Discounted Cash Flow Applications 2014 Level 1 CFA® Quantitative Methods IFT Notes for the CFA® Exam

Discounted Cash Flow Applications

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Contents 1. Introduction ........................................................................................................................................ 2 2. Net Present Value and Internal Rate of Return..................................................................................... 2 3. Portfolio Return Measurement .......................................................................................................... 13 4. Money Market Yields ......................................................................................................................... 23 Summary ............................................................................................................................................... 30 Next Steps ............................................................................................................................................. 32

This document should be read in conjunction with the corresponding reading the 2014 Level I CFA® Program curriculum.

Some of the graphs, charts, tables, examples, and figures are copyright 2013, CFA Institute. Reproduced and republished with permission from CFA Institute. All rights reserved. Required disclaimer: CFA Institute does not endorse, promote, or warrant the accuracy or quality of the products or services offered by Irfanullah Financial Training. CFA Institute, CFA®, and Chartered Financial Analyst® are trademarks owned by CFA Institute.

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1. Introduction Much of the work of investment analysts involves evaluating investment opportunities and identifying projects that are worth pursuing. The concepts of time value of money that we saw in the previous reading find extensive applications in real life to evaluate investments.

This reading covers: 

Decision tools: Net present value, internal rate of return (IRR)



Decision rules for NPV and IRR



Conflicting situations under IRR, and what rule to use in such cases



How is the return on a portfolio measured: money-weghted return and time-weighted return



Calculation of different yield measures of money market instruments

2. Net Present Value and Internal Rate of Return Net present value (NPV) and internal rate of return (IRR) are often applied across different areas of finance such as: 

Capital budgeting: Allocation of funds to relatively long-range projects or investments



Capital structure: Choice of long-term financing for the investments the company wants to make.



Working capital management: Management of the company’s short-term assets (such as inventory) and short-term liabilities (such as money owed to suppliers).

2.1 Net Present Value and the Net Present Value Rule

The net present value (NPV) of an investment is the present value of its cash inflows minus the present value of its cash outflows. The initial investment is a cash outflow. The net present value Copyright © Irfanullah Financial Training. All rights reserved.

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rule is a method for choosing among alternative investments. It is an important decision tool to evaluate projects and decide which ones are worth investing. The steps in calculating NPV and applying the NPV rule are as follows: 

Identify all the cash flows associated with an investment (both inflows and outflows)



Determine the appropriate discount rate or opportunity cost, r, for the investment project. r is also known as the hurdle rate.



Using the discount rate r, find the present value of each cash flow.



Sum the present values of all cash flows, including the initial investment. This value represents the investment’s net present value.



Apply the NPV rule: accept the project if the investment’s NPV is positive; reject the project/investment if NPV is negative. If an investor has two or more projects for investment, but can only invest in one (i.e. mutually exclusive projects), the investor must choose the project with the highest positive NPV.

In calculating the NPV, we use an estimate of the opportunity cost of capital as the discount rate. NPV decision rule for independent projects: Accept project if NPV > 0 Reject project if NPV < 0

The opportunity cost of capital is the alternative return that investors forgo by investing the cash elsewhere (cost of capital will be discussed in detail in Corporate Finance.) When NPV is positive, the investment adds value because it more than covers the opportunity cost of the capital needed to undertake it. The following formula can be used to calculate NPV: Equation 1: Net Present Value NPV = present value of cash inflows – present value of cash ouflows CF

t NPV = ∑N t=0 (1+r)t

where Copyright © Irfanullah Financial Training. All rights reserved.

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CFt = the expected net cash flow at time t, N = the investment’s projected life r = the discount rate or opportunity cost of capital

Consider a project which requires an initial outlay of $750,000. It is expected to produce $200,000 in the first year, $300,000 in the second year, and $400,000 in the third year. The cost of capital for this project is 10%. What is the NPV? Should the project be accepted?

Using equation 1 for NPV, CF1

CF2

CF3

NPV = CF0 + [(1+r)1] + [(1+r)2] + [(1+ NPV = −750,000 +

200000 (1.1)1

+

300000 (1.1)2

+

r)3

]

400000 (1.1)3

NPV = -19.72

While this method is conceptually simple, it can be tedious if the number of cash flows is large. It is much easier to use the financial calculator. The steps are outlined below:

Keystrokes

Explanation

Display

[2nd] [QUIT]

Return to standard mode

0

[CF] [2nd] [CLR WRK]

Clear CF Register

CF = 0

750 [+/-] [ENTER]

Initial Outlay (in 000’s)

CF0 = -750

[↓] 200 [ENTER]

Enter CF at t = 1

C01 = 200

[↓] [↓] 300 [ENTER]

Enter CF at t = 2

C02 = 300

[↓] [↓] 400 [ENTER]

Enter CF at t = 3

C03 = 400

[↓] [NPV] [10] [ENTER]

Enter discount rate

I = 10

[↓] [CPT]

Compute NPV

-19.722

In this case, the NPV is negative, which means that the present value of all future cash flows is less than the initial investment. Hence, this project should not be accepted.

