In this post we will cover the topic of sensitivity analysis and uncertainty related to expected return of an investment.
When evaluating an investment, it’s important to do as much data gathering and analysis as possible. Good analysis will help you have a greater understanding of the associated costs and returns. However, even the most well-prepared analysis must face the uncertainty of the future. No matter how large your spreadsheet is, no one can predict the future with certainty.
Uncertainty means that some of your assumptions regarding costs, reinvestment, expenses, returns, etc. may be over- or under-estimated. No. It is almost certain your assumption will be wrong. But if we’re certain that we’ll be wrong, how can we make a useful evaluation?
With uncertainty or sensitivity analysis we can use statistical models to tell us which of the parameters in our model will have the biggest impact on the investment evaluation. Furthermore, statistical analysis will help us have a better understanding of the variance in outcomes
Simply put, sensitivity analysis is nothing more than setting up several “What-if” scenarios, testing those assumptions and see how they impact the outcome of the evaluation. For example:
What if the revenue is 15% lower than expected?
What is the impact if our margin is 5 base points lower than expected?
What happens if we’re only able to raise 60% of the funds we need?
Defining a what if scenario allows you to reconsider your assumptions. It’s likely that you have a gut feeling which of the parameters will influence your evaluation most. But to increase confidence you can run the numbers.
There are many ways to do sensitivity analysis. For small projects, simply running the numbers manually in an excel sheet will work just fine. For larger projects you may want to consider professional software.
Below you can find one example of a basic sensitivity analysis and a more advanced sensitivity analysis.
Sensitivity Analysis Example (Basic)
For the basic sensitivity analysis, I use an example of an equity valuation of a Singaporean mobile app. For this project, it was important to make assumptions about all related costs on an annual basis. The result was an equity valuation using the discounted cash flow model.
In my base model, the valuation of the project was approximately SG $35.6M.
To do sensitivity analysis, I described 10 additional scenarios varying one or more of the eight different parameters. I ran the numbers for each scenario and laid them out in a separate excel sheet. As you can see, the equity valuation now ranges between SG -$8.5M and SG $35.6M with an average value of SG $18.8M.
Sensitivity Analysis Example (Advanced)
For the advanced sensitivity analysis, I use a share valuation of AMD I made in February 2019. This share valuation is a lot less detailed than the equity valuation for the Singaporean app as I use only three value drivers: growth rate, operating margin and re-investment rate.
Growth rate: expected average annual revenue growth for the next 5 years, then converging to inflation rate by year 10.
Operating margin: targeted pre-tax operating margin, converging from the current margin to target margin by year 10
Sales to capital ratio: the required reinvestment cost to support the revenue growth
The share valuation is based on the current value of the company and number of outstanding shares.
Using Oracle Crystal Ball we create assumptions for each of the three parameters by determining the distribution and its factors. Then we run 1,000 to 20,000 trials. The software will use random values for the three parameters and calculate the share value.
As output, we get an expected distribution of share value as well as the associated statistical information. In addition, Crystal Ball also provides additional information on which of the three parameters impact the valuation most. In this case, operating margin is the dominant parameter.
The return on an investment is perhaps the most obvious aspect of the investment decision for corporate finance. However, there’s more to evaluating an investment than just looking at the earnings. In this post we will talk more in detail about how to evaluate an investment in light of its potential returns.
First Things First: Cash flow
Before we get into the process of evaluating returns, we must consider how we are measuring returns. There are various ways of quantifying earnings, but which one makes sense?
We use cash flow to calculate the returns of an investment. Cash flow allows us to both quantify the returns and evaluate the timing of the returns. Furthermore, cash is something we can actually spend as opposed to earnings which are merely available on the books.
Incremental, because you want to evaluate returns related to the investment
Time-weighted, because early returns are more convenient than later returns
What Impacts Returns?
The next question to ask is what actually impacts returns. Turns out there are several factors:
Revenues may be overestimated
Revenues may be delayed
Operating costs and capital expenditure may be underestimated
Tax rate may increase
Interest rates may increase
Risk premiums and default spreads may increase
There may be unanticipated opportunity costs
Synergy within the company may be overestimated
All of the variables above will have an impact on the evaluation of the return of an investment and must be considered before approving a new investment
Estimating Cash flow Returns
The two main drivers of estimating the return are projected revenues and projected expenses.
