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Sidi

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4

What’s the meaning of discount rate ‘r’ in NPV?

Hi there,

One simple finance doubt that has me struggling: meaning of discount rate 'r' in an Equity-only and in a Debt&Equity scenario.

Specifically:

  1. What is the meaning of ‘r’ in a basic NPV case where the firm investment is only in Equity?
    If I understand correctly, it’s the interest I would make by investing in a basic benchmark. Personally, it’s not clear if this benchmark is:
    a) an investment in a risk-free bond; b) the average rate of return of the company.

  2. And what changes when the firm investment is a Debt-Equity mix?
    Is 'r': a) The WACC, or is it still b) an investment benchmark?

Background
My question derives from this doubt: according to Modigliani-Miller, the value of a company in a no-tax scenario should not be impacted by the financing mean. Therefore NPV1 should be equal to NPV2

Thank you in advance for your help!

T

Hi there,

One simple finance doubt that has me struggling: meaning of discount rate 'r' in an Equity-only and in a Debt&Equity scenario.

Specifically:

  1. What is the meaning of ‘r’ in a basic NPV case where the firm investment is only in Equity?
    If I understand correctly, it’s the interest I would make by investing in a basic benchmark. Personally, it’s not clear if this benchmark is:
    a) an investment in a risk-free bond; b) the average rate of return of the company.

  2. And what changes when the firm investment is a Debt-Equity mix?
    Is 'r': a) The WACC, or is it still b) an investment benchmark?

Background
My question derives from this doubt: according to Modigliani-Miller, the value of a company in a no-tax scenario should not be impacted by the financing mean. Therefore NPV1 should be equal to NPV2

Thank you in advance for your help!

T

(edited)

4 answers

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Hi!

simple answer: the discount rate r corresponds to the cost of capital.

  • For equity, the cost of capital corresponds to the opportunity cost of not investing it. This opportunity is the rate of return of riskless investments in the market. Government bonds of strong countries could be an example.
  • For debt financing, the cost of capital corresponds to the weighted average between the cost of equity and the interest rate paid on the debt.

Cheers, Sidi

Hi!

simple answer: the discount rate r corresponds to the cost of capital.

  • For equity, the cost of capital corresponds to the opportunity cost of not investing it. This opportunity is the rate of return of riskless investments in the market. Government bonds of strong countries could be an example.
  • For debt financing, the cost of capital corresponds to the weighted average between the cost of equity and the interest rate paid on the debt.

Cheers, Sidi

(edited)

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Hi there

SHORT ANSWER

1. In general b) assuming you are referring to the equity return. However the rate of return could be related to a specific project of the company and not to the overall company. If the project is more or less risky than the risk of the company, the rate of the project would be different from the rate of the company

2. a)

LONG ANSWER

The discount rate is normally approximated with the WACC of the company or project (Weighted Average Cost of Capital), which can be computed as follows:

WACC = Re x E/V + Rd x (1 - corporate tax rate) x D/V.

where:

Re= Cost of Equity

Rd= Cost of Debt

E= Equity

D= Debt

V= Enterprise Value (E+D)

When there is no debt, the formula reduces to WACC = Re.

In terms of the meaning of the discount rate Re: you can calculate it as follows:

Re=Rrf+b*(Rm-Rrf)

where:

Re=Expected return of the security

Rrf=Risk free rate (eg US 10 year bond yeld)

Rm=Expected market return

b=beta of the security.

The beta of the security is an indicator of how much the price of a particular stock moves up and down compared with how much the entire stock market moves up and down. A stock with a beta of 1.5 would move up by 15% if the market moved up by 10% and move down by 15% if the market moved down by 10%.

You are correct in terms of the Modigliani Miller theorem, so far that there are no taxes or costs related to bankruptcy, asymmetric information, agency costs and assuming an efficient market, the value of the firm is unaffected by the way it is financed.

Best,

Francesco

Hi there

SHORT ANSWER

1. In general b) assuming you are referring to the equity return. However the rate of return could be related to a specific project of the company and not to the overall company. If the project is more or less risky than the risk of the company, the rate of the project would be different from the rate of the company

2. a)

LONG ANSWER

The discount rate is normally approximated with the WACC of the company or project (Weighted Average Cost of Capital), which can be computed as follows:

WACC = Re x E/V + Rd x (1 - corporate tax rate) x D/V.

where:

Re= Cost of Equity

Rd= Cost of Debt

E= Equity

D= Debt

V= Enterprise Value (E+D)

When there is no debt, the formula reduces to WACC = Re.

In terms of the meaning of the discount rate Re: you can calculate it as follows:

Re=Rrf+b*(Rm-Rrf)

where:

Re=Expected return of the security

Rrf=Risk free rate (eg US 10 year bond yeld)

Rm=Expected market return

b=beta of the security.

The beta of the security is an indicator of how much the price of a particular stock moves up and down compared with how much the entire stock market moves up and down. A stock with a beta of 1.5 would move up by 15% if the market moved up by 10% and move down by 15% if the market moved down by 10%.

