Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Time Value of Money

Because of assumed inflation, the purchasing power of a Dollar in the future should be less than it is today. Investing is the means by which one can offset this loss of purchasing power. With compound interest, a dollar invested today has the potential to grow exponentially. The Future Value is computed as follows:

The Future Value of an investment made today, which has a current or , PV, is a function of its return, r, over N periods.

If provided the FV, one can simply manipulate the equation to derive the PV:

This process is known as as the FV is discounted at the assumed , r, to derive the PV.

Net Present Value

When deriving the NPV of a series of future cash flows, one would expand the formula as follows:

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Note how each period cash flow, CFn, is discounted to the present. The , NPV, is the sum of the discounted present values of the individual future cash flows.

The discount rate, r, in a should be the of a project or the required return of the investor. The , TV, reflects either the return of principal (in the case of a investment) or the projected Future Value of the project/investment. In the case of the latter, the Terminal Value (or projected Future Value) represents the discounted present value of ALL future cash flows of the asset. It assumes, therefore, that the asset has an infinite life.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Example:

Cash Flow Discount Rate Discount Factor Discounted PV 1 $ 100 5% 0.9523810 $ 95.24 2 $ 100 5% 0.9070295 $ 90.70 3 $ 100 5% 0.8638376 $ 86.38 4 $ 100 5% 0.8227025 $ 82.27 5 $ 100 5% 0.7835262 $ 78.35 TV $ 1,000 5% 0.7835262 $ 783.53

PV = $ 1,216.47

Total cash flows earned from the investment are $500 (i.e., $100 each year for five years). At the end of year 5, the original principal of $1,000 is returned to the investor. The Discounted Cash Flow approach and the NPV answer the following question: how much is this cash flow stream worth on a present value basis if the required rate of return is 5%?

Answer: the investor should pay no more than $1,216.47.

Note in the table how each individual cash flow, including the Terminal Value, is discounted to the present at the discount rate, r (i.e., 5%). The “discount factor” is the term:

The discount factor is multiplied by the cash flow that corresponds with the period to derive the discounted present value of the individual cash flow. The sum of the individual discounted present values is the Net Present Value of the cash flow stream.

Interpretation: the Net Present Value is the most that an investor should pay, up front, for this cash flow stream. The NPV will be quite sensitive to the discount rate. Consequently, one could view the discount rate from the perspective of or “hurdle rate”. In other words, based on an initial investment of $1,216.47, one could compare the investment reflected in this cash flow stream against other investments with similar risk parameters.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Internal Rate of Return

Situation: you have a friend who recently won the lottery. He/she has the option of receiving an upfront, lump sum payment of $875,000 or an annuity stream of $100,000 each year for ten years. How do you advise your friend?

Analysis: you are provided a Net Present Value (i.e., $875,000) and a cash flow stream of $100,000 annually for ten years. This would appear to be a Net Present Value problem in which you need to solve for the discount rate.

Essentially, that is correct. However, using NPV would involve an exhaustive iterative process (i.e., trial and error) in which you attempt multiple discount rates until the present value is determined.

The more effective and accurate method involves using (“IRR”), which is the discount rate that equates the net present value with the future cash flow stream.

Example of IRR calculations using multiple Net Present Value assumptions:

t = 0 $ (875,000) $ (850,000) $ (825,000) $ (800,000) $ (775,000) $ (750,000) 1 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 2 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 3 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 4 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 5 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 6 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 7 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 8 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 9 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 10 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 $ 100,000 IRR = 2.5046% 3.0696% 3.6598% 4.2775% 4.9248% 5.6045%

Interpretation: the IRR is the rate of equivalence of the projected cash flow streams. In other words, your friend should be indifferent as to the lump sum payment or the annuity cash flow stream IF AND ONLY IF the money is invested at the IRR.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

 If you elect the lump sum payment of $875,000 and invest for ten years at 2.5046%, you would earn the cash flow stream as depicted. The two methods are financially equivalent after accounting for the .

 If you believe you can earn more than 2.5046% (i.e., the IRR), you should elect to receive the lump sum payment.