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Worked Example 1 As an analyst covering the IBD Corporation, you are evaluating its research and development (R&D) program for the current year. Management has announced that it intends to invest $5 million in R&D. Incremental net cash flows are forecasted to be $525,000 per year in perpetuity. IBD Corporation’s opportunity cost of capital is 10 percent. 1. State whether IBD’s R&D program will benefit shareholders, as judged by the NPV rule. 2. Evaluate whether your answer to Part 1 changes if RAD Corporation’s opportunity cost of capital is 15 percent rather than 10 percent. Solution to 1: Since the net cash flows of $525,000 are constant, it can be considered as a perpetuity. The present value of the perpetuity is NPV = −$5,000,000 +

525,000 0.10

525,000 0.1

. The project’s NPV is calculated as:

= $250,000

With an opportunity cost of 10 percent, the present value of the program’s cash inflows is $5,250,000. The program’s cost is an immediate outflow of $5,000,000; therefore, its net present value is $250,000. As NPV is positive, you conclude that IBD Corporation’s R&D program will benefit shareholders. Solution to 2: With an opportunity cost of capital of 15 percent, NPV is computed as above using a 15% discount rate instead of 10 percent. NPV = −$5,000,000 +

$525,000 0.15

= −1,500,000

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With a higher opportunity cost of capital, the present value of the inflows is smaller and the program’s NPV is smaller. At 15 percent, the NPV is negative. This means that the present value of future cash flows is less than the cost. Hence the project should not be undertaken. If NPV = 0, a project generates just enough cash flow to compensate shareholders for the opportunity cost of making the investment. When a company undertakes a zero-NPV project, the company becomes larger, but shareholders’ wealth does not increase.

2.2 Net Present Value and the Net Present Value Rule

The internal rate of return (IRR) is the discount rate that makes the net present value equal to zero. It is a single number which represents the return generated by an investment.

Consider a very simple project where the initial investment is $100. One year later the amount received from this project is $110. There are no other cash flows. The internal rate of return is 10%. It is ‘internal’ because it depends only on the cash flows of the investment; no external data are needed.

The formal definition of IRR is as follows: IRR can be calculated through the equation show below:

Equation 2: Internal Rate of Return NPV = CF0 + [CF1 /(1+IRR)1] + [CF2/(1+IRR)2] + … + [CFN/(1+IRR)N] = 0 where CFt = the expected net cash flow at time t, NPV = net present value of the investment IRR = internal rate of return

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In the example given above, we have an initial investment and then one cash flow; hence the equation can be applied easily: −100 +

110 =0 1 + 𝐼𝑅𝑅

Solving this equation shows that IRR = 0.1 or 10%.

Another way of describing IRR is that it is the rate such that the present value of future cash flows is equal to the initial investment:

Equation 3: Internal Rate of Return (Alternate Formula) CF0 = [CF1 /(1+IRR)1] + [CF2 /(1+IRR)2] + … + [CFN /(1+IRR)N] where CF0 = initial investment made at time t=0 CFt = the expected net cash flow at time t IRR = internal rate of return

Now consider another project where the initial investment is $150,000. Estimated cash flows for the following three years are $50,000, $100,000 and $40,000 respectively. What is the IRR? If the opportunity cost of capital is 12%, should the project be accepted?

We can set up an equation with the initial outflow equal to the present value of future cash flows and solve for the IRR: CF0 = [

CF1 CF2 CF3 ]+ [ ]+ [ ] 1 2 (1 + IRR) (1 + IRR) (1 + IRR)3

Plugging in the values, we get: 150,000 = [

50,000 100,000 40,000 ]+ [ ]+ [ ] 1 2 (1 + IRR) (1 + IRR) (1 + IRR)3

While it is theoretically possible to solve the above equation, it is much simpler to use the financial calculator.

Keystrokes

Explanation

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Display

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[2nd] [QUIT]

Return to standard mode

0

[CF] [2nd] [CLR WRK]

Clear CF Register

CF = 0

150 [+/-] [ENTER]

Initial outlay (in 000’s)

CF0 = -150

[↓] 50 [ENTER]

Enter CF at t = 1

C01 = 50

[↓] [↓] 100 [ENTER]

Enter CF at t = 2

C02 = 100

[↓] [↓] 40 [ENTER]

Enter CF at t = 3

C03 = 40

[↓] [ÌRR] [CPT]

Compute IRR

13.11%

We can decide whether or not to proceed with a project by using the IRR rule: ‘accept projects or investments for which the IRR is greater than the opportunity cost of capital.’ Simply put, the IRR rule uses the opportunity cost of capital as a hurdle rate. IRR decision rule for independent projects: Accept project if IRR > r Reject project if IRR < r where r = hurdle rate or opportunity cost of capital

For the above project the internal rate of return (13.11%) exceeds the opportunity cost of capital (12%), hence it should be accepted. Worked Example 2 In the previous IBD Corporation example, the initial outlay is $5 million and the program’s cash flows are $525,000 in perpetuity. Now you are interested in determining the program’s internal rate of return. Address the following: 1. Write the equation for determining the internal rate of return of this R&D program. 2. Calculate the IRR.