Estimating operating income
The revenues include all income related to the investment.
The expenses include the capital expenditure required to initiate the project (such as buying land or machines) as well as fixed and variable operating expenses. Also take into account the depreciation, amortization, G&A costs and the taxes to be paid on the operating income as these will directly affect the cash flows.
The result of is an estimate of after-tax operating income related to the project. To get the cash flow view of the project we add back the non-cash charges, less the capital expenditures and change in non-cash work capital such as inventory and accounts receivable.
Cash flow to firm = + After-Tax Operating Income + Depreciation + Amortization + Tax benefits received - Capital Expenditures - Change in non-cash work capital
From cash flow to incremental cash flow
To calculate the incremental cash flows on the project, we include the pre-project sunk investment, less the tax on depreciation and add back tax on allocated G&A.
Now that we have an overview of the incremental cash flows to the firm, it’s time to factor in the time value of the cash flows. As said, we consider the timing of the cash flows because early returns are more convenient than later returns.
Net Present Value (NPV)
To calculate the time-weighted cash flow, we use Net Present Value of the cash flows. The NPV is the sum of the present value of all cash flows on the project including the initial investment.
The cash flows are discounted at the appropriate hurdle rate. Use cost of capital if the cash flows return to the firm, or use cost of equity of the cash flows return to equity investors.
If the Net Present Value is higher than Zero, then the project is acceptable.
Internal Rate of Return (IRR)
The internal rate of return is the discount rate for which the net present value is zero.
If the Internal Rate of Return is higher than the hurdle rate, then the project is acceptable
If a project or investment is finite and short life, then you can calculate a salvage value. It is the expected proceeds from selling all of the investment in the project at the end of the project life. Typically the salvage value is sum of the book value of fixed assets and the working capital.
For a project with undetermined or very long life, it is sometimes not reasonable to estimate the present value of all cash flows. It is reasonable to compute a terminal value which constitutes the present value of all cash flows beyond the estimated cash flows.
To calculate the terminal value, it is reasonable to use the inflation rate as growth rate. The terminal value in year n is then:
TV(n) = (Cash Flow in Year n+1) / (cost of capital – growth rate)
Uncertainty and Returns
In the beginning of the post we outlined the various factors that may impact the revenue. At the beginning of a project, we forecast revenues and costs as best as possible. But since we can never be sure about the future, there is always a certain degree of uncertainty. What can we do about it?
A simplistic approach to this problem is to calculate how quickly we earn back our money. Based on the projected cash flows we can determine the timing of investment payback.
Sadly, this method does not help us understand, value and mitigate the uncertainty. What degree of inaccuracy of which assumptions will are NPV and IRR most sensitive to?
With sensitivity analysis and what-if questions we can determine this. As sensitivity analysis can get complicated, we will cover this topic in a different post.
Side Costs and Benefits
Most projects or investments create side costs and benefits to the business that are not directly linked to the project.
The opportunity cost is allocated resources of a firm to a project that could otherwise have been allocated to another project. It represents the loss of opportunity of a project that could not be invested in.
Within the organization, multiple projects could create synergies that are not captured in the traditional capital budgeting analysis.
The returns on a project should incorporate these cost and benefits.
Debt is a deferred repayment of a sum of money. It is a legal commitment to make fixed payments which have the benefit of being tax deductible. On the flip side, failure to make payments can lead to default or loss of control of the firm.
Debt includes all short-term and long-term interest bearing liabilities such as loans or bonds. Perhaps surprisingly, it should also include any lease obligations as those obligations also match the definition of debt. After all, monthly rent is not only a (1) legal commitment to make (2) fixed payments but also impacts your (3) working capital. Furthermore, (4) failure to make payments may lead to loss of control of the firm.
The cost of servicing debt matters in two major decisions of corporate finance:
Investment Decision: what is the cost of debt?
Finance Decision: what is the optimal and right kind of debt?
In this post we will talk about the impact of debt on the investment decision and we’ll cover the finance decision in a different post.
