Sebastian Perrier

In order to calculate the NET PRESENT VALUE of an investment project, we need to know what is the discount rate at which to discount the Cash Flows. This discount rate is called the weighted average cost of capital (WACC), which is the rate that a company is expected to pay on average to all its security holders to finance its assets.

HOW TO CALCULATE THE DISCOUNT RATE “r” or “WACC”?

First, calculate the COST OF EQUITY (Ke):

1. Unique Risk: investors expect a determine level of return for each project and its related risk. This unique risk is calculated using historical data; you can calculate the “standard deviation” of the returns that investors got by investing in XX shares.
2. Diversification: however, unique risk can be eliminated by diversification; i.e. you buy stocks of umbrellas and ice-creams.
3. Market Risk: but experience has shown that risk can be mitigated until a level where, no matter how more you diversify your portfolio, you’ll still have the same level of “market risk” which cannot be avoided (e.g. earthquake, macro economic cycles, government policies, etc). So, assuming investors are intelligent and they have a diversified portfolio, the only risk they have to care is market risk.
4. Beta: but how is market risk calculated? With Beta. Beta is a measure of the extent to which the return of the shares of a specific company “moves together” with the returns of market indexes which are 100% diversified portfolios (e.g. S&P, NASDAC, ISEC, etc). Beta > 1 means that the share is volatile, Beta < 1 means stable, and Beta=0 means that it moves identically as the market index.
5. Capital Asset Pricing Model (CAPM): If we use the two following benchmarks a) the return of risk-free shares with a Beta=0 (like US Gov Bonds) and b) the return of market index – 100% diversified portfolio – we can draw the “Security Market Line“. Graphing Return on the y axis and Beta on the x axis, the return of any share will fall into this Security Market Line. Therefore, following the equation of a line, to calculate a company’s expected return on equity (or its cost of equity): Ke= Rf + B [Rm-Rf]

In other words, expected return on a share will consist of a minimum (R_f) [risk free, like GovBonds at 3%], plus a premium for risk [Rm-Rf] “market index return minus risk free” , that is proportionate to beta value of that industry (the risk of that industry, or portfolio). The expected return on any investment, or portfolio, will consist of the risk-free rate of return, and a premium for risk, which is proportional to beta.

Second, if the project also has debt, also calculate the COST OF DEBT (Kd):

To calculate the cost of debt is easyier, its the interest that the bank charges for the issued debt. But note that debt financing has a “tax shield”, because taxes are calculated after subtracting interest cost to profit.

Therefore: WACC= Kd (1-tax rate) . %D + Ke . %E

Click here to download example.

Calculation for one period:

• FV=PV + PV. r -> FV=PV(1+r) -> PV=FV/(1+r)

Calculation for multiple periods:

• PV=[FCF1/(1+r) + FCF2/(1+r)'2 + ... + FCFn/(1+r)'n]

Calculation of NPV:

• NPV = PV – Initial Investment

Calculation of IRR:

• Solve “r” (the discount rate) that gives an NPV of zero

NOTES:

• FCF1, FCF2, FCFn – Projected Yearly Free Cash Flows
• r – Interest Rate or Cost of Capital (carefully use the project’s WACC, not the company’s one bc risks are different)
• I – Initial Investment
• n – Project Life (years, months, etc, but put r according to period specified)

Mathematical Problems with the IRR

(a) Negative Cash Flows and Multiple Rates of Return; e.i. when cash flows are +-+, gives multiple IRR, and thus mathematical calculation gives an error.

(b) The Re-investment Problem: bc when you borrow, WACC can change due to different risks.

(c) Ranking Differences: cannot decide in terms of size of projects and timing of cash.

Investment Appraisal and Inflation

(a) Adjust projected cash flows, to account for anticipated fall in value of money

(b) Re-state the discount rate, to allow for anticipated impact of expected inflation

APPENDIX – Future Values and Discount Rate Table