Solution to 1:

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Finding the IRR is equivalent to finding the discount rate that makes the NPV equal to 0. Because the program’s cash flows are a perpetuity, you can set up the NPV equation as: NPV = −investment + present value of future cash flows = 0 NPV = −$5,000,000 + $525,000 / IRR = 0 or as Initial outlay = present value of future cash flows 5,000,000 =

525,000 IRR

Solution to 2: We can solve for IRR as: IRR =

$525,000 $5,000,000

= 0.105 or 10.5 percent.

Inferences based on the computed IRR: 

If the opportunity cost of capital is also 10.5 percent, the R&D program just covers its opportunity costs and neither increases nor decreases shareholder wealth.



If it is less than 10.5 percent, the IRR rule indicates that management should invest in the program because it more than covers its opportunity cost.



If the opportunity cost is greater than 10.5 percent, the IRR rule indicates that management should reject the R&D program.



For a given opportunity cost, the IRR rule and the NPV rule lead to the same decision in this example.

Worked Example 3

Note: this example has been reproduced from the curriculum, but the solution is presented differently.

The Japanese company Kageyama Ltd. is considering whether or not to open a new factory to manufacture capacitors used in cell phones. The factory will require an investment of ¥1,000 million. The factory is expected to generate level cash flows of ¥294.8 million per year in each Copyright © Irfanullah Financial Training. All rights reserved.

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of the next five years. According to information in its financial reports, Kageyama’s opportunity cost of capital for this type of project is 11 percent. 1. Determine whether the project will benefit Kageyama’s shareholders using the NPV rule.

2. Determine whether the project will benefit Kageyama’s shareholders using the IRR rule.

Solution to 1:

The NPV can be calculated using the financial calculator.

Keystrokes

Explanation

Display

[2nd] [QUIT]

Return to standard mode

0

[CF] [2nd] [CLR WRK]

Clear CF Register

CF = 0

1000 [+/-] [ENTER]

Initial Outlay (in millions)

CF0 = -1000

[↓] 294.8 [ENTER]

CF at t = 1

C01 = 294.8

[↓] 5 [ENTER]

Same cash flow repeats 5

F01 = 5

[↓] [NPV] [11] [ENTER]

times Enter discount rate

I = 11

[↓] [CPT]

Compute NPV

89.55

Because the project’s NPV is positive ¥89.55 million, the project should be accepted.

Solution to 2: This project’s positive NPV tells us that the internal rate of return must be greater than the cost of capital (11 percent). The actual IRR can be calculated as follows:

Keystrokes

Explanation

Display

[2nd] [QUIT]

Return to standard mode

0

[CF] [2nd] [CLR WRK]

Clear CF Register

CF = 0

1000 [+/-] [ENTER]

Initial Outlay (in millions)

CF0 = -1000

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[↓] 294.8 [ENTER]

CF at T = 1

C01 = 294.8

[↓] 5 [ENTER]

Same cash flow repeats 5 times

F01 = 5

[↓] [ÌRR] [CPT]

Compute IRR

14.50%

Because the IRR of 14.50 percent is greater than the opportunity cost of the project, the project should benefit Kageyama’s shareholders. Whether it uses the IRR rule or the NPV rule, Kageyama makes the same decision: build the factory.

3. Problems with the IRR Rule

As shown in the above examples, IRR and NPV rules give the same accept or reject decision when projects are independent. Independent projects are those projects where the decision to accept or reject does not depend on any other project. However, there can be situations where projects are not independent as there are not enough resources to undertake all projects. In other words, a firm needs to decide between project A and Project B. These are called mutually exclusive projects. So, what rule must be applied to determine which project should be undertaken?

NPV/IRR decision rule for mutually exclusive projects: If there is no conflict: NPV rule: project with the highest NPV must be selected. IRR rule: project with the highest IRR must be selected If there is conflict (explained below – due to change in size of project or timing of cash flows): Always followthe NPV rule and select the project with the highest NPV.

When deciding between projects which are not independent, the NPV and IRR rules do not always lead to the same decision. It is possible that the NPV of one project is higher than the IRR of the other project is higher. This is called a ranking conflict and is due to a theoretical limitation of IRR, whereby interim cash flows are assumed to be reinvested at the IRR rate and not at the cost of capital. Copyright © Irfanullah Financial Training. All rights reserved.

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Suppose the IRR for a project is 15% and we reinvest the cash generated by the project at a rate lower than 15%. In this case, we will realize a return that is less than 15%.

IRR and NPV can rank projects differently when: 1. The size or scale of the project differs 2. Timing of the projects’ cash flows differs Note the emphasis on the word ‘can’. Differences in size and/or timing of cash flows do not always lead to a conflict between IRR and NPV.

To illustrate the first point (difference in project size or scale), consider a company with $100 million available to invest. It has two investment projects and as shown below:

Project

CF0

CF1

CF2

CF3

IRR (%)

NPV at 10%

A

-100

120

0

0

20

10

C

-100

0

0

170

19

28

The IRR rule ranks project A first because of the higher IRR. The NPV rule, however, ranks project C first because of the higher NPV. In such a situation, we should follow the NPV rule. This is because the NPV represents the expected addition to shareholder wealth from an investment. The maximization of shareholder wealth is a basic financial objective of a company and hence, the NPV rule must be given preference.

