV annuity factor = (362,375)/45.76196 = $7919 We find that by increasing the annual savings from $5,000 to $7,919 starting in year 11 and continuing to year 30, she will be successful in accumulating enough income for retirement. How are Present Values, Future Value and Cash Flows connected? The cash flow additivity principle allows us to add amounts of money together, provided they are indexed to the same period. The last example on retirement savings illustrates cash flow additivity: we were planning to accumulate a sum of money from three separate sources and we needed to determine what the total amount would be so that the accumulated sum could be compared with the client's retirement cash outflow requirement. Our example involved uneven cash flows from two separate annuity streams and one single lump sum that has already accumulated. Comparing these inputs requires each amount to be indexed first, prior to adding them together. In the last example, the annuity we were planning to accumulate in years 11 to 30 was projected to reach $362,375 by year 30. The current savings initiative of $5,000 a year projects to $72,433 by year 10. Right now, time 0, we have $10,000. In other words, we have three amounts at three different points in time. According to the cash flow additivity principle, these amounts could not be added together until they were either discounted back to a common date, or compounded ahead to a common date. We chose t = 30 in the example because it made the calculations the simplest, but any point in time could have been chosen. The most common date chosen to apply cash flow additivity is t = 0 (i.e. discount all expected inflows and outflows to the present time). This principle is frequently tested on the CFA exam, which is why the technique of drawing timelines and choosing an appropriate time to index has been emphasized here.

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Quantitative Methods - Net Present Value and the Internal Rate of Return This section applies the techniques and formulas first presented in the time value of money material toward real-world situations faced by financial analysts. Three topics are emphasized: (1) capital budgeting decisions, (2) performance measurement and (3) U.S. Treasury-bill yields. Net Preset Value NPV and IRR are two methods for making capital-budget decisions, or choosing between alternate projects and investments when the goal is to increase the value of the enterprise and maximize shareholder wealth. Defining the NPV method is simple: the present value of cash inflows minus the present value of cash outflows, which arrives at a dollar amount that is the net benefit to the organization. To compute NPV and apply the NPV rule, the authors of the reference textbook define a five-step process to be used in solving problems: 1.Identify all cash inflows and cash outflows. 2.Determine an appropriate discount rate (r). 3.Use the discount rate to find the present value of all cash inflows and outflows. 4.Add together all present values. (From the section on cash flow additivity, we know that this action is appropriate since the cash flows have been indexed to t = 0.) 5.Make a decision on the project or investment using the NPV rule: Say yes to a project if the NPV is positive; say no if NPV is negative. As a tool for choosing among alternates, the NPV rule would prefer the investment with the higher positive NPV. Companies often use the weighted average cost of capital, or WACC, as the appropriate discount rate for capital projects. The WACC is a function of a firm's capital structure (common and preferred stock and long-term debt) and the required rates of return for these securities. CFA exam problems will either give the discount rate, or they may give a WACC. Example: To illustrate, assume we are asked to use the NPV approach to choose between two projects, and our company's weighted average cost of capital (WACC) is 8%. Project A costs $7 million in upfront costs, and will generate $3 million in annual income starting three

years from now and continuing for a five-year period (i.e. years 3 to 7). Project B costs $2.5 million upfront and $2 million in each of the next three years (years 1 to 3). It generates no annual income but will be sold six years from now for a sales price of $16 million. For each project, find NPV = (PV inflows) - (PV outflows). Project A: The present value of the outflows is equal to the current cost of $7 million. The inflows can be viewed as an annuity with the first payment in three years, or an ordinary annuity at t = 2 since ordinary annuities always start the first cash flow one period away. PV annuity factor for r = .08, N = 5: (1 - (1/(1 + r)N)/r = (1 - (1/(1.08)5)/.08 = (1 - (1/ (1.469328)/.08 = (1 - (1/(1.469328)/.08 = (0.319417)/.08 = 3.99271 Multiplying by the annuity payment of $3 million, the value of the inflows at t = 2 is ($3 million)*(3.99271) = $11.978 million. Discounting back two periods, PV inflows = ($11.978)/(1.08) 2 = $10.269 million. NPV (Project A) = ($10.269 million) - ($7 million) = $3.269 million. Project B: The inflow is the present value of a lump sum, the sales price in six years discounted to the present: $16 million/(1.08)6 = $10.083 million. Cash outflow is the sum of the upfront cost and the discounted costs from years 1 to 3. We first solve for the costs in years 1 to 3, which fit the definition of an annuity. PV annuity factor for r = .08, N = 3: (1 - (1/(1.08)3)/.08 = (1 - (1/(1.259712)/.08 = (0.206168)/.08 = 2.577097. PV of the annuity = ($2 million)*(2.577097) = $5.154 million. PV of outflows = ($2.5 million) + ($5.154 million) = $7.654 million. NPV of Project B = ($10.083 million) - ($7.654 million) = $2.429 million. Applying the NPV rule, we choose Project A, which has the larger NPV: $3.269 million versus $2.429 million.

