# Expected Returns

## What is Equity Risk and Expected Returns?

To see how risk is scene in finance, we will explain the analysis in three steps. First, we will identify risk in terms of the distribution of real returns around an expected return. Second, we will distinguish risk that is specific to an investment or a few investments and risk that affects a much larger sample of investments. We will reason that when the large institutional investor is properly diversified, it is only the latter risk, called market risk that will be rewarded. Third, we will look at different models for computing this market risk and the expected returns that go with this risk.

## I. Measuring Risk

Investors who invest in an asset demand a return over the time horizon that they own the asset. The real return that they gain or lose over this time frame may by significantly different from the expected return, and this is where the risk comes in. Consider an investor with a 1-year time line buying a 1-year Treasury bill (or any other risk-free one-year bond) with a 3% expected return.

At the end of the 1-year time line, the actual return that this investor would make on this investment will always be 3%, which is equal to the expected return. The return distribution for this investment is shown in Image 1.1.

This is a riskless investment, at least in nominal terms.

To see a difference, think about an investor who invests in Granite Construction. This investor, having completing his research, may presume that he will make an expected return of 30% on Granite Construction over his 1-year investment period. The actual return over this period will unmistakably not be equal to 30%; it could be much higher or a great deal lower. The distribution of returns on this investment is illustrated in Image 1.2:

Image 1.2: Probability Distribution on Risky Investment

In addition to the expected return, an investor now has to think about the following. First, the difference of the actual returns around the expected return is noted by the variance or standard deviation of the distribution; the larger the spread of the actual returns from expected returns, the bigger the variance. Second, the bias towards positive or negative returns is noted by the skewness of the distribution. The distribution above is positively skewed, since there is a greater chance of large positive returns than large negative returns. Third, the profile of the tails of the distribution is measured by the kurtosis of the distribution; fatter tails lead to higher kurtosis. In investment terms, this captures the tendency of the price of this investment to "pop" in either direction.

With a normal distribution, returns are symmetric and investors do not have to be concerned with skewness and kurtosis, since there is no skewness and a normal distribution has a kurtosis of zero. In this instance, one could state that investments can be identified only two ways - (1) the 'expected return' on the investment is the reward, and (2) the variance in anticipated returns is the risk on the investment. Image 1.3 illustrates the return distributions on two investments with symmetric returns.

Image 1.3: Return Distribution Comparisons

### Expected Return

In this setting, an investor faced with an alternative between two investments with identical standard deviations but different expected returns, will generally select the investment with the higher expected return.

Generally, when distributions are neither symmetric nor normal, it is possible, though not likely, that investors will select an investments based only on the expected return and the variance, if they possess utility functions1 that allow them to do so. It is far more likely, however, that they prefer positive skewed distributions to negatively skewed ones, and distributions with a lower chance of pops (lower kurtosis) over those with a higher chance of pops (higher kurtosis). In this likelihood, investors will swap the good (higher expected returns and more positive skewness) against the bad (higher variance and kurtosis) in making investments. Of the various risk and return models, one (the capital asset pricing model or the CAPM) clearly notes that preferences be made simply in terms of expected returns and variances. Although it pays no attention to the skewness and kurtosis, it is unclear how much of an issue these factors in determining expected returns.

The assumption we make when we use historical variances is that past returns are a good indication of future returns. When this belief is violated, as is the case when the investment's characteristics have altered widely over time, the historical estimates may not be good measures of risk.

Example 1.1: Calculation of standard deviation using historical returns: Granite Construction

We collected the data on the returns we would have made on a monthly basis for every month from July 2015 to August 2021 on an investment in Granite Construction stock. To compute the returns, we looked at the price change in each month (with Pricet being the price at the end of month t) and dividends if any during the month (Dividends t):

Returnt = (Pricet - Pricet-1 + Dividendst)/Pricet-1

The monthly returns are graphed in Image 1.4:

Image 1.4: Returns on Granite Construction

Granite Construction's returns identify the risk that an investor in the stock would have dealt with over the time frame, with November 2020 being the best month (with a return of 27.50%) and October 2019 representing the worst month (with a return of -26.90%).