To illustrate the second point (difference in cash flow timings), consider Projects A and D shown in the table below: Project

Investment at t = 0

Cash flow at t = 1

IRR (%)

NPV at 10%

A

-100

120

20%

10

D

1,000

1,150

15%

45

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The IRR rule ranks Project A first because of the higher IRR. The NPV rule, however, ranks Project D first because of the higher NPV. The NPV rule assumption about reinvestment rates is more realistic and more economically relevant because it incorporates the market-determined opportunity cost of capital as a discount rate. Hence, in such a situation as well, the NPV rule should be given preference when deciding between projects.

To summarize, whenever there is a conflict in ranking between the IRR rule and the NPV rule, the NPV rule should be used to decide between mutually exclusive projects.

NPV/IRR Decision Rule Decision Rule Independent projects

NPV and IRR both give the same result – whether to accept or reject the project

Mutually exclusive projects

Use NPV if there is a conflicting decision

3. Portfolio Return Measurement Portfolio performance measurement involves calculating returns in a logical and consistent manner. For instance, determining the returns of a portfolio helps in comparing and ranking different mutual funds. In this section, we cover the following return measures and identify situations when these measures are to be used: 

Holding period return



Money-weighted rate of return (MWRR)



Time-weighted rate of return (TWRR)

Each method has its advantages and disadvantages. The choice of the method depends on what purpose the performance return number will be used for: that is, to determine the manager’s performance, or how the investor’s money performed (personal performance).

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3.1 Holding Period Return Holding Period Return (HPR) is the return that an investor earns over a specified holding period. The formula for calculating HPR for an investment that makes one time cash payment at the end of the holding period is given below: Equation 4: Holding Period Return HPR =

P1 – P0 + D1 P0

where P0 = initial investment P1 = price received at the end of the holding period D1= cash paid by the investment at the end of the holding period

As a simple illustration, assume we buy a stock for $50. Six months later, the stock price goes up to $53 and we receive a dividend of $2. Using equation 4, HPR =

53 – 50 + 2 50

= 0.1 = 10%.

When the performance needs to be measured over multiple periods, or when the portfolio is subject to inflows and withdrawals, use the money-weighted rate of return or the time-weighted rate of return.

3.2 Money-weighted Rate of Return

The money-weighted rate of return accounts for the timing and amount of all cash flows into and out of a portfolio. It is simply the internal rate of return. Let us consider a simple example to illustrate this point. At time t=0, an investor buys a share for $20. At the end of the year 1, he receives a dividend of $0.50 and purchases another share for $22.50. At the end of the year 2, he sells both shares for $23.50 after receiving another dividend of $0.50 per share. What is the money-weighted return?

Since the money-weighted return is the IRR, we can use a financial calculator. The first step is to determine the net cash flows for every period. This is illustrated in the table below: Copyright © Irfanullah Financial Training. All rights reserved.

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Time

Outlay

0

$20.00 to purchase the

Proceeds

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Net Cash Flow -20.00

first share 1

$22.50 to purchase the

$0.50 dividend received

second share

on first share

2

$1.00 received ($0.50 x

-22.50

+48.00

2 shares) $47.00 received from selling 2 shares @ $23.50 per share

After entering these cash flows, use the calculator’s IRR function to find the money-weighted rate of return as 9.39%.

Now consider another example where the initial investment is $100. At the end of the quarter, there is an investment of $25. At the end of six months, there is an investment of $50. At the end of the year, the portfolio is worth $200. There are no other inflows or outflows. What is the money-weighted return?

For this example, we consider each quarter as one period. The data input is as follows: CF0 = -100, CF1 = -25, CF2 = -50, CF3 = 0, CF4 = +200

Note the following important points: 

The periods have to be equally spaced. For this problem, we define a period as one quarter.



There is no cash flow at the end of the third period; it must be explicitly specified as 0.



Money being invested or going into a portfolio is shown as a negative number; money coming out is shown as positive.



In this case, assume the value of the portfolio at the end of the investment period is taken out; so, it is shown as a positive number.

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Compute the IRR. We get IRR i= 4.13%, but this is for one quarter. Annualize the IRR as: (1 + 0.0413)4 – 1 = 0.1756 = 17.56%.

Drawback of money-weighted return: 

Clients determine when and how much money is given. The timing is beyond the investment manager’s control, which can affect the money-weighted rate of return.

3.3 Time-weighted Rate of Return

The time-weighted rate of return measures the compound rate of growth of $1 initially invested in the portfolio over a stated measurement period. 

Preferred performance measure



Overcomes the drawback of money-weighted return by reducing the impact of cash flows: both inflows and withdrawals from the portfolio.



Every period is weighted equally



Useful to compare the performance of managers

The time-weighted return can be calculated using the following steps:

1. Price the portfolio immediately prior to any significant addition or withdrawal of funds. 2. Break the overall evaluation period into subperiods based on the dates of cash inflows and outflows. 3. Calculate the holding period return on the portfolio for each subperiod. 4. Link or compound holding period returns to obtain an annual rate of return for the year (the time-weighted rate of return for the year). If the investment is for more than a year, take the geometric mean of the annual returns to obtain the time-weighted rate of return over that measurement period.