up so that it is tempting to pick a choice that seems intuitively better (i.e. by people who are guessing), but this is wrong by NPV lion) with a payoff of $16 million, which is more than the combined $15 million payoff of Project A. Don\'t rely on what feels b

The Internal Rate of Return The IRR, or internal rate of return, is defined as the discount rate that makes NPV = 0. Like the NPV process, it starts by identifying all cash inflows and outflows. However, instead of relying on external data (i.e. a discount rate), the IRR is purely a function of the inflows and outflows of that project. The IRR rule states that projects or investments are accepted when the project's IRR exceeds a hurdle rate. Depending on the application, the hurdle rate may be defined as the weighted average cost of capital. Example: Suppose that a project costs $10 million today, and will provide a $15 million payoff three years from now, we use the FV of a single-sum formula and solve for r to compute the IRR. IRR = (FV/PV)1/N -1 = (15 million/10 million)1/3 - 1 = (1.5) 1/3 - 1 = (1.1447) - 1 = 0.1447, or 14.47% In this case, as long as our hurdle rate is less than 14.47%, we green light the project. NPV vs. IRR Each of the two rules used for making capital-budgeting decisions has its strengths and weaknesses. The NPV rule chooses a project in terms of net dollars or net financial impact on the company, so it can be easier to use when allocating capital. However, it requires an assumed discount rate, and also assumes that this percentage rate will be stable over the life of the project, and that cash inflows can be reinvested at the same discount rate. In the real world, those assumptions can break down, particularly in periods when interest rates are fluctuating. The appeal of the IRR rule is that a discount rate need not be assumed, as the worthiness of the investment is purely a function of the internal inflows and outflows of that particular investment. However, IRR does not assess the financial impact on a firm; it only requires meeting a minimum return rate. The NPV and IRR methods can rank two projects differently, depending on thesize of the investment. Consider the case presented below, with an NPV of 6%:

Project

Initial outflow

Payoff after one year

IRR

NPV

A

$250,000

$280,000

12%

+$14,151

B

$50,000

$60,000

20%

+6604

By the NPV rule we choose Project A, and by the IRR rule we prefer B. How do we resolve the conflict if we must choose one or the other? The convention is to use the NPV rule when the two methods are inconsistent, as it better reflects our primary goal: to grow the financial wealth of the company. Consequences of the IRR Method In the previous section we demonstrated how smaller projects can have higher IRRs but will have less of a financial impact. Timing of cash flows also affects the IRR method. Consider the example below, on which initial investments are identical. Project A has a smaller payout and less of a financial impact (lower NPV), but since it is received sooner, it has a higher IRR. When inconsistencies arise, NPV is the preferred method. Assessing the financial impact is a more meaningful indicator for a capital-budgeting decision.

