# Discounted cash flow methods – NPV v/s IRR

Slide 1: Discounted Cash Flow Applications Volume 1 Study Session 1 Reading 6

Slide 2: Agenda

The agenda of this session is to cover Reading 6 as prescribed by the CFA Level I Exam Curriculum. You can go through this reading from Volume 1 of CFA Level I training made available to you by CFA Institute for completing the CFA program.

Slide 2: Agenda

The agenda of this session is to cover Reading 6 as prescribed by the CFA Level I Exam Curriculum. You can go through this reading from Volume 1 of CFA Level I training made available to you by CFA Institute for completing the CFA program.

In this lesson we will start by understanding the concept of discounted cash flow analysis using the two most widely used techniques i.e. NPV and IRR. Next we will be discussing NPV and IRR and their rules. Next we will learn the distinguishing features of the IRR and NPV rules. An investment manager often faces the task of investing funds for the short-term and therefore to understand the choices available they need to understand the calculation of money market yields. We would also cover the calculation and interpretation of a holding period return. Then we would discuss the calculation and differences between the money-weighted and time-weighted rate of return of a portfolio and then evaluate the performance of portfolios based on these measures. This lesson also covers the calculation and interpretation of bank discount yield, holding period yield, and money market yield. Lastly we will be covering the conversion calculations among the various yields such as money market and effective annual yield.

As per the learning outcome statements after completion of this lesson the candidate should be able to calculate and interpret the net present value and the internal rate of return of an investment; contrast the NPV rule to the IRR rule, and identify problems associated with the IRR rule; calculate and interpret a holding period return or total return; calculate, interpret, and distinguish between the money-weighted and time weighted rates of return of a portfolio, and evaluate the performance of portfolios based on these measures; calculate and interpret the bank discount yield, holding period yield, effective annual yield, and money market yield for a U.S. Treasury bill and convert among holding period yields, money market yields, effective annual yields, and bond equivalent yields.

Slide 3: Discounted Cash Flow Analysis:

Let us now start with gaining an understanding of Discounted Cash Flow Analysis or DCF. DCF is a valuation method used to estimate the attractiveness of an investment opportunity. Discounted cash flow analysis uses future free cash flow projections and discounts them to arrive at a present value, which is used to evaluate the potential for investment. A DCF analysis yields the overall value of a business including both debt and equity. It is calculated using the formula highlighted in the slide.

There are three major financial decisions within any organization they are discussed as follows:

- Capital Budgeting: It is the process in which a business determines whether projects such as building a new plant or investing in a long-term venture are worth pursuing. It is used to describe how managers plan significant outlays on projects that have long-term implications such as the purchase of new equipment and the introduction of new products. Popular methods of capital budgeting include net present value or NPV, internal rate of return or IRR, discounted cash flow and payback period.
- Capital Structure: Capital structure is a mix of a company's long-term debt, specific short-term debt, common equity and preferred equity. The capital structure is how a firm finances its overall operations and growth by using different sources of funds. It refers to a firm's debt-to-equity ratio, which provides insight into how risky a company is. Usually a company more heavily financed by debt poses greater risk, as this firm is relatively highly levered.
- Working Capital Management: It is the management of the company’s short term assets and short term liabilities.Working capital management ensures a company has sufficient cash flow in order to meet its short-term debt obligations and operating expenses.

Let us now forward to the next slide and discuss NPV and IRR techniques of discounted cash flows.

Slide 4: NPV and Its Rule

In the last slide we discussed the concept of DCF analysis where we came across the various tools used in DCF such as NPV and IRR. Let us now begin with the discussion of NPV.

Net Present Value or NPV is the difference between the present value of cash inflows and the present value of cash outflows. It is used in capital budgeting to analyze the profitability of an investment or project.

NPV compares the value of a dollar today to the value of that same dollar in the future, taking inflation and returns into account. If the NPV of a prospective project is positive, it should be accepted. However, if NPV is negative, the project should probably be rejected because cash flows will also be negative.For example, an investment of $1,000 today at 10 percent will yield $1,100 at the end of the year. A positive net present value means a better return, and a negative net present value means a worse return.

To calculate NPV first identify all the cash flows associated with the investment i.e. all inflows and outflows associated with the project. After identifying the cash flows the appropriate discount rate or opportunity cost is determined r for the investment project based on expected return. After determining the r the present value of cash flow i.e. inflow and outflow is assessed. The increased NPV has a positive inflow and decreased NPV has a negative outflow. Sum all the present value of inflow and outflow. Apply the NPV formula. If the investment’s NPV is positive an investor should undertake it and if the NPV is negative the investor should not undertake it.