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Equation 5: Calculate TWRR as the Geometric Mean of N Annual Returns 1

TWRR = [(1 + r1 ) ∗ (1 + r2 ) ∗ … (1 + rN)]N − 1 where ri = time-weighted return for year i

Consider the example we discussed above with the following cash flows:

Time

Outlay

0

$20.00 to purchase the first share

1

$22.50 to purchase the second share

2

Proceeds

$0.50 dividend received on first share $1.00 dividends ($0.50 x 2 shares); $47.00 from selling 2 shares @ $23.50 per share

Calculating the TWRR for this example is relatively simple because cash flows only occur at the start/end of every year. During period 1 (which is also year 1), the portfolio value increased from $20.00 to $23.00 (including the dividend received on first share). Each $1.00 invested at t= 0 is worth $1.15 at t= 1, a holding-period return of 15%.

At the start of period 2 (year 2), the portfolio value was $23.00 + 22.50 = $45.50. Note that this is the value of the portfolio after the second share has been purchased. At the end of the period, the portfolio is worth $48.00. This includes the price and dividend for both shares. An increase in the portfolio value from $45.50 to $48.00. Each $1.00 invested at t=1 is worth $1.0549 at t=2, a holding-period return of 5.49%. Using equation 5, the TWRR is calculated as: (1.1500 x 1.0549)1/2 – 1 = 0.1014 = 10.14%.

Now, consider another example where cash flows occur that the start/end of every quarter. Here each quarter is considered a subperiod. The return for each subperiod has already been calculated and is shown below: Copyright © Irfanullah Financial Training. All rights reserved.

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We need to link the returns for the first four quarters (subperiods) in order to determine the return for the year 1. This is done as follows: (1 + 0.10) (1 – 0.05) (1 + 0.15) (1 – 0.10) = 1.0818

Linking the returns for the quarters in the year 2 gives: (1 – 0.20) (1 + 0.30) (1 + 0.20) (1 + 0) = 1.2480 Next we find the geometric mean: (1.0818 x 1.2480) 1/2 = 1.1619

The TWRR is 16.19%.

An example showing the calculation of time-weighted return is shown below. Worked Example 4: Time-Weighted Rate of Return Note: this example has been reproduced from the curriculum. Strubeck Corporation sponsors a pension plan for its employees. It manages part of the equity portfolio in-house and delegates management of the balance to Super Trust Company. As chief investment officer of Strubeck, you want to review the performance of the in-house and Super Trust portfolios over the last four quarters. You have arranged for outflows and inflows to the portfolios to be made at the very beginning of the quarter. The table below summarizes the inflows and outflows as well as the two portfolios’ valuations. In the table, the ending value is the portfolio’s value just prior to the cash inflow or outflow at the beginning of the quarter. The amount invested is the amount each portfolio manager is responsible for investing.

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Quarter 1

2

3

4

In House Account Beginning value

4,000,000

6,000,000

5,775,000

6,720,000

Beginning of Period inflow

1,000,000

-500,000

225,000

-600,000

Amount invested

5,000,000

5,500,000

6,000,000

6,120,000

Ending value

6,000,000

5,775,000

6,720,000

5,508,000

10,000,000

13,200,000

12,240,000

5,659,200

2,000,000

-1,200,000

-7,000,000

-400,000

Amount invested

12,000,000

12,000,000

5,240,000

5,259,200

Ending value

13,200,000

12,240,000

5,659,200

5,469,568

Super Trust Account Beginning value Beginning of Period inflow

Based on the information given, address the following. 1. Calculate the time-weighted rate of return for the in-house account. 2. Calculate the time-weighted rate of return for the Super Trust account

Solution to 1:

To calculate the time-weighted rate of return for the in-house account, we compute the quarterly holding period returns for the account and link them into an annual return. The in-house account’s time-weighted rate of return is 27 percent, calculated as follows: 1Q HPR: r1 = 2Q HPR: r2 = 3Q HPR: r3 = 4Q HPR: r4 =

($6,000,000−$5,000,000) $5,000,000

= 0.20

($5,775,000 − $5,500,000) $5,500,000 ($6,720,000−$6,000,000) $6,000,000 ($5,508,000−$6,120,000) $6,120,000

= 0.05

= 0.123 = −0.10

TWRR = (1 + r1)(1 + r2)(1 + r3)(1 + r4) − 1 = (1.20)(1.05)(1.12)(0.90) − 1 = 0.27 or 27%.

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Solution to 2:

The account managed by Super Trust has a time-weighted rate of return of 26 percent, calculated as follows: 1Q HPR: r1 = 2Q HPR: r2 = 3Q HPR: r3 = 4Q HPR: r4 =

$12,000,000

= 0.10

($12,240,000−$12,000,000) $12,000,000 ($5,659,200 − $5,240,000) $5,240,000 ($5,469,568 − $5,259,200) $5,259,200

= 0.02

= 0.08 = 0.04

TWRR = (1.10)(1.02)(1.08)(1.04) − 1 = 0.26 or 26%

If a client gives an investment manager more funds to invest at an unfavorable time, the manager’s money-weighted rate of return will tend to be depressed. Similarly, if a client adds fund at a favorable time, the money-weighted rate of return will tend to be elevated. The timeweighted rate of return removes these effects.