Project Investment

Income in future periods t1

t2

t3

t4

t5

IRR

NPV

A

$100k

$125k

$0

$0

$0

$0

25.0%

$17,925

B

$100k

$0

$0

$0

$0

$200k

14.9%

$49,452

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Quantitative Methods - Money Vs. Time-Weighted Return Money-weighted and time-weighted rates of return are two methods of measuring performance, or the rate of return on an investment portfolio. Each of these two approaches has particular instances where it is the preferred method. Given the priority in today's environment on performance returns (particularly when comparing and evaluating money managers), the CFA exam will be certain to test whether a candidate understands each methodology. Money-Weighted Rate of Return A money-weighted rate of return is identical in concept to an internal rate of return: it is the discount rate on which the NPV = 0 or the present value of inflows = present value of outflows. Recall that for the IRR method, we start by identifying all cash inflows and outflows. When applied to an investment portfolio: Outflows 1. The cost of any investment purchased 2. Reinvested dividends or interest 3. Withdrawals Inflows 1.The proceeds from any investment sold 2.Dividends or interest received 3.Contributions Example: Each inflow or outflow must be discounted back to the present using a rate (r) that will make PV (inflows) = PV (outflows). For example, take a case where we buy one share of a stock for $50 that pays an annual $2 dividend, and sell it after two years for $65. Our moneyweighted rate of return will be a rate that satisfies the following equation: PV Outflows = PV Inflows = $2/(1 + r) + $2/(1 + r)2 + $65/(1 + r)2 = $50 Solving for r using a spreadsheet or financial calculator, we have a money-weighted rate of return = 17.78%.

concept of money-weighted return, but any computations should not require use of a financial calculator It's important to understand the main limitation of the money-weighted return as a tool for evaluating managers. As defined earlier, the money-weighted rate of return factors all cash flows, including contributions and withdrawals. Assuming a money-weighted return is calculated over many periods, the formula will tend to place a greater weight on the performance in periods when the account size is highest (hence the label money-weighted). In practice, if a manager's best years occur when an account is small, and then (after the client deposits more funds) market conditions become more unfavorable, the moneyweighted measure doesn't treat the manager fairly. Here it is put another way: say the account has annual withdrawals to provide a retiree with income, and the manager does relatively poorly in the early years (when the account is larger), but improves in later periods after distributions have reduced the account's size. Should the manager be penalized for something beyond his or her control? Deposits and withdrawals are usually outside of a manager's control; thus, a better performance measurement tool is needed to judge a manager more fairly and allow for comparisons with peers - a measurement tool that will isolate the investment actions, and not penalize for deposit/withdrawal activity. Time-Weighted Rate of Return The time-weighted rate of return is the preferred industry standard as it is not sensitive to contributions or withdrawals. It is defined as the compounded growth rate of $1 over the period being measured. The time-weighted formula is essentially a geometric mean of a number of holding-period returns that are linked together or compounded over time (thus, time-weighted). The holding-period return, or HPR, (rate of return for one period) is computed using this formula:

= ending market value, ow received at period end (deposits subtracted, withdrawals added back)

1

For time-weighted performance measurement, the total period to be measured is broken into many sub-periods, with a sub-period ending (and portfolio priced) on any day with significant contribution or withdrawal activity, or at the end of the month or quarter. Sub-

periods can cover any length of time chosen by the manager and need not be uniform. A holding-period return is computed using the above formula for all sub-periods. Linking (or compounding) HPRs is done by (a) adding 1 to each sub-period HPR, then (b) multiplying all 1 + HPR terms together, then (c) subtracting 1 from the product: Compounded time-weighted rate of return, for N holding periods = [(1 + HPR1)*(1 + HPR2)*(1 + HPR3) ... *(1 + HPRN)] - 1. The annualized rate of return takes the compounded time-weighted rate and standardizes it by computing a geometric average of the linked holding-period returns.

d rate)1/Y - 1

Example: Time-Weighted Portfolio Return Consider the following example: A portfolio was priced at the following values for the quarter-end dates indicated:

Date

Market Value

Dec. 31, 2003

$200,000

March 31, 2004

$196,500

June 30, 2004

$200,000

Sept. 30, 2004

$243,000

Dec. 31, 2004

$250,000

On Dec. 31, 2004, the annual fee of $2,000 was deducted from the account. On July 30, 2004, the annual contribution of $20,000 was received, which boosted the account value to $222,000 on July 30. How would we calculate a time-weighted rate of return for 2004? Answer: For this example, the year is broken into four holding-period returns to be calculated for each quarter. Also, since a significant contribution of $20,000 was received intra-period, we will need to calculate two holding-period returns for the third quarter, June 30, 2004, to July 30, 2004, and July 30, 2004, to Sept 30, 2004. In total, there are five HPRs that must be

computed using the formula HPR = (MV1 - MV0 + D1 - CF1)/MV0. Note that since D1, or dividend payments, are already factored into the ending-period value, this term will not be needed for the computation. On a test problem, if dividends or interest is shown separately, simply add it to ending-period value. The ccalculations are done below (dollar amounts in thousands): Period 1 (Dec 31, 2003, to Mar 31, 2004): HPR = (($196.5 - $200)/$200) = (-3.5)/200 = -1.75%. Period 2 (Mar 31, 2004, to June 30, 2004): HPR = (($200 - $196.5)/$196.5) = 3.5/196.5 = +1.78%. Period 3 (June 30, 2004, to July 30, 2004): HPR = (($222 - $20) - $200)/$200) = 2/200 = +1.00%. Period 4 (July 30, 2004, to Sept 30, 2004): HPR = ($243 - $222)/$222 = 21/222 = +9.46%. Period 5 (Sept 30, 2004, to Dec 31, 2004): HPR = (($250 - $2) - $243)/$243 = 5/243 = +2.06% Now we link the five periods together, by adding 1 to each HPR, multiplying all terms, and subtracting 1 from the product, to find the compounded time- weighted rate of return: 2004 return = ((1 + (-.0175))*(1 + 0.0178)*(1 + 0.01)*(1 + 0.0946)*(1 + 0.0206)) - 1 = ((0.9825)*(1.0178)*(1.01)*(1.0946)*(1.0206)) - 1 = (1.128288) - 1 = 0.128288, or 12.83% (rounding to the nearest 1/100 of a percent). Annualizing: Because our compounded calculation was for one year, the annualized figure is the same +12.83%. If the same portfolio had a 2003 return of 20%, the two-year compounded number would be ((1 + 0.20)*(1 + 0.1283)) - 1, or 35.40%. Annualize by adding 1, and then taking to the 1/Y power, and then subtracting 1: (1 + 0.3540)1/2 - 1 = 16.36%. Note: The annualized number is the same as a geometric average, a concept covered in the statistics section.

Example: Money Weighted Returns Calculating money-weighted returns will usually require use of a financial calculator if there are cash flows more than one period in the future. Earlier we presented a case where a money-weighted return for two periods was equal to the IRR, where NPV = 0. Answer: For money-weighted returns covering a single period, we know PV (inflows) - PV (outflows) = 0. If we pay $100 for a stock today, and sell it in one year later for $105, and collect a $2 dividend, we have a money-weighted return or IRR = ($105)/(1 + r) + ($2)/(1 + r) - $100 = $0. r = ($105 + $2)/$100 - 1, or 7%. Money-weighted return = time-weighted return for a single period where the cash flow is received at the end. If the period is any time frame other than one year, take (1 + the result), raised to the power 1/Y and subtract 1 to find the annualized return.

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Quantitative Methods - Calculating Yield Calculating Yield for a U.S. Treasury Bill A U.S. Treasury bill is the classic example of a pure discount instrument, where the interest the government pays is the difference between the amount it promises to pay back at maturity (the face value) and the amount it borrowed when issuing the T-bill (the discount). T-bills are short-term debt instruments (by definition, they have less than one year to maturity), and there is zero default risk with a U.S. government guarantee. After being issued, T-bills are widely traded in the secondary market, and are quoted based on the bank discount yield (i.e. the approximate annualized return the buyer should expect if holding until maturity). A bank discount yield (RBD) can be computed as follows:

Formula 2.10 RBD = D/F * 360/t Where: D = dollar discount from face value, F = face value, T = days until maturity, 360 = days in a year

By bank convention, years are 360 days long, not 365. If you recall the joke about banker's hours being shorter than regular business hours, you should remember that banker's years are also shorter. For example, if a T-bill has a face value of $50,000, a current market price of $49,700 and a maturity in 100 days, we have: RBD = D/F * 360/t = ($50,000-$49,700)/$50000 * 360/100 = 300/50000 * 3.6 = 2.16% On the exam, you may be asked to compute the market price, given a quoted yield, which can be accomplish by using the same formula and solving for D:

Formula 2.11 D = RBD*F * t/360

Example: Using the previous example, if we have a bank discount yield of 2.16%, a face value of $50,000 and days to maturity of 100, then we calculate D as follows: D = (0.0216)*(50000)*(100/360) = 300 Market price = F - D = 50,000 - 300 = $49,700 Holding-Period Yield (HPY) HPY refers to the un-annualized rate of return one receives for holding a debt instrument until maturity. The formula is essentially the same as the concept of holding-period return needed to compute time-weighted performance. The HPY computation provides for one cash distribution or interest payment to be made at the time of maturity, a term that can be omitted for U.S. T-bills.

Formula 2.12 HPY = (P1 - P0 + D1)/P0 Where: P0 = purchase price, P1 = price at maturity, and D1= cash distribution at maturity Example: Taking the data from the previous example, we illustrate the calculation of HPY: HPY = (P1 - P0 + D1)/P0 = (50000 - 49700 + 0)/49700 = 300/49700 = 0.006036 or 0.6036% Effective annual yield (EAY) EAY takes the HPY and annualizes the number to facilitate comparability with other investments. It uses the same logic presented earlier when describing how to annualize a compounded return number: (1) add 1 to the HPY return, (2) compound forward to one year by carrying to the 365/t power, where t is days to maturity, and (3) subtract 1. Here it is expressed as a formula:

Formula 2.13 EAY = (1 + HPY)365/t - 1 Example: Continuing with our example T-bill, we have:

EAY = (1 + HPY)365/t - 1 = (1 + 0.006036)365/100 - 1 = 2.22 percent. Remember that EAY > bank discount yield, for three reasons: (a) yield is based on purchase price, not face value, (b) it is annualized with compound interest (interest on interest), not simple interest, and (c) it is based on a 365-day year rather than 360 days. Be prepared to compare these two measures of yield and use these three reasons to explain why EAY is preferable. The third measure of yield is the money market yield, also known as the CD equivalent yield, and is denoted by rMM. This yield measure can be calculated in two ways: 1. When the HPY is given, rMM is the annualized yield based on a 360-day year:

Formula 2.14 rMM = (HPY)*(360/t) Where: t = days to maturity For our example, we computed HPY = 0.6036%, thus the money market yield is: rMM = (HPY)*(360/t) = (0.6036)*(360/100) = 2.173%. 2. When bond price is unknown, bank discount yield can be used to compute the money market yield, using this expression:

Formula 2.15 rMM = (360* rBD)/(360 - (t* rBD)

Using our case: rMM = (360* rBD)/(360 - (t* rBD) = (360*0.0216)/(360 - (100*0.0216)) = 2.1735%, which is identical to the result at which we arrived using HPY. Interpreting Yield This involves essentially nothing more than algebra: solve for the unknown and plug in the known quantities. You must be able to use these formulas to find yields expressed one way when the provided yield number is expressed another way.

Since HPY is common to the two others (EAY and MM yield), know how to solve for HPY to answer a question. EAY = (1 + HPY)365/t - 1

HPY = (1 + EAY)t/365 - 1

rMM = (HPY)*(360/t)

HPY = rMM * (t/360)

Bond Equivalent Yield The bond equivalent yield is simply the yield stated on a semiannual basis multiplied by 2. Thus, if you are given a semiannual yield of 3% and asked for the bond equivalent yield, the answer is 6%.