Reviewing the summary statistics, the average monthly return on Granite Construction over the 59 months was 1.02%. In fact, we started the period, in July 2015, with a stock price of \$31.12 and ended the period on August 02, 2021 with a stock price of \$37.78 However, the stock did pay an annual dividend of \$0.52 in 2015 and \$0.52 in 2021. To gauge the volatility or risk in the stock, we calculated the standard deviation in monthly returns over this period to be 10.09%; the variance in monthly returns was 31.76%.2 To adjust monthly values to annualized ones:

Annualized Standard Deviation = 10.09% *√12 = 34.95%
Annualized Variance = 31.76% * 12 = 381.12%2

Looking at Granite Construction's standard deviation in stock returns without comparing stock returns of other companies gives us little or no evidence whether Granite is a risky investment or not.

## II. Rewarded and Unrewarded Risk

Risk, as previously defined in the previous section, comes from the difference of actual returns from expected returns. This difference, however, can happen for any number of reasons, and these reasons can be identified into two categories - those specific to the investment under consideration (called company specific risks) and those relevant to most or all investments (market risks).

### Risk!

When a company makes an investment, in a new asset or a project, the return on that investment can be influenced by a number of variables, many of which are not under the complete power of the company. Much of the risk extends directly from the investment, a fraction from competition, some from changes in the industry, some from movements in exchange rates and a lot from macroeconomic factors. A piece of this risk, nonetheless, will be reduced by the company by adding multiple investments or projects and another piece by investors as they hold diversified portfolios.

The first source of risk is project-specific; a single project may have higher or lower cashflows than anticipated, either because the company miscalculated the cashflows for that project or because of reasons exclusive to that project. If companies take a large number of identical projects, it can be reasoned that most of this risk can be diversified away in the ordinary course of business. For example, Granite Construction, while considering a new construction project, exposes itself to estimation error - it may under or over estimate the cost and time of completing the project, and may also err in its estimates of revenues from both time and cost of materials. Since Granite Construction completes hundreds of construction projects a year, it can be argued that some or most of this project risk can be diversifiable across projects during the course of the year.

The second source of risk is competitive risk, where the earnings and cashflows on a project are influenced (positively or negatively) by the actions of competitors. Whereas a high-quality project analysis will support the anticipated response of competitors into estimates of profit margins and growth, the tangible responses taken by competitors can diverge from these expectations. In many cases, this element of risk will influence more than a single project, and is harder to diversify away in the day-to-day operations by the company. Granite Construction, for instance, in its review of revenues from its transportation division may err in its judgments of the strength and weaknesses of competitors like Flat Iron, Inc. While Granite Construction cannot diversify away its competitive risk, stockholders in Granite Construction can, if they are willing to hold stock in the competitors.3

The third source of risk is industry-specific risk -- the things that effect the earnings and cashflows of a specific industry. There are three sources of industry-specific risk. The first is technology risk, which reflects the effects of technologies that change or evolve in ways different from those expected when a project was originally analyzed. The second source is legal risk, which reflects the effect of changing laws and regulations.

The third is commodity risk, which highlights the impact of price changes in commodities and services that are used or produced disproportionately by a specific industry. Granite Construction, for instance, in evaluating the outlook of its water division is likely to be exposed to all three risks; to technology risk, as the lines between engineering and infrastructure are increasing blurred by companies like Fluor Engineering, to legal risk, as the laws governing building and safety change and to commodity risk, as the costs of building and construction materials change over time. A company cannot diversify away its industry-specific risk without diversifying across industries, either with new projects or through acquisitions. Stockholders in the company should be able to diversify away industry-specific risk by holding portfolios of stocks from different industries.

The fourth source of risk is international risk. A company faces this type of risk when it generates revenues or has costs outside its domestic market. In such cases, the earnings and cashflows will be impacted by volatile exchange rate instability or by political developments. Granite Construction, for instance, is clearly exposed to this risk with its construction division in Mexico. Some of this risk may be diversified away by the company in the everyday course of business by investing in projects in different countries whose currencies may not all move in the same direction. Jacob Engineering, for instance, operates in many different countries and should be able to diversify away some (though not all) of its exposure to international risk. Companies can also reduce their exposure to the exchange rate component of this risk by borrowing in the local currency to fund projects. Investors should be able to reduce their exposure to international risk by diversifying globally.