Slide 5: IRR and Its Rule

In slide 4 we explained to you NPV and its rule that is applied to evaluate the time value of money. The other popular DCF technique is Internal Rate of Return or IRR which we shall now discuss in this slide.

IRR is the discount rate that makes NPV zero. It is the rate that equals the present value of a future steam of cash flows to the initial investment. If the IRR is higher than the desired rate of return on investment, then the project is a desirable one. IRR can be used to rank several prospective projects a firm is considering. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first.

The rule of the IRR states to “Accept project or investments for which the IRR is greater than the opportunity cost of capital”.

This means that the IRR uses the opportunity cost as the hurdle rate which the project must exceed for the project to be accepted. Simply stated the IRR rule states that if the internal rate of return on a project or investment is greater than the minimum required rate of return or the cost of capital then the decision would generally be to go ahead with it. Conversely, if the IRR on a project or investment is lower than the cost of capital, then the best course of action may be to reject it. The IRR rule may not always be enforced because of its inconsistent results with various projects when a company may prefer a larger project with a lower IRR.

Slide 6: How are NPV and IRR different?

In the above two slides we discussed the NPV and IRR rules. Let’s now discuss the differences between the two rules in terms of assessing the investment or a project.

The net present value of an investment is the present value of its cash inflows minus the present value of its cash outflows. The internal rate of return (IRR) is that discount rate which makes net present value equal to zero.

According to the NPV rule the company should accept projects where the NPV is positive and reject where the NPV is negative. This means that a positive NPV indicates cash inflow outweighing cash outflows on a present value basis and thus the project should be accepted. It also signifies that positive cash flows are sufficient to repay the initial investment along with the opportunity cost associated with the project. If the company is to choose between two mutually-exclusive projects, the one with the higher NPV should be chosen.

According to the IRR rule the company should accept those projects where the IRR is greater than the discount rate used and reject those where the IRR is less than the discount rate. An IRR greater than the discount rate suggests that the project will repay more than the opportunity costs incurred.

Slide 7: Problems of IRR Rule

There are few limitations of the IRR rule that will be discussed in this slide. There are three problems associated with IRR rule.

The first problem is the Reinvestment Problem.

Calculation of the IRR assumes that all project cash flows can be reinvested to earn a rate of return exactly equal to the IRR itself. In other words, a project with an IRR of say 6 Persent assumes that all cash flows can be reinvested to earn exactly 6 Persent.

The second problem is the Scale Problem.

One of the situations in which IRR is likely to contradict NPV, is when there are two, mutually-exclusive projects of greatly differing scale. One that requires a relatively small investment and returns relatively small cash flows, compared to another that requires a much larger investment and returns much larger cash flows.

The third and last problem is the Timing Problem.

The other situation in which IRR is likely to contradict NPV, is that of two, mutually-exclusive projects whose cash flows are timed very differently - one that receives its largest cash flows early in the project versus another that receives its largest cash flows late in the project.

Slide 8: How is Portfolio Return Measured?

Let us now discuss some ways in which we can measure the portfolio returns. A portfolio is a pool of assets bought to provide returns within a stipulated risk as per a stated mandate or objective of the portfolio.

The first type of measure is Holding Period Return or HPR.

As an investor it becomes quite important to assess the success of the investments. There are two ways of measuring the performance. The first is performance measurement which involves calculating returns in a logical and consistent manner. The second is the performance appraisal that provides a basis for accurate performance measurement.

Holding period return is a very basic way to measure how much return you have obtained on a particular investment. The returns on an investment may be shown on an annual, quarterly, or monthly basis. The return is equal to the income and other gains such as appreciation earned from the asset, divided by the original cost of the asset. The holding period return can be calculated for any asset, including a bond, an individual stock, or a complete portfolio. The formula to calculate HPY is given in the slide.

Slide 9: How is Portfolio Return Measured?

Let us now discuss the other two performance measurement concepts, the Money and Time weighted rate of return. Let’s discuss them one by one.

First we shall discuss Money-Weighted Rate of Return.

It is called the Money-weighted rate of return because it accounts for the timing and amount of all cash flows in and out of the portfolio. Money weighted rate of return is the rate of return on an investment, expressed as a percentage of the total amount invested. It reflects the size and timing of cash flows, so it is an effective measure for returns on a portfolio. Using money-weighted rate of return investors can easily determine if they are making a consistent month on month return. It is also ideal for comparing investment performance over time regardless of the size of the investment.