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Understanding the Yield Curve The yield curve is a favorite market indicator of analysts and investors around the world, but what can it tell us? How can we use the yield curve to analyze current market conditions and project future market conditions? The yield curve can tell us a lot about what investors’ expectations for interest rates are and whether they believe the economy is going to be expanding or contracting. [VIDEO] Understanding the Yield Curve

The Yield Curve The yield curve is a graph that plots the relationship between yields to maturity and time to maturity for a group of bonds. Along the x-axis of a yield-to-maturity graph, we see the time to maturity for the associated bonds, and along the y-axis of the yield-to-maturity graph, we see the yield to maturity for the associated bonds. When you hear people talking about the yield curve, they are most likely talking about the yield curve for U.S. Treasuries. However, virtually any group of bonds or other fixed-rate securities that come from the same asset class and share the same credit quality can be plotted on a yield curve. For this discussion, we will be referring to the yield curve for U.S. Slope of the Yield Curve The slope of the yield curve provides analysts and investors with the important information they are looking for. Typically, you will see one of the following three slopes on your yield curve: – Normal yield curve – Flat yield curve – Inverted yield curve Normal Yield Curve: A normal yield curve tells us that investors believe the Federal Reserve is going to be raising interest rates in the future. Typically, the Federal Reserve only has to raise interest rates when the economy is expanding and the Fed is worried about inflation. Therefore, a normal yield curve often precedes an economic upturn.

Image of a Normal Yield Curve in relation to the S&P 500 on 17 March 2003—Chart courtesy of StockCharts.com Flat Yield Curve: A flat yield tells us that investors believe the Federal Reserve is going to be cutting interest rates. Typically, the Federal Reserve only has to cut interest rates when the economy is contracting and the Fed is trying to stimulate growth. Therefore, a flat yield curve is often a sign of an economic slowdown.

Image of a Flat Yield Curve in relation to the S&P 500 on 9 March 2006—Chart courtesy of StockCharts.com Inverted Yield Curve: An inverted yield curve tells us that investors believe the Federal Reserve is going to be dramatically cutting interest rates. Typically, the Federal Reserve has to dramatically cut interest rates during a recession. Therefore, an inverted yield curve is often a sign that the economy is in, or is headed for, a recession.

Image of an Inverted Yield Curve in relation to the S&P 500 on 31 January 2007—Chart courtesy of StockCharts.com

2.3 Bond Prices: Multi-Period Case

T he cash flows from most fixed-income securities extend beyond one period. The valuation of these cash flows uses two principles: that of discounting and that of value additivity. Value additivity says that you can simply add up the different discounted cash flows. The discounting principles you have learned say that you must be careful to add together only present values at the same time. With these principles in mind, it is easy to move to multiple periods and to multiple cash flows. One complication that arises is that different interest rates may be used to discount cash flows at different periods. For example, you may be willing to lend money for one year at 7%, but may require a higher interest rate, say, 8% per year, to lend money for two years. The extra return would compensate you for not being able to use the money for that extra year. The interest rate used to discount cash flows in period t, rt , is called the spot interest ratefor t periods. We will quote interest rates relative to one period (which is generally one year). That is, rt per period means that interest accrues (or is paid) at the rate of rt % per period for t periods. For example, consider a zero-coupon bond that pays its face value, F, in two years. You can value this bond given the two-year spot interest rate. The arbitrage-free price of this pure discount bond is the present value of the face amount F discounted at the two-year spot interest rate.

Example: Present Value using Spot Rates of Interest The present value of $100 at the end of two years when two-year spot interest rates are 6% per year is:

As you can see, the only differences between a cash flow in one period and a cash flow in two periods are (1) we use the two-period interest rate, and (2) we discount the cash flows "twice." If the cash flow occurs in t periods, we use the t-period spot interest rate and discount the cash flow for t periods. With many periods, however, you can also receive intermediate payments. For example, a coupon bond makes a payment every period in addition to the face value, which is paid at maturity. These more complex cash flows can also be valued using what you have learned in Chapter 1. First, you know how to discount each cash flow. Second, you know that you can add up the discounted cash flows, as in the derivation of the derivation of the annuity formula (see appendix A, in chapter 1).