The final source of risk is market risk : macroeconomic factors that impact nearly every company and all projects, to some degree. For example, movements in interest rates will have an effect on the value of projects previously taken and those yet to be taken both directly, through the discount rates, and indirectly, through the cashflows. Other variables that shape all investments include the term structure (the difference between short and long term rates), the risk preferences of investors (as investors become more risk averse, more risky investments will lose value), inflation, and economic growth. While expected values of all these variables enter into project analysis, unanticipated movements in these variables will change the values of these investments. Neither investors nor companies can diversify away this risk because all risky investments bear some exposure to this risk.

### Why Diversification Reduces or Eliminates Company-Specific Risk

Why do we single out the different types of risk? Risk that impact one of a few companies, i.e., company specific risk, can be lower or even removed by investors as they hold more diverse portfolios due to two reasons.

The first is that all investment in a diversified portfolio is a much lower percentage of that portfolio. So, any risk that improves or lowers the value of only that investment or a small set of investments will have only a minimal impact on the total portfolio.

The second is that the results of company-specific actions on the prices of individual assets in a portfolio can be either good or bad for each investment for any time period. So, in large portfolios, it can be stated that this risk will average out to be zero and therefore not affect the total value of the portfolio.

In contrast, risk that impacts nearly all assets in the market will continue to persist even in large and diversified portfolios. For example, higher interest rates will generally lower the values of nearly all assets in a portfolio. Image 1.5 sums up the different pieces of risk and the actions that can be taken by the company and its investors to lessen or remove this risk.

Image 1.5: Risk Analysis

While the intuition for diversfication and eliminating risk is straightforward, the advantages of diversification can also be graphed statistically. In the previous section, we presented the standard deviation as the measure of risk in an asset and computed the standard deviation for an individual stock (Granite Construction). When you combine two investments that do not move up and down together in a portfolio, the standard deviation of the two investments can be greater than the standard deviation of the individual stocks in the portfolio. To observe how the math of diversification works, let's look at a portfolio of two assets. Asset A has an expected return of µA and a variance in returns of σ2A, while asset B has an expected return of µB and a variance in returns of σ2B. The correlation in returns between the two assets, which measures how the assets move up and down together, is ρAB.2 The expected returns and variance of a two-asset portfolio can be written as a function of these inputs and the proportion of the portfolio going to each asset.

μportfolio = wA μA + (1 - wA) μB

σ2portfolio = wA2 σ2A + (1 - wA)2σ2B + 2 wA wB ρAB σA σB

where

wA = Proportion of the portfolio in asset A

The last term in the variance formulation is sometimes written in terms of the covariance in returns between the two assets, which is

σAB = ρAB σA σB

The savings that accrue from diversification are a function of the correlation coefficient. Other things remaining equal, the higher the correlation in returns between the two assets, the smaller are the potential benefits from diversification. The next example shows the benefits from diversification.

### The Capital Asset Pricing Model

The risk and return model that has been in use the longest and is still the recognized standard in the majority of most real world analyses is the capital asset pricing model (CAPM). While it has faced its fair share of disapproval over the years, it offers a constructive footing for our conversation of risk and return models.

1. Assumptions

While diversification has its appeal in terms of lessening the exposure of investors to company specific risk, nearly all investors limit their diversification to owning only a few assets. Even large mutual funds are unwilling to own more than a few hundred stocks, and a lot of them hold as few as 8 to 18 stocks. There are two reasons for this unwillingness. The first is that the added benefits of diversification become less significant as the portfolio grows more diversified - the twenty-first asset added will usually grant a much lower reduction in company specific risk than the fifth asset added, and may not cover the costs of diversification, which include transactions and monitoring costs. The second is that countless investors (and funds) trust that they can identify undervalued investments and so choose not to own those investments that they judge to be correctly or overvalued.

The capital asset pricing model assumes that there are no transactions costs, all assets are traded and that investments are infinitely divisible (i.e., you can buy any fraction of a unit of the asset). It also assumes that there is no private information and that investors therefore cannot find under or overvalued assets in the market place. By making these assumptions, it eliminates the factors that cause investors to stop diversifying. With these assumptions in place, the logical end limit of diversification is to hold every traded risky asset (stocks, bonds and real assets included) in your portfolio, in proportion to their market value4. This portfolio of every traded risky asset in