Estimating the Cost of Debt
The first important note is that the cost of debt must always be calculated in the same currency as the cost of equity!
To estimate the cost of debt, there are several options:
Yield to maturity on a long-term corporate bond as interest rate if the firm has issued bonds and those bonds are traded
Default spread associated with a rating if the firm has a rating determined by an external rating agency
Interest rate on a recent long-term borrowing if the firm has recently taken on a loan.
Try to estimate a synthetic rating for the firm
Actual and Synthetic Ratings
During the financial crisis of 2008 the credit rating agencies received a lot of public attention. Nearly everyone now has heard about Standard & Poor’s and Moody’s. These two organizations are world leader in attributing ratings to all kinds of companies. The exact algorithm of the rating is unknown, so it is challenging to determine the actual rating for an unrated company.
Estimate a Synthetic Rating
However, we can estimate a synthetic rating by calculating the interest coverage ratio. The ratio is very simple. It calculates how many times the interest on a debt can be covered with the earnings of a firm.
Interest Coverage Ratio = (Earnings Before Income and Tax) / (Interest Expenses)
Using a reference table we can then determine a typical default spread for a company with a certain interest coverage ratio.
From Synthetic Rating to Cost of Debt
To determine the cost of debt from the synthetic rating, we return to the description of hurdle rate. The hurdle rate is the riskfree rate with added risk premium. In the case of a synthetic rate, we can say the cost of debt is then the riskfree rate + default spread associated with the synthetic rating. However, we should factor in that interest expense on debt is tax deductible. So,
Pre-tax cost of debt = riskfree rate of local currency + default spread of synthetic rating
After-tax cost of debt = (pre-tax cost of debt) x (1 – tax rate)
Be careful when using actual ratings from different rating agencies. Some may or may not include the country risk in their evaluation. That case, remember that the total risk for a firm is the sum of the riskfree rate, the country risk, and the firm risk.
Market Value of Debt
For the calculation of the cost of capital, it is preferred to use market value for both the cost of equity and cost of debt. For public companies, the market value of equity is determined by its stock prices. However, it’s a bit more challenging to determine the market value of debt.
One way to determine the market value of debt is to consider the entire debt as one coupon bond. The coupon payment is set equal to the interest expenses on all debt, and maturity set to the (face-value) weighted average maturity of debt. Then, it’s easy to calculate the present value of the annuity. Add to that the coupon bond at current cost of debt for the firm.
You can find an example by Mr. Aswath Damodaran here.
Market Value of Operating Leases
As mentioned, we should include any operating leases in the debt calculation. To calculate the value of the operating leases, simply calculate the present value of the lease payments at a rate that reflects their risk. This rate is usually the pre-tax cost of debt.
The CAPM, or Capital Asset Pricing Model, is one of the models that describe the relationship between market risk and expected returns for an asset. We can use CAPM to determine how much of the expected return can be explained by the associated market risks. It is an important tool to determine the cost of equity for a firm. The cost of equity is used to determine the hurdle rate in the investment decision.
The formula of CAPM is:
Expected Return = Riskfree Rate + Beta x (Expected Return on Market Portfolio – Risk-free Rate)
The difference between expected return on market portfolio and the risk-free rate is called the market risk premium. It is the premium that investors demand for an investment riskier than the risk-free rate.
Thus, to use the model we need three inputs:
The current risk-free rate
The expected market risk premium
The beta of the analyzed asset
A risk-free asset is an asset where the future return is both certain and known. That means the actual return is equal to the expected return. Of course strictly speaking there are no risk-free assets but for practical purposes we assume that for some very safe investments the risk is so low that it can be ignored.
The conventional practice to estimate risk-free rates is to use the government bond rate, with government being the one that controls issuing the currency. For example, for US dollar we use the rate on a ten-year US treasury bond.
Note that not all government securities are risk-free. Some governments face risk of default so the rates on the bonds they issue will not be risk-free.
In case of a government with default risk, you can include the local currency default spread in your calculation. The risk-free rate will then be the government bond rate adjusted for (subtracted by) the default spread for the local currency as determined by the currency rating.