For example, consider the timeline in Figure 2.1 for a newly issued 30-year Treasury bond with a face value of $10,000 and a coupon rate equal to 10% compounded semiannually. Figure 2.1 30-Year Coupon Bond

Here, we know the present value of each cash flow. Value additivity says that the value of the bond must equal the sum of all the present values, so we also know the value of the bond. There is another way to see why value additivity must hold, and this comes from arbitrage. The cash flows from the 30-year Treasury bond are exactly the same as a portfolio of 59 zero-coupon bonds with face values equal to $500 plus one zerocoupon bond with a face value equal to $10,500. The first zero-coupon bond has maturity equal to 6 months, the second 12 months, and so on. Since it is possible to reconstruct the Treasury bond from the zero-coupon bonds, the value of the Treasury bond must equal the value of the collection (or portfolio) of zerocoupon bonds. If the Treasury bond had a higher price, you would sell it and buy all the necessary zero-coupon bonds, giving you a pure arbitrage profit. You can calculate the value of this bond using the procedure in Chapter 1, in the Time Value of Money topic 1.2. You should be able to verify that if the interest rate is 7%, the value of the bond is $13,741.71. Example: Coupon Bond Suppose you want to value a 10-year U.S. Treasury bond with a face value of $10,000 and a coupon rate equal to 6% payable semiannually. The bond is issued today. With a face value of $10,000 and a coupon rate equal to 6% per year, the bond will pay $300 ( = 10,000 x 0.06 x 1/2 ) every six months for ten years and then pay the face value of $10,000 at the end of ten years. The cash flows are shown in the timeline in Figure 2.2. Figure 2.2 10-Year Coupon Bond

In other words, the holder of this bond receives 20 separate cash payments of $300 each and one cash payment of $10,000 over the ten years. The arbitrage-free price of each coupon payment (denoted by C = $300 for thetth coupon payment) is:

The arbitrage-free price of the face value is:

Value additivity (or lack of arbitrage) then says that the price of this bond is:

in the above each rt is the six-month interest rate. The stream of coupon payments makes up an ordinary annuity, so the value of the bond equals the value of an ordinary annuity plus the present value of the face value. However, you cannot apply the annuity formula directly because it assumes that all the spot interest rates are the same. In Chapter 3, (see Overview (Topic 3.1)), you will see how to value bonds when interest rates for maturities differ. For now, if we make the simplifying assumption that all interest rates are constant, we can apply the annuity formula. You can use the software in Bond Tutor to calculate the value of any annuity. Example: Present Value of a Coupon Bond Assume for the bond in Figure 2.2 that each spot interest rate is 6%. What is the value of this coupon bond? To compute this value you calculate the present value of the coupon payments and add to this number the present value of the bond’s face value. The coupon payments form an ordinary annuity. Using Bond Tutor, you can calculate that the present value of $1 for ten periods at 6% compounded twice per period is 14.8775:

The present value of all future coupon payments for this bond (to the nearest dollar) is then: $4,463 = $300 x 14.8775 The present value of $1 at the end of 20 periods at 3% per period is $0.5537, which is computed as follows:

Thus, the present value of the face amount is: $5,537 = $10,000 x 0.5537 As a result, the value of the bond is $10,000. Note that this equals the face value. If a bond’s price equals its face value, it is said to be trading "at par." Otherwise, it may trade at a "discount from par" or at a "premium to par."

What happens if the market interest rate falls below or rises above 6%? In the Bond Tutor subject two titled, Bond Values and Interest Rates, you can see what happens in each of these cases. For example, suppose we want to see how the value of the bond is altered when exposed to a range of interest rates from 1% to 10%, with a step size of 1%. In the interactive calculator below, enter the following values:

You want to look at a sensitivity analysis in step sizes of 1% (i.e., 10 steps between 1% to 10%). The exposure profile is the following: Click on numeric button and verify the following bond values:

You can see that the bond value declines as interest rates increase. When the interest rate equals the coupon rate, the value equals the face amount of $10,000. You can also see the principle of value additivity at work, by selecting to plot Cash Flows

and viewing the components both numerically and graphically. For example, for the first ten coupon payments, the present value of each component is:

To see the relative weighting of coupon versus face value, click on the Composition button to view Period 0 (i.e., present time) :

The display appears as follows: The numbers supporting the above graph for Time 0 are:

That is, the present value of the face value is $5,536.76 and the present value of the future coupon payments is $4,463.24. The valuation principles developed thus far apply to any point in time. In the next topic, we calculate the future value of a fixed-income security.