You are correct in terms of the Modigliani Miller theorem, so far that there are no taxes or costs related to bankruptcy, asymmetric information, agency costs and assuming an efficient market, the value of the firm is unaffected by the way it is financed.

Best,

Francesco

(edited)

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Hi

If the investment is equity, then the appropriate discount rate to use is the cost of equity, (re). This can be calculated using CAPM (capital asset pricing model). re = rf + Beta * MRP

Here we would need to know:

  • rf = The risk-free rate (virtually zero right now), typically you can use the rate for a 10-year treasury if it’s a US based project). Ideally the tenor of the bond used would match the duration of the project).
  • Beta for the equity (which is the correlation of the project relative to the market - it captures the degree to which the equity returns moves relative to market risk - i,e. how much riskier than the market the project is). Beta is calculated as the covariance of the equity returns with market returns, divided by the variance of market returns. If you don't know the returns for your equity, then you can use a proxy stock in the same space as your company. (To do this properly, there are potentially further adjustments you need to make to differentiate between a levered beta and unlevered beta if you are using comparables, to control for their capital structure, but we probably don’t need to go into that detail here.)
  • MRP = market risk premium which is the market return less the risk-free rate. Here you should use the broadest definition for the market available. Ideally this would be something broad like Willshire 5000, but S&P500 also fine for US

If we are looking at capital structure with debt and equity, then the correct discount rate to use is the WACC.

Bear in mind that the M-M proposition only works in a world where there are no taxes. This is a simplification of the real world, where we of course do pay taxes. In this instance, the NPV is higher when using debt relative to using equity. The additional PV component arises from the interest that is tax deductible and is equal to the PV of ‘tax shields’ (i.e. the tax we avoided by paying interest) and is equal to the tax rate multiplied by the value of the debt:

NPV2 (levered) = NPV1 (unlevered) + Tax rate * Debt

With no taxes, NPV1 would indeed equal NPV2.

Hope this is helpful!

Feel free to follow up with any further questions.

Best,

Adam

Hi

If the investment is equity, then the appropriate discount rate to use is the cost of equity, (re). This can be calculated using CAPM (capital asset pricing model). re = rf + Beta * MRP

Here we would need to know:

  • rf = The risk-free rate (virtually zero right now), typically you can use the rate for a 10-year treasury if it’s a US based project). Ideally the tenor of the bond used would match the duration of the project).
  • Beta for the equity (which is the correlation of the project relative to the market - it captures the degree to which the equity returns moves relative to market risk - i,e. how much riskier than the market the project is). Beta is calculated as the covariance of the equity returns with market returns, divided by the variance of market returns. If you don't know the returns for your equity, then you can use a proxy stock in the same space as your company. (To do this properly, there are potentially further adjustments you need to make to differentiate between a levered beta and unlevered beta if you are using comparables, to control for their capital structure, but we probably don’t need to go into that detail here.)
  • MRP = market risk premium which is the market return less the risk-free rate. Here you should use the broadest definition for the market available. Ideally this would be something broad like Willshire 5000, but S&P500 also fine for US

If we are looking at capital structure with debt and equity, then the correct discount rate to use is the WACC.

Bear in mind that the M-M proposition only works in a world where there are no taxes. This is a simplification of the real world, where we of course do pay taxes. In this instance, the NPV is higher when using debt relative to using equity. The additional PV component arises from the interest that is tax deductible and is equal to the PV of ‘tax shields’ (i.e. the tax we avoided by paying interest) and is equal to the tax rate multiplied by the value of the debt:

NPV2 (levered) = NPV1 (unlevered) + Tax rate * Debt

With no taxes, NPV1 would indeed equal NPV2.

Hope this is helpful!

Feel free to follow up with any further questions.

Best,

Adam

Hi Adam. Wow, thank you for your in-depth analysis. If I understand correctly the impact on a case scenario: a) if I have to discount an equity-only project, I use Re (cost of equity) from the CAPM; b) If I have to discount a mixed debt-equity project, I use the WACC. Only thing I don't understand: in a no-tax case scenario, NPV1 and NPV2 should be equal, therefore also WACC should equal Re. Does this make sense (or at least is it possible)? Or, in general: which one is typically higher between WACC and Re? Thank you! — Anonymous A on Apr 09, 2020

Yes if there is no taxes then the value is the same in both instances. We would use cost of equity for NPV1 and WACC for NPV2. (Note that when we calculate WACC, we would set tax rate equal to zero). With no taxes, NPV1 and NPV2 would indeed be equal. The mechanism for making them equal is that for NPV2, the cash flow is lower (since we pay a contracted interest payment to debt holders), but we also have a lower discount rate (since WACC is lower than Re because Rd is lower than Re and we take a weighted average of the two). The lower cash flow combined with the lower discount rate leads to equality of NPV2 with NPV1 and hence the “irrelevance of capital structure” result. — Adam on Apr 09, 2020

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