If the valuation of the portfolio prior to any significant inflows or outflows is costly (as mentioned in Step 1), then we can often obtain a reasonable approximation of the time-weighted rate of return by valuing the portfolio at frequent, regular intervals. The more frequent the valuation, the more accurate the approximation. Worked Example 5: Time-Weighted and Money-Weighted Rates of Return Side by Side Note: this example has been reproduced from the curriculum, but the solution is presented differently. Your task is to compute the investment performance of the Walbright Fund during 2003. The facts are as follows: 

On 1 January 2003, the Walbright Fund had a market value of $100 million.

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

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During the period 1 January 2003 to 30 April 2003, the stocks in the fund showed a capital gain of $10 million.



On 1 May 2003, the stocks in the fund paid a total dividend of $2 million. All dividends were reinvested in additional shares.



Because the fund’s performance had been exceptional, institutions invested an additional $20 million in Walbright on 1 May 2003, raising assets under management to $132 million ($100 + $10 + $2 + $20).



On 31 December 2003, Walbright received total dividends of $2.64 million. The fund’s market value on 31 December 2003, not including the $2.64 million in dividends, was $140 million.

The fund made no other interim cash payments during 2003. Based on the information given, address the following: 1. Compute the Walbright Fund’s time-weighted rate of return. 2. Compute the Walbright Fund’s money-weighted rate of return. 3. Interpret the differences between the time-weighted and money-weighted rates of return.

Solution to 1:

Because interim cash flows were made on 1 May 2003, we must compute two interim total returns and then link them to obtain an annual return. The table below lists the relevant market values on 1 January, 1 May, and 31 December as well as the associated interim four-month (1 January to 1 May) and eight-month (1 May to 31 December) holding period returns.

Date

Value and Return

1 January 2003

Beginning portfolio value = $100 million

1 May 2003

Portfolio value = $110 million Dividends received before additional investment = $2 million; Holding period return =

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2 + 10 100

= 12%

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New investment = $20 million Beginning market value for next period = $132 million 1 September 2003

No activity

31 December 2003

Dividends received = $2.64 million Ending portfolio value = $140 million Holding period return =

132

= 8.06%

Now we must geometrically link the four- and eight-month returns to compute an annual return. We compute the time-weighted return as follows: Time-weighted return = 1.12 × 1.0806 − 1 = 0.2103

In this instance, we compute a time-weighted rate of return of 21.03 percent for one year. The four-month and eight-month intervals combine to equal one year.

Solution to 2:

To compute the MWRR, enter the net cash flows and use the IRR function on the calculator. The cash flow input: CF0 = -100, CF01 = -20, CF2 = 0, CF3 = 142.64. It is important to recognize that there are three four-month periods during the year. That is why, we explicitly enter CF2 as 0. The calculator gives a four-month IRR of 6.28 percent. The quick way to annualize this is to multiply by 3. A more accurate way is (1.0628) 3 − 1 = 0.20 or 20 percent.

Solution to 3:

In this example, the time-weighted return (21.03 percent) is greater than the money-weighted return (20 percent). The Walbright Fund’s performance was relatively poorer during the eightmonth period, when the fund owned more shares, than it was overall. This fact is reflected in a lower money-weighted rate of return compared with time-weighted rate of return, as the money-

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weighted return is sensitive to the timing and amount of withdrawals and additions to the portfolio.

The table below summarizes the most important points related to the MWRR and TWRR.

Money-weighted vs. Time-weighted Rate of Return Money-Weighted Rate of Return

Time-Weighted Rate of Return

Simply the IRR

Compound rate of growth of $1 initially invested in the portfolio over a stated measurement period

Return depends on timing and amount of cash

The return does NOT depend on the timing

flows

and amount of cash flows

Not an appropriate performance measure if

Appropriate performance measure if the

the portfolio manager does not control the

portfolio manager does not control the

timing and amount of investment

timing and amount of investment

Appropriate measure if portfolio manager has

Not an appropriate measure if portfolio

control over the timing and amount of

manager has control over the timing and

investment

amount of investment

4. Money Market Yields The money market is the market for short-term debt instruments (one year maturity or less). Some instruments require the issuer to repay the lender the amount borrowed plus interest. Others are pure discount instruments that pay interest as the difference between the amount borrowed and the amount paid back. U.S. T-bills are pure discount instruments and are quoted on a bank discount yield basis.

The following yield measures are referenced when dealing with money market instruments: 

Bank discount yield



Holding period yield

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Discounted Cash Flow Applications



Effective annual yield



Money market yield



Bond equivalent yield

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As a basis for understanding these yield measures, let us consider the following instrument: a T bill with a face value (or par value) of $100,000 and 150 days until maturity which is selling for $98,000.