It is calculated by finding the rate of return that will set the present values of all cash flows and terminal values equal to the value of the initial investment.

The next and last tool for portfolio returns measurement is Time Weighted rate Of Return.

The time-weighted rate of return is the preferred performance measure as it is not sensitive to contributions or withdrawals. It is measured as the compounded growth rate of $1 initially invested in the portfolio over the period being measured. In contrast to the money-weighted rate of return it is not affected by the cash withdrawals or additions to the portfolio. There are three steps to compute the time-weighted rate of return:

Step 1 - Price the portfolio immediately before any significant addition or withdrawal of funds.

Step 2 - Calculate the holding period return on the portfolio for each sub period.

Step 3- Compound the holding period returns to obtain an annual rate of return for the year. If the investment is for more than a year the take the geometric mean of the annual returns to obtain time-weighted rate of return.

Slide 10: What are Money Market Yields

So far in our discussion of internal rate of return and NPV we referred to the opportunity cost of capital. Both IRR and NPV are used to calculate return on an equity investment. In this slide we begin with the discussion of discounted cash flow analysis in actual money markets by considering short-term debt markets. The Yield is used to calculate the returns on debts or bonds. In this slide and the next slide we would discuss various types of yields.

First lets us discuss Bank Discount Yield or BDY.

Discount yield is most frequently used to calculate the yield on short-term bonds and treasury bills sold at a discount. This yield calculation uses a 30-day month and 360-day year to simplify calculations. BDY assumes simple interest; the yield is also based on the face value of the bill and not the purchase price. BDY is used by the Federal Reserve to calculate yields on T-bills with less than one year of maturity, typically the 3 or 6 month bill.

Now we’ll discuss Holding Period Yield or HPY.

Holding period yield is calculated as the sum of all income and capital growth divided by the value at the beginning of the period being measured is a very basic way to measure how much return you have obtained on a particular investment. This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it difficult to compare returns on different investments with different time frames.

Let’s now move to second slide and discuss two other types of yields.

Slide11: What are Money Market Yields?

This slide covers the third and fourth type of money market yields.

The third type of Money Market Yield is Effective Annual Yield or EAY.

EAY takes the HPY and annualizes the number to facilitate comparability with other investments. Effective yield is a more accurate measure of the investor's return than calculating a simple annual interest rate because effective yield takes compounding into account.

Another measure of yield is the money market yield also known as the CD equivalent yield and is denoted as manifested in slide. This yield measure can be calculated in two ways:

1. When the HPY is given, money market yield is the annualized yield based on a 360-day year. The formula is shown in the slide.

2. When bond price is unknown, bank discount yield can be used to compute the money market yield, using this expression. The formula to calculate yield is shown in the slide.

Let’s discuss all four types of yields with the help of an example. This should clear most of your doubts.

Slide 12: Yield Calculated for a U.S. Treasury bill

For T-bill which has a face value of $50,000, a current market price of $49,700 and a maturity in 100 days we will calculate all four types of yield.

To calculate Bank Discount Yield we will first calculate D or discount which is equal to difference between face value ‘F’ $50,000 and the current market rate of the bond $49,700. ‘t’ is maturity period of the bond and is 100 days. Input all values in the formula to calculate BDY. The calculation is shown in the slide.

To calculate Holding Period Yield input P1 or price at maturity as $50000 and P0 or purchase price or discounted price or market price as $49,700. D1 or cash distribution at maturity is zero. Input all values in the formula to calculate the results. Calculation is shown in slide.

Once we have calculated Holding Period Return or HPY we can easily calculate Effective Annual yield and money Market yield. Calculation for both EAY and money market yield is shown in the slide.

Slide 4: NPV and Its Rule

In the last slide we discussed the concept of DCF analysis where we came across the various tools used in DCF such as NPV and IRR. Let us now begin with the discussion of NPV.

Net Present Value or NPV is the difference between the present value of cash inflows and the present value of cash outflows. It is used in capital budgeting to analyze the profitability of an investment or project.

NPV compares the value of a dollar today to the value of that same dollar in the future, taking inflation and returns into account. If the NPV of a prospective project is positive, it should be accepted. However, if NPV is negative, the project should probably be rejected because cash flows will also be negative.For example, an investment of $1,000 today at 10 percent will yield $1,100 at the end of the year. A positive net present value means a better return, and a negative net present value means a worse return.