Market Risk Premium
The second input required by the CAPM is the (market) Equity Risk Premium. This is the premium of an investment relative to a risk-free investment. Thus, (1) the premium is greater than 0, (2) the premium increases with risk aversion of investors and (3) the premium increases with the riskiness of the investment.
There are two ways of estimating the Equity Risk Premium. Either (1) use historical data or (2) use future expectations as input.
Historical Equity Risk Premium
Historical data has as main advantage that factual information is available. If you choose to use historical data, make sure to use a long enough time window, as well as ensure it’s consistent with the risk-free rate and use a compounded average. However, always be aware that historical data is backward looking, noisy and subject to selection bias.
The implied ERP model employs a basic discounted cash flow model. It equates the current value of the market with the expected future cash flow growth and solves the equation for the expected return rate.
The inputs needed to calculate the implied ERP are:
Current value of the market
Expected cash flow growth of the market
Terminal growth rate of the market
To practically calculate the implied ERP, you need a proxy for the market. One option is to use the S&P 500 index as proxy for the market. The S&P 500 is a stock market index based on the market cap of 500 large companies having common stock listed on the NYSE, NASDAQ, or the Cboe BZX Exchange.
To calculate the expected cash flow growth, Mr. Damodaran relies on analysts to forecast future retained earnings as well as future returns to shareholder in the form of dividends and buybacks. The time window is set to the next 5 years
The terminal growth rate is set equal to the risk-free rate.
The implied expected return on the stocks is then subtracted by the risk-free rate to come to the implied equity risk premium. In the example of using the S&P 500 as market proxy, we have the implied equity risk premium for the US.
In the example above, we assume the S&P 500 is a good proxy for the US market in general. But what about other countries?
If we follow the theory of the implied equity risk premium, we should calculate the implied ERP for any country using a proxy similar to the S&P 500. However, this is not very practical. Mr. Damodaran proposes a composite way of estimating ERP for countries. The composite way adds the risk of a specific country relative to the risk of a mature market.
First, you estimate an equity risk premium for a mature market either by using the backward looking historical ERP or a forward looking implied ERP.
Then, define what you consider to be a mature market. One option would be to go by country’s local currency rating and associated default spread. In that case, any AAA rated country is considered mature.
Finally, estimate the additional risk premium for non-mature markets. There are two options:
Default spread for the country, estimated based on the sovereign ratings or the CDS market
Scaled up default spread, where you adjust the default spread upwards for the additional risk in equity markets.
The beta of a firm or stock (asset) measures its exposure to the market risk. It indicates both the volatility of the asset as well as how the volatility correlates to the general market. The beta is normalized around 1, meaning that the weighted average of all betas of firms in the market is 1.
Beta > 1: either returns that are more volatile than the market, or returns that are not very correlated with the market
Example: cutting-edge technology companies are typically more volatile than the market because they have a faster innovation pace
Beta = 1: the returns are as volatile as the market, or correlated strongly with the market
Example: large and mature companies tend to follow the market behavior
Beta < 1: the returns are less volatile than the market, or are not very correlated with the market
Example: firms that operate in stable industries which produce common-use products will typically have a low beta
Beta <= 0: the asset is market risk inducing
Example: the price of gold typically correlates inversely with the market and thus firms operating in this industry will have a negative beta.
What Impacts Beta?
There are three main factors that will impact the beta of an asset.
Firstly, the industry a firm is operating in will impact its beta. Since the beta is a measurement of how exposed a firm is to market risk, firms that produce goods or services that are heavily dependent on the season will have a higher beta. For example: a firm that makes ice-cream cones will have greater volatility in earnings than a company that makes toilet paper. Ice cream is in high demand during the summer, but low demand during the winter, whereas toilet paper is used during all seasons. That higher volatility in earnings will result in a higher beta.
Secondly, the operating leverage will impact a firm’s beta too. Companies with a higher operating leverage, or fixed operating costs, will see their earnings vary more. For example, a ‘bad summer’ will result in lower ice cream sales. A business that has invested in a ice cream stall with inside seating and air conditioning will have higher fixed costs than someone who’s just selling at the side of the street. When revenues are lower, the higher fixed costs means lower earnings.