4.1

Bank Discount Yield

The bank discount yield is computed as follows:

Equation 6: Bank Discount Yield BDY = (D/F) * (360/t) where BDY = the annualized yield on a bank discount basis D = the dollar discount, which is equal to the difference between the face value of the bill and the purchase price F = face value of the bill P0 = purchase price t = the actual number of days remaining to maturity

For the T-bill described above, the dollar discount, D, is $2,000. The yield on a bank discount basis is 4.8 percent as shown below:

BDY =

$2,000 $10000

∗(

360 150

) = 0.048 = 4.8%

The bank discount formula takes the T-bill’s dollar discount from face or par as a fraction of face value, and then annualizes by the factor 360/150. Copyright © Irfanullah Financial Training. All rights reserved.

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At times, we might be given the bank discounted yield and be asked to calculate the price of the T-bill. Given a bank discount yield of 4.8%, we can solve for the dollar discount, D, as follows:

D=

𝐵𝐷𝑌∗𝐹∗𝑡 360

= 0.048 × $100,000 × 150/360 = $2,000.

Once we have computed the dollar discount, the price for the T-bill is its face value minus the dollar discount: F − D = $100,000 − $2,000 = $98,000. Why the bank discount yield is not an accurate measure of investors’ return: 

The yield is based on the face value of the bond, not on its purchase price.



The yield is annualized based on a 360-day year rather than a 365-day year.



The bank discount yield annualizes with simple interest, which ignores the opportunity to earn interest on interest (compound interest).

4.2

Holding Period Return

It is the return an investor will earn by holding the instrument to maturity. It is also known as holding period yield (HPY).

Equation 7: Holding Period Yield HPY =

P1 – P0 + D1 P0

where P0 = initial purchase price of the instrument P1 = price received for the instrument at maturity; D1 = cash distribution by paid by the instrument at maturity (interest)

Note: This concept has been discussed ealier in section 3.1 and is being reproduced here.

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Consider the T-bill from above which is selling for $98,000. The face value (or par value) $100,000 and 150 days remain till maturity. There is no cash distribution and so D 1 = 0.

HPY =

4.3

(100,000 – 98,000 – 0) 98,000

= 0.0204 = 2.04%

Effective Annual Yield

The effective annual yield is calculated as follows:

Equation 8: Effective Annual Yield EAY = (1 + HPY)365/t – 1 where HPY = holding price yield 365 = number of days in a year t = the actual number of days remaining to maturity Plugging in the numbers for our T-bill we have: EAY = (1 + 0.0204)365/150 – 1 = 0.0504 = 5.04%

EAY is conceptually similar to effective annual return (EAR) which was discussed in the reading on time value of money: EAR = (1 + periodic interest rate) m – 1

Each period in the EAR expression is essentially a holding period; therefore, HPY and the periodic interest rate are synonymous. Recall that the exponent, ‘m’, in the EAR expression represents the numbers of periods in a year. 365/t in the EAY expression can be interpreted in exactly the same way.

EAY addresses all three weaknesses associated with the bank discount yield. Copyright © Irfanullah Financial Training. All rights reserved.

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Comparison between bank discount yield and effective annual yield Bank Discount Yield

Effective Annual Yield

1. The yield is based on the face value of

1. The yield is based on purchase price.

the bond, not on its purchase price. 2. The yield is annualized based on a 360-

2. The yield is annualized based on a 365-

day year rather than a 365-day year. 3. The bank discount yield annualizes with

day year. 3. The effective annual yield annualizes

simple interest, which ignores the

using an exponent and hence includes

opportunity to earn interest on interest

interest on interest (compound interest).

(compound interest).

Bank discount yield is less than the effective annual yield

4.4

Money Market Yield

The money market yield (also known as the CD equivalent yield) is computed by annualizing the HPY assuming simple interest and a 360-day year. The formula is as follows:

Equation 9: Money Market Yield MMY =

HPY∗360 t

where HPY = holding price yield 365 = number of days in a year t = the actual number of days remaining to maturity

Plugging in the numbers for our T-bill we have: MMY =

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0.0204∗360 150

= 0.0490 = 4.90%

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Worked Example 6: Using the Appropriate Discount Rate Note: this example has been reproduced from the curriculum. You need to find the present value of a cash flow of $1,000 that is to be received in 150 days. You decide to look at a T-bill maturing in 150 days to determine the relevant interest rate for calculating the present value. You have found a variety of yields for the 150-day bill as shown below:

Holding period yield

2.04%

Bank discount yield

4.80%

Money market yield

4.90%

Effective annual yield

5.04%

Which yield or yields are appropriate for finding the present value of the $1,000 to be received in 150 days?

Solution:

The holding period yield is appropriate, and we can also use the money market yield and effective annual yield after converting them to a holding period yield. 

Holding period yield (2.0408 percent). This yield is exactly what we want. Because it applies to a 150-day period, we can use it in a straightforward fashion to find the present value of the $1,000 to be received in 150 days. (Recall the principle that discount rates must be compatible with the time period.) The present value is

$1,000

= $980.

1.02048

Now we can see why the other yield measures are inappropriate or not as easily applied: 

Bank discount yield (4.80 percent). We should not use this yield measure to determine the present value of the cash flow. As mentioned earlier, the bank discount yield is based on the face value of the bill and not on its price.