 If you believe you will earn less than 2.5046%, you should elect to receive the annuity cash flow stream.

One can also interpret the IRR as the “break-even” rate.

NPV and IRR are closely related. In fact, if one undertook an exhaustive, iterative NPV approach to solving the problem, the answer would be quite close to the IRR calculation.

Fundamentally, the discount rate used in NPV would be equal to the IRR if the cash flows are reinvested at that rate. When the cash flows are reinvested at the cost of capital, NPV and IRR will provide different answers.

In the example on page 2, the cost of capital is 5%, but the IRR of the cash flow stream is 4.33%.

Estimating Discount Rates

When valuing financial assets, the common methodology involves CAPM (Capital Asset Pricing Model). The cost of equity capital for the company is essentially the required rate of return for the investor. In other words, the discount rate applied to the future stream of cash flows for a company should be equivalent to the minimal return required by shareholders. Consequently,

ke = cost of equity capital, or the required rate of return of the investor RFR = Risk Free Rate Mkt = assumed or expected return of the market ᵦ = Beta of the

We will examine each component of this framework.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

The expected return is generally the average historical return of an asset over a long period of time. The distribution of historical returns around the Mean reflects the volatility of the asset’s returns, and the entire distribution of returns reflects the range of potential outcomes. During the investment process, each investor assesses the relative probability of earning the expected return. That process involves an evaluation of the asset’s expected return and its expected volatility.

Assets with higher volatilities have a wider range of expected outcomes. For this reason, higher volatility is associated with a lower probability of earning the expected, or Mean, return. Therefore, the first component to computing a reasonable cost of capital is the risk-free rate (“RFR”) because this rate reflects the opportunity cost associated with NOT investing.

Theoretically, the RFR assumes zero volatility, and investors often use a cash proxy for this cost. Long term equity investors might prefer to use a longer dated Treasury (e.g., 10 year Treasury Note) because Treasuries represent an alternative to equities. It should be noted that Treasuries, unlike cash, do not have zero volatility. Clearly, the term “risk-free” only refers to credit risk, not interest rate risk. Essentially, the RFR is the rate of return associated with an investment (i.e., cash or equivalent) in which return of principal is effectively guaranteed.

The CAPM framework then assesses the return above the RFR threshold that an investor would require. This is known as the “risk premium” (or “ERP” for Equity Risk Premium) and is captured by the term, (Mkt – RFR), i.e., the difference between the expected return of the market and the RFR. If the RFR = 3% and the expected return of the market is 9%, the Equity Risk Premium would be 6%.

Interpretation: the ERP assumes investors would only invest in the stock market if they could earn a return premium over the RFR. The additional return, or premium, theoretically compensates the investor for the higher volatility of relative to bonds or cash.

The final adjustment involves including an idiosyncratic factor. A stock’s Beta measures its sensitivity to the overall market and is a proxy for relative volatility. Thus, for a stock with a Beta of 1.05, it has exhibited 105% of the volatility of the market. If the market is up 10%, this stock should increase 10.5%. If the market declines 10%, the stock should fall 10.5%.

 Stocks with a Beta less than 1.00 are considered less volatile than the market. Therefore, investors would require a lower relative rate of return. The company’s cost of capital would also be lower on a relative basis.

 Stocks with a Beta greater than 1.00 are considered more volatile than the market. Therefore, investors would require a higher relative rate of return. The company’s cost of capital would also be higher on a relative basis.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

What is the Terminal Value component of a Discounted Cash Flow?

The standard formula for a five period DCF:

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Arguably, the easiest way to understand discounted cash flow methodology is to study a bond. Bonds are loans, and they are categorized as “fixed income” securities. Why? The cash flows of bonds are generally fixed at issue. Additionally, the bond’s principal is returned to the investor (i.e., lender) at maturity. If an investor buys a 5 year, $1,000 par value bond, the principal value of $1,000 is returned to the investor at maturity. At that point, the debt instrument will cease to exist.