To calculate NPV first identify all the cash flows associated with the investment i.e. all inflows and outflows associated with the project. After identifying the cash flows the appropriate discount rate or opportunity cost is determined r for the investment project based on expected return. After determining the r the present value of cash flow i.e. inflow and outflow is assessed. The increased NPV has a positive inflow and decreased NPV has a negative outflow. Sum all the present value of inflow and outflow. Apply the NPV formula. If the investment’s NPV is positive an investor should undertake it and if the NPV is negative the investor should not undertake it.

Slide 5: IRR and Its Rule

In slide 4 we explained to you NPV and its rule that is applied to evaluate the time value of money. The other popular DCF technique is Internal Rate of Return or IRR which we shall now discuss in this slide.

IRR is the discount rate that makes NPV zero. It is the rate that equals the present value of a future steam of cash flows to the initial investment. If the IRR is higher than the desired rate of return on investment, then the project is a desirable one. IRR can be used to rank several prospective projects a firm is considering. Assuming all other factors are equal among the various projects, the project with the highest IRR would probably be considered the best and undertaken first.

The rule of the IRR states to “Accept project or investments for which the IRR is greater than the opportunity cost of capital”.

This means that the IRR uses the opportunity cost as the hurdle rate which the project must exceed for the project to be accepted. Simply stated the IRR rule states that if the internal rate of return on a project or investment is greater than the minimum required rate of return or the cost of capital then the decision would generally be to go ahead with it. Conversely, if the IRR on a project or investment is lower than the cost of capital, then the best course of action may be to reject it. The IRR rule may not always be enforced because of its inconsistent results with various projects when a company may prefer a larger project with a lower IRR.

Slide 6: How are NPV and IRR different?

In the above two slides we discussed the NPV and IRR rules. Let’s now discuss the differences between the two rules in terms of assessing the investment or a project.

The net present value of an investment is the present value of its cash inflows minus the present value of its cash outflows. The internal rate of return (IRR) is that discount rate which makes net present value equal to zero.

According to the NPV rule the company should accept projects where the NPV is positive and reject where the NPV is negative. This means that a positive NPV indicates cash inflow outweighing cash outflows on a present value basis and thus the project should be accepted. It also signifies that positive cash flows are sufficient to repay the initial investment along with the opportunity cost associated with the project. If the company is to choose between two mutually-exclusive projects, the one with the higher NPV should be chosen.

According to the IRR rule the company should accept those projects where the IRR is greater than the discount rate used and reject those where the IRR is less than the discount rate. An IRR greater than the discount rate suggests that the project will repay more than the opportunity costs incurred.

Slide 7: Problems of IRR Rule

There are few limitations of the IRR rule that will be discussed in this slide. There are three problems associated with IRR rule.

The first problem is the Reinvestment Problem.

Calculation of the IRR assumes that all project cash flows can be reinvested to earn a rate of return exactly equal to the IRR itself. In other words, a project with an IRR of say 6 Persent assumes that all cash flows can be reinvested to earn exactly 6 Persent.

The second problem is the Scale Problem.

One of the situations in which IRR is likely to contradict NPV, is when there are two, mutually-exclusive projects of greatly differing scale. One that requires a relatively small investment and returns relatively small cash flows, compared to another that requires a much larger investment and returns much larger cash flows.

The third and last problem is the Timing Problem.

The other situation in which IRR is likely to contradict NPV, is that of two, mutually-exclusive projects whose cash flows are timed very differently - one that receives its largest cash flows early in the project versus another that receives its largest cash flows late in the project.

Slide 8: How is Portfolio Return Measured?

Let us now discuss some ways in which we can measure the portfolio returns. A portfolio is a pool of assets bought to provide returns within a stipulated risk as per a stated mandate or objective of the portfolio.

The first type of measure is Holding Period Return or HPR.

As an investor it becomes quite important to assess the success of the investments. There are two ways of measuring the performance. The first is performance measurement which involves calculating returns in a logical and consistent manner. The second is the performance appraisal that provides a basis for accurate performance measurement.

Holding period return is a very basic way to measure how much return you have obtained on a particular investment. The returns on an investment may be shown on an annual, quarterly, or monthly basis. The return is equal to the income and other gains such as appreciation earned from the asset, divided by the original cost of the asset. The holding period return can be calculated for any asset, including a bond, an individual stock, or a complete portfolio. The formula to calculate HPY is given in the slide.