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

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Money market yield (4.90 percent). To use the money market yield, we need to convert it to the 150-day holding period yield by dividing it by (360/150). After obtaining the holding period yield



0.04898 360 150

= 0.020408, we use it to discount the $1,000 as above.

Effective annual yield (5.0388 percent). This yield has also been annualized, so it must be adjusted to be compatible with the timing of the cash flow. We can obtain the holding period yield from the EAY as follows: (1.050388)150/365− 1 = 0.020408

Recall that when we found the effective annual yield, the exponent was 365/150, or the number of 150-day periods in a 365-day year. To shrink the effective annual yield to a 150-day yield, we use the reciprocal of the exponent that we used to annualize.

4.5

Bond Equivalent Yield

We frequently need to convert periodic rates to annual rates. Bond investors compute IRRs for bonds, known as yield to maturity (YTM). An approach used in the U.S. bond markets, however, is to double the semiannual YTM. The yield to maturity calculated this way, ignoring compounding, is known as the bond-equivalent yield. As a simple example, say the semi-annual YTM on a bond is 4.00%. The bond equivalent yield (YTM) will then be 4.00 x 2 = 8.00%. Note that the BEY is less than the EAY which is 1.04 2 – 1 = 0.0816 or 8.16%.

Bond-equivalent yield = 2* semi-annual YTM

Converting between yield measures

The table below summarizes how to convert between different yield measures:

How to convert between different yield measures Conversion

Formula / Method

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HPY∗360

HPY to MMY

MMY =

HPY to EAY

EAY = (1 + HPY)365/t – 1

HPY to BDY (bank

Given the face value and HPY, compute price and the discount.

discount yield)

Using the discount, face value and number of days to maturity,

t

compute the bank discount yield. BDY to EAY

From the BDY, compute the discount. From the discount, compute the HPY. From the HPY, compute the EAY.

BDY to MMY

From the BDY, compute the discount and the price From the discount and the price, compute the HPY. From the HPY, compute the MMY.

The curriculum outlines a formula for converting a bank discount yield to a money market yield, but it is easier and more intuitive to use the process shown in the table.

Going back to the T-bill example, assume the bank discount yield is given as 4.80%. Here is the three step process: 1. Compute the discount: 0.048 × $100,000 × 150/360 = $2,000 and the price: 100,000 – 2,000 = 98,000. 2. From the discount and price, we can compute the HPY: 2,000/98,000 = 0.024. 3. From the HPY, compute the MMY: 0.024 x 360/150 = 0.0490 = 4.90%.

Summary 

The net present value (NPV) of an investment is the present value of its cash inflows minus the present value of its cash outflows. o For an independent project, accept the project if NPV is positive. o For mutually exclusive projects, accept the project with the higher positive NPV.



IRR is the discount rate that makes the net present value equal to zero. o For an independent project, accept if IRR is greater than the cost of capital.

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o For mutually exclusive projects, accept the project with the higher IRR as long as IRR is greater than the cost of capital. 

At times NPV and IRR might rank projects differently. This is because IRR makes the incorrect assumption that interim cash flows are reinvested at the IRR rate. o Ranking conflicts may occur when there are differences in the scale or timing of cash flows



When there is a conflict in rankings from the IRR rule and NPV rule, the NPV rule should be used to decide between mutually exclusive projects.



The money-weighted rate of return is the internal rate of return on a portfolio; it accounts for the timing and amount of all cash flows into and out of the portfolio.



The time-weighted rate of return removes the effects of timing and amounts of withdrawals and additions to the portofolio and reflects the compound rate of growth of one unit of currency invested over a stated measurement period.



Time-weighted rate of return is the standard in the investment management industry.



Money-weighted rate of return can be appropriate if the investor exercises control over additions and withdrwals to the portfolio.



Holding Period Return (HPR) is the return an investor earns over a specified holding period. To calculate HPR for an investment that makes a one time cash payment at the end of the holding period, the formula is: HPR = (P1 – P0 + D1)/P0 o For a U.S. Treasury bill, HPY = D/P0 where D is the dollar discount.



The bank discount yield for U.S. Treasury bills (and other money-market instruments sold on a discount basis) is given by BDY = (F − P)/F × 360/t = D/F × 360/t, where F is the face amount to be received at maturity, P is the price of the Treasury bill, t is the number of days to maturity, and D is the dollar discount.



The effective annual yield (EAY) is (1 + HPY) 365/t − 1. The money market yield is given by MMY = HPY × 360/t, where t is the number of days to maturity.



We can convert back and forth between holding period yields, money market yields, and effective annual yields by using the holding period yield, which is common to all the calculations.



The bond equivalent yield of a yield stated on a semiannual basis is that yield multiplied by two.

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Next Steps 

Make sure you are comfortable using the financial calculator to calculate the NPV and IRR.



Work through the examples presented in the curriculum.



Solve the practice problems in the curriculum.



Solve the IFT Practice Questions associated with this reading.



Review the learning outcomes presented in the curriculum. Make sure that you can peform that actions implied by learning outcome.

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