# Discount Rate

Discount rate is the rate of interest used to determine the present value of the future cash flows of a project. For projects with average risk, it equals the weighted average cost of capital but for project with different risk exposure it should be estimated keeping in view the project risk.

Capital budgeting techniques such as net present value, internal rate of return, discounted payback period and profitability index are based on the concept of time value of money. In net present value, we discount the future incremental cash flows of a project to time 0 and then subtract the initial investment to see if we the project adds value or not. In internal rate of return technique, the IRR is compared with a rate called the hurdle rate which represents the cost of capital of the company and the risk of the project and the project is accepted only if the IRR is higher than the discount rate. The result we get from all the techniques is very sensitive to the discount rate which makes it arguably the most important input in capital budgeting process.

## WACC

The weighted average cost of capital (WACC) is a good starting point in determining the appropriate discount rate. WACC is the marginal composite cost of all the company’s sources of capital, i.e. debt, preferred stock and equity. It is calculated using the following formula:

$$ wacc=w_e\times r_e+w_p\times r_p+w_d\times r_d\times(1-t) $$

Where w_{e}, w_{p} and w_{d} are the target weights of common stock, preferred stock and debt in the company’s capital structure respectively. Similarly, r_{e}, r_{p} and r_{d} represent the cost of equity, cost of preferred stock and cost of debt respectively. The cost of debt is multiplied with a factor of (1 – t) where t represents the tax rate. It is because interest payments are tax deductible which results in a decrease in effective cost of debt.

Cost of equity can be calculated using capital asset pricing model or Gordon growth model. Cost of preferred stock equals preferred dividends divided by current price of preferred stock and cost. Cost of debt is estimated based on the debt’s yield to maturity.

## Risk-adjusted discount rate

While WACC is a good starting point in determining the discount rate, it is useful only when the project has the same risk as that of the average project of the company which is rarely the case. A better approach is to notch the discount rate up and down keeping in view the project risk. A risk-adjusted discount rate can be determined through application of the capital asset pricing model and pure play approach.

Capital asset pricing model was developed to estimate the required rate of return on equity as equal to the sum of the risk-free rate plus the product of the company’s equity beta coefficient and market risk premium. Following is the formula for CAPM:

$$ r=r_f+\beta_e\times MRP=r_f+\beta_e\times(r_m-r_f) $$

Where r_{f} is the risk-free rate, β_{e} is the equity beta, MRP is market risk premium equals the difference between market rate of return r_{m} and risk-free rate.

The same formula can be used to estimate the hurdle rate for a project by substituting the equity beta with project beta which can be estimated using the pure play method as explained below:

$$ h=r_f+\beta_p\times MRP=r_f+\beta_p\times(r_m-r_f) $$

All the other inputs are the same expect for the beta. In determining the hurdle rate for a project using CAPM, we must use the project beta or asset beta.

We first need to find out a publicly traded company which purely engages in the projects very similar to the project we are considering. For example, if Apple wants to evaluate whether they should start developing their own processors, they can look at Intel or Qualcomm, for example. Because Intel and Qualcomm focus is developing computer processors, their beta coefficient is a better representation of the associated risk. Apple needs to find Intel or Qualcomm’s equity beta coefficient and then unlever it to remove the effect of their capital structure. This can be achieved using the following formula:

$$ \beta_a=\frac{\beta_e}{1+\frac{D}{E}\times(1-t)} $$

Where β_{a} is the project beta we should use, β_{e}, D/E and t are the equity beta, debt-to-equity ratio and tax rate of the pure play company.

The project beta should be plugged into the CAPM equation to get the appropriate discount rate.

The discount rate determined using this approach will be higher or lower than the weighted average cost of capital. It will be higher where the project is riskier and vice versa. It offers a better measurement of value added by a project.

## Example

Let’s continue with Apple’s example above.

Since the processor project has a risk profile different than the average risk of Apple’s other project, we should use the risk-adjusted discount rate determined based on the pure play approach. Qualcomm is a good pure play company. It’s equity beta is 1.49, D/E ratio is 0.81. Current interest rate on 10-year Treasury bonds is 2.86% and the expected return on the market is 7.6%. Assume a marginal tax rate of 30%.

We first need to find the unlevered beta for Qualcomm:

$$ \beta_a=\frac{1.49}{1+0.81\times(1-30\%)}=0.95 $$

Using capital asset pricing model, we can work out the appropriate discount rate as follows:

$$ h=r_f+\beta_p\times(r_m-r_f)=2.86\%+0.95\times(7.6\%-2.86\%)=7.3\% $$

Let’s compare it with Apple’s WACC calculated below:

Apple’s cost of equity determined using CAPM based on Apple’s own beta of 1.1:

$$ r=r_f+\beta_e\times(r_m-r_f)=2.86\%+1.1\times(7.6\%-2.86\%)=8.074\% $$

Assuming Apple’s has after-tax cost of debt 3.5 and debt to equity is 0.52, Apple’s WACC can be calculated as follows:

$$ wacc=(1-0.52)\times8.07\%+0.52\times3.5\%=5.69\% $$

If we use Apple’s WACC to determine the processor project we would be overstating the NPV because the WACC is understating the project risk. The risk-adjusted discount rate approach based on the pure play method is a theoretically better approach.