Thirdly, the financial leverage will impact a firm’s beta as well. A firm with debt has to repay interest fees which are deducted from the firm’s earnings. The interest payments are fixed, regardless of the revenue. Therefore a “bad sales season” will have a larger impact on the firm’s earnings.
Lastly, it’s important to note that the beta of a firm is the market-value weighted average of the businesses of the firm. A large corporate which operates in many industries will have a different beta than the industry-focused firms it competes with in those markets. This principle also applies to your personal investment portfolio; the beta of your total investment will be the weighted average of the betas of your assets.
Measuring Beta: Top-Down or Bottom-Up
When measuring beta, you can either choose a backward looking (top-down) or a forward looking (bottom-up ) approach.
The bottom-up beta can be estimated using the following steps:
Find the businesses or industries the firm operates in and determine their impact on sales or operating income
Find the unlevered betas of other firms in these businesses
Determine the weighted average of the unlevered betas
Lever up using the firm’s debt/equity ratio
The main advantage of a bottom-up approach is that this approach reflects the current or even future mix of the businesses that the firm is in.
The top-down beta can be estimated by regressing the stock returns against market returns. The formula is:
Rj = a + b x Rm
Rj is the stock returns
Rm is the market returns.
b, or slope, corresponds to the beta of a stock and measures its riskiness
a, or intercept, is a performance measurement of the stock.
R², or R-squared, is an estimate of the proportion of the variance attributed to market risk
The key challenge of using regression to estimate the beta is the data selection.
Choosing a larger period to evaluate a stock will yield a larger data set but may not accurately reflect the change a firm is going through during that period. Similarly, choosing shorter interval periods between returns will yield a larger data set, but may be affected by noise caused by lack of trading
Unlevered and Levered Beta
The term (un)levered beta usually refers to the debt of a firm. All top-down beta estimates are levered because the estimate is based on stock prices. Stock prices are set by the market and take into account the mix of equity and debt of a firm.
To calculate the unlevered beta (UB) you can use the following formula:
UB = Levered Beta / (1 + (1 – tax rate) x (D/E))
Where the tax rate is the marginal tax rate for the firm, and D/E is the average debt equity ratio during the regression period
Beta is the weighted average of the risk exposure of all the firm’s businesses to the general market
Market Equity Risk Premium is the weighted average of market risk premium of all the markets the firm operates in
Practically, we can choose to calculate the risk of operating in a mature market like the US and add risk associated with operating in a different country. In that case, we can say:
Expected Return = Country Gov’t Bond Rate + (Levered Beta) x (Mature Market Risk Premium + Country Risk Premium)
Country Government Bond Rate is the government in control of issuing the currency
If there’s a default risk associated with the government bond rate, adjust the borrowing rate for the default risk
What About Private Companies?
The methods to estimating the cost of equity discussed above are very useful when evaluating public companies. However, for private companies things are more complicated. The lack of stock prices and historical returns make it difficult to calculate how exposed the company is to market risk (beta).
There are two main ways to estimate a beta for non-traded assets: use comparable firms (bottom-up) or use accounting earnings (top-down).
When evaluating the levered betas of comparable firms, it’s important to still deleverage and releverage using the private firm’s debt equity ratio. Again, it’s difficult to estimate the market value of the firm’s equity or debt, so it’s reasonable to use book value instead.
Last but not least, remember that beta is a measure of risk added on to a diversified portfolio. However, the owners of most private companies are not diversified. Therefore using beta only to come to the cost of equity will underestimate the cost of equity of the private firm.
To adjust for the added risk of an undiversified portfolio, we can use R-squared of the regression as it measures the proportion of risk that is market risk.
Beta = Market Beta / Correlation of sector with market
The correlation factor of industries like technology are relatively low compared to more mature industries. If you dig into it, you will quickly find that a private technology company is a risky business!
Risk is a central element of the business world. Risky but successful projects can give investors an incredible return on their investment. But risky and unsuccessful projects have the potential to leave you with nothing. Estimating the risk associated with an investment is therefore an essential part of business practice.
In this post we cover the first aspect of the Investment Decision for corporate finance: hurdle rate. The hurdle rate reflects the riskiness of an investment and the mix of debt and equity to fund it. Simply put, the hurdle rate sets the minimum expectation of return for a specific investment.