Slide 9: How is Portfolio Return Measured?

Let us now discuss the other two performance measurement concepts, the Money and Time weighted rate of return. Let’s discuss them one by one.

First we shall discuss Money-Weighted Rate of Return.

It is called the Money-weighted rate of return because it accounts for the timing and amount of all cash flows in and out of the portfolio. Money weighted rate of return is the rate of return on an investment, expressed as a percentage of the total amount invested. It reflects the size and timing of cash flows, so it is an effective measure for returns on a portfolio. Using money-weighted rate of return investors can easily determine if they are making a consistent month on month return. It is also ideal for comparing investment performance over time regardless of the size of the investment.

It is calculated by finding the rate of return that will set the present values of all cash flows and terminal values equal to the value of the initial investment.

The next and last tool for portfolio returns measurement is Time Weighted rate Of Return.

The time-weighted rate of return is the preferred performance measure as it is not sensitive to contributions or withdrawals. It is measured as the compounded growth rate of $1 initially invested in the portfolio over the period being measured. In contrast to the money-weighted rate of return it is not affected by the cash withdrawals or additions to the portfolio. There are three steps to compute the time-weighted rate of return:

Step 1 - Price the portfolio immediately before any significant addition or withdrawal of funds.

Step 2 - Calculate the holding period return on the portfolio for each sub period.

Step 3- Compound the holding period returns to obtain an annual rate of return for the year. If the investment is for more than a year the take the geometric mean of the annual returns to obtain time-weighted rate of return.

Slide 10: What are Money Market Yields

So far in our discussion of internal rate of return and NPV we referred to the opportunity cost of capital. Both IRR and NPV are used to calculate return on an equity investment. In this slide we begin with the discussion of discounted cash flow analysis in actual money markets by considering short-term debt markets. The Yield is used to calculate the returns on debts or bonds. In this slide and the next slide we would discuss various types of yields.

First lets us discuss Bank Discount Yield or BDY.

Discount yield is most frequently used to calculate the yield on short-term bonds and treasury bills sold at a discount. This yield calculation uses a 30-day month and 360-day year to simplify calculations. BDY assumes simple interest; the yield is also based on the face value of the bill and not the purchase price. BDY is used by the Federal Reserve to calculate yields on T-bills with less than one year of maturity, typically the 3 or 6 month bill.

Now we’ll discuss Holding Period Yield or HPY.

Holding period yield is calculated as the sum of all income and capital growth divided by the value at the beginning of the period being measured is a very basic way to measure how much return you have obtained on a particular investment. This calculation is on a per-dollar-invested basis, rather than a time basis, which makes it difficult to compare returns on different investments with different time frames.

Let’s now move to second slide and discuss two other types of yields.

Slide11: What are Money Market Yields?

This slide covers the third and fourth type of money market yields.

The third type of Money Market Yield is Effective Annual Yield or EAY.

EAY takes the HPY and annualizes the number to facilitate comparability with other investments. Effective yield is a more accurate measure of the investor's return than calculating a simple annual interest rate because effective yield takes compounding into account.

Another measure of yield is the money market yield also known as the CD equivalent yield and is denoted as manifested in slide. This yield measure can be calculated in two ways:

1. When the HPY is given, money market yield is the annualized yield based on a 360-day year. The formula is shown in the slide.

2. When bond price is unknown, bank discount yield can be used to compute the money market yield, using this expression. The formula to calculate yield is shown in the slide.

Let’s discuss all four types of yields with the help of an example. This should clear most of your doubts.

Slide 12: Yield Calculated for a U.S. Treasury bill

For T-bill which has a face value of $50,000, a current market price of $49,700 and a maturity in 100 days we will calculate all four types of yield.

To calculate Bank Discount Yield we will first calculate D or discount which is equal to difference between face value ‘F’ $50,000 and the current market rate of the bond $49,700. ‘t’ is maturity period of the bond and is 100 days. Input all values in the formula to calculate BDY. The calculation is shown in the slide.

To calculate Holding Period Yield input P1 or price at maturity as $50000 and P0 or purchase price or discounted price or market price as $49,700. D1 or cash distribution at maturity is zero. Input all values in the formula to calculate the results. Calculation is shown in slide.

Once we have calculated Holding Period Return or HPY we can easily calculate Effective Annual yield and money Market yield. Calculation for both EAY and money market yield is shown in the slide.