The Terminal Value growth rate, gTV , is considered zero. As the Cash Flows are fixed, there is also no interim period growth. The only relevant variable for computing the Present Value of a bond is the required rate of return, r, also known as the “ to maturity”.

Given the fact that $1,000 is returned at maturity, bonds offer no hedge against inflation. Clearly, $1,000 five years hence should be worth less than $1,000 today. For this reason, the return on bond investments is vulnerable to rising inflation. The return of principal at maturity is also the “Terminal Value” of the bond. For a bond issued at Par ($1,000 face, or Present Value), a return of the principal at maturity (i.e., Terminal Value) represents no growth.

The discount rate in any discounted cash flow valuation model should be the required return of the investor. At the time of issue of a bond, the rate (i.e., the rate which determines the Cash Flow) is usually equal to the discount rate (i.e., the yield to maturity). The Present Value of the bond should, therefore, equal the Par Value of the bond.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Consider:

 $1,000 par value  5 year maturity  5% annual coupon  5% yield to maturity

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The correct answer, subject to rounding, is $1,000, or Par. If the yield to maturity (i.e., required rate of return of the bond investor) equals the cash flow rate AND the Terminal Value equals the initial investment, the Present Value must be Par.

Now, let’s consider the Present Value of a bond, issued at Par, when interest rates immediately drop 100 basis points. Note: the Terminal Value of the bond and the interim period Cash Flows remain the same. The only factor that changes is the discount rate, or yield to maturity. Since the prevailing market interest rate has fallen, the investor should be willing to pay more than Par for this bond to receive the same fixed cash flows. In other words, the discount factor for each interim period cash flow (i.e., coupon payment) and for the terminal value (i.e., return of principal) has become smaller. Because the denominator has decreased, the Present Value of each Cash Flow and the Present Value of the Terminal Value both increase.

The Present Value of a bond for which the yield to maturity is less than the cash flow (i.e., coupon) rate should be greater than Par.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Consider:

 $1,000 par value  5 year maturity  5% annual coupon  4% yield to maturity

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Indeed, the Present Value of the bond increased $44.518 over Par because the yield to maturity is 4.0%. Note the increase in the Present Value of the Terminal Value as well as each of the interim cash flows.

Another way to consider the Premium over Par: the excess cash flow over the life of the bond based on the yield to maturity is $50, or $10 per year. This represents the $10 annual excess cash flow versus a 4% coupon bond. The discounted present value of this excess cash flow of $50 over the life of the bond is $44.518.

Now, let’s consider the Present Value of a bond, issued at Par, when interest rates immediately rise 100 basis points. Note: the Terminal Value of the bond and the interim period Cash Flows remain the same. The only factor that changes is the discount rate, or yield to maturity. Since the prevailing market interest rate has risen, the investor should be willing to pay less than Par for this bond to receive the same fixed cash flows. In other words, the discount factor for each interim period cash flow (i.e., coupon payment) and for the terminal value (i.e., return of principal) has become larger. Because the denominator has increased, the Present Value of each Cash Flow and the Present Value of the Terminal Value both decrease.

The Present Value of a bond for which the yield to maturity is greater than the cash flow (i.e., coupon) rate should be less than Par.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Consider:

 $1,000 par value  5 year maturity  5% annual coupon  6% yield to maturity

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Indeed, the Present Value of the bond decreased by $42.124 from Par because the yield to maturity is 6.0%. Note the decrease in the Present Value of the Terminal Value as well as each of the interim cash flows.

Another way to consider the Discount from Par: the reduced cash flow over the life of the bond based on the yield to maturity is $50, or $10 per year. This represents the $10 annual reduction in cash flow versus a 6% coupon bond. The discounted present value of this reduction in cash flow of $50 over the life of the bond is $42.124.

Now, let’s apply the discounted cash flow approach to an asset whose cash flows are NOT fixed. The analyst must consider multiple variables: a) interim period growth rate of cash flows; b) required rate of return of the investor; and c) terminal value growth rate. Further, let’s consider an asset whose Terminal Value is not pre-determined. The investor would assume that, at the end of the interim cash flow period, the asset would retain some value. The investor would further assume that the asset should be worth more than it is today. Given the uncertainties associated with both the ability to grow cash flows and the Terminal Value, however, the investor would require a higher rate of return than for a bond.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Let’s consider:

 Initial Cash Flow is $50  g, growth rate of cash flow for interim period: 10% each year

 gTV ,growth rate of cash flow beyond the interim period: 5% (also known as the Terminal Value growth rate)  Required rate of return = 12%

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

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Interpretation:

For an asset expected to generate initial cash flow of $50, interim cash flow growth of 10% annually, and an expectation for terminal value cash flow growth of 5%, an investor with a required rate of return of 12% should be willing to pay $803.316.

Alternative Interpretation:

For an asset expected to generate initial cash flow of $50, interim cash flow growth of 10% annually, and whose Terminal Value in year 5 is expected to be $1,098.075, an investor with a required rate of return of 12% should be willing to pay $803.316.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

The Terminal Value formula is equivalent to that of the Present Value of a perpetuity. This is conceptually accurate as the Terminal Value should represent the discounted present value of ALL future cash flows of the asset. It assumes, therefore, that the asset has an infinite life, unlike a bond where the Terminal Value represents the return of principal and the full redemption of the loan.

The numerator is determined by growing the final cash flow estimate in the interim period by the Terminal Value growth rate assumption. The denominator represents the spread between the required rate of return and the Terminal Value growth rate assumption. As a company’s share price reflects the discounted present value of future cash flows, the Terminal Value is the present value of the asset at the end of the interim (i.e., forecast) period.

Incorrect Interpretation

The DCF valuation approach is extremely sensitive to assumptions. Therefore, it would be a mistake to consider the Terminal Value of $1,098.075 as some form of predicted or forecasted price. Instead, the investor should consider the Terminal Value estimate at the end of the forecast period to be the equivalent of an estimate of the future value of the asset provided ALL assumptions are maintained, including acquisition at the Present Value estimate.

Sensitivity

The Terminal Value is extremely sensitive to the differential between the required rate of return and the Terminal Value growth rate assumption. In the example, the required rate of return is 12%, and the Terminal Value growth rate assumption is 5%. This provides a spread of 7%.

The discounted present value of the Terminal Value is $623.077. If the Present Value of the asset is $803.316, the Terminal Value represents 77.6% of the Present Value estimate, indicating that much of the valuation depends on the asset achieving a sustainable, or constant, growth rate of 5%.

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Research Memorandum 07.06.2019 Primer on NPV, IRR & DCF RVR

Let’s consider a Scenario in which the Terminal Value growth rate assumption is assumed to be 6%.

The numerator increases to $77.597. The denominator decreases to 6%. This should result in a much larger Terminal Value.

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The discounted present value of the Terminal Value increases to $733.8465. The estimated Present Value of the asset now increases to $914.085. The discounted present value of the Terminal Value now represents 80.28% of the entire present value of the asset. The Present Value estimate of the asset increased by more than $100 because of a mere 1.0% increase in the Terminal Value growth rate assumption.

The Multiples Approach

At $1,293.288, the Terminal Value represents a 16.67x multiple of the Cash Flow estimate of $77.597. Is this reasonable? It depends on the company, but a cash flow multiple in excess of 10.0x would be considered “rich” by most standards. Moreover, the Terminal Value estimate assumes a constant growth rate, implying the 16.67x multiple is warranted if and only if the 6% growth rate can be maintained.

What to Use as a Terminal Value Growth Rate Assumption

As companies mature, their cash flow growth rates tend to converge to the Mean for all companies. The Mean cash flow growth rate tends to coalesce around 1.0 – 1.5x multiple of GDP. If the nominal US GDP growth rate is 5.0%, this implies a Terminal Value growth rate assumption of 5.0 – 7.5%. A constant, or sustainable, growth rate of cash flow assumption in excess of 5.0%, however, is difficult to justify given historical data. There are a few exceptions, however.

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