As mentioned in Fundamentals of Corporate Finance, the Risk Premium captures the difference between a safe investment and a risky investment. Thus, we can describe the hurdle rate as follows: Hurdle Rate = Riskless Rate + Risk Premium.
The description outlines three major questions.
What is risk and riskless?
How do we measure risk?
How to quantify risk into a risk premium?
Risk and Riskless
Mathematically, the variance in actual returns around expected returns is the measurement of the risk of an investment. Simply put, it describes the relationship between an expectation and an actual return. The larger the difference between the actual and expected returns, the riskier the investment.
A riskless investment is an investment with a certain and known rate of return, and with no chance of default. Strictly speaking there are no riskless investments. In practical terms US treasury securities are considered riskless because the US government is considered the best possible issuer of securities in the market.
Even though riskless strictly speaking does not exist, the level of risk is so small it can be ignored. The downside of a riskless investment is that the rate of return is very low and that the returns are exposed to inflation risk.
Rewarded and Unrewarded Risk
There are two types of risk: rewarded and unrewarded risk.
Unrewarded risk, or market risk, is a risk that affects all investments across the market. It is unrewarded because taking the risk does not yield any advantage over the rest of the market. For example: risks associated with compliance to safety standards are unrewarded risks because all market players must comply.
Rewarded risk, or firm specific risk, is a risk that is specific to the investment. It is rewarded because it may give you a competitive advantage in the market.
A smart investor should diversify their investment portfolio so that the total firm-specific risk is averaged out across the portfolio. It’s reasonable to assume that any investor holds a “diversified” portfolio, thus investors price only market risk.
When we talk about market risk, it’s important to understand what we mean by ‘market’. The market pertains to not only the country where we sell our goods or services, but also the specific industry we compete in.
Considering the above, we can anticipate some challenges when trying to estimate the market risk.
A lot of businesses and corporations are operating worldwide. That means the risk of the firm depends on which markets they are exposed to and by how much. A firm which collects 90% of their revenue in the US market will be less risky than a firm that performs the same business in South Africa. The exact market risk for a firm depends on the specific mix of markets it operates in.
As a rule of thumb we could argue it’s more sensible to determine the risk based on where the firm is operating as opposed to where it is incorporated.
Furthermore, large corporations are active in different fields. A corporation such as McDonalds is in the business of not only selling food, but also real-estate. The industry risk associated with each business is different. The exact market risk for a firm will depend on the specific mix of industries it is active in
Measuring and Quantifying Risk
There are several methods to measure market risk.
CAPM or Capital Asset Pricing Model
APM: Asset Pricing Model
While the CAPM is limited in many ways, it has survived as the default model for risk in equity valuation. The alternative models do a better job explaining past returns but are less effective in predicting future returns. Furthermore, they are much more complex and require a lot more information than CAPM. Lastly, the difference in expected returns predicted by the alternative models and CAPM is not significant enough to justify the additional work.
As it is an extensive topic, we will cover the CAPM in a different post.
The CAPM will give us the cost of equity related to an investment. However, a firm can also raise funds for an investment project from debt.
Simply put, debt is a deferred repayment of a sum of money. It is a commitment to make fixed payments with as main advantage that it is tax deductible. However, failure to make payments can lead to default or loss of control of the firm.
Debt includes short-term and long-term interest-bearing liabilities (i.e. loans or corporate bonds) or any lease obligations.
When we talk about the Cost of Debt, we talk about the cost associated with raising funds from debt.
We will cover debt in more detail in a different post.
Cost of Capital
The weighted average cost of equity and cost of debt is called the cost of capital. To determine the weight for equity and debt, the preferred method is to use market value. It is relatively easy to determine the market value for equity, but can be more difficult for private companies.
Determining the market value of debt is challenging for both public and private companies. Hence most practitioners opt for the book value of debt as a proxy for market value.
Choosing the Hurdle Rate
Finally, the choice of hurdle rate still depends on the preference of the managers. As said, the hurdle rate is the internal benchmark for projects. Using the cost of capital as hurdle rate is a common, solid choice but not mandatory.
If your firm is a start-up which is looking for angel investors to provide equity it is reasonable to use cost of equity as a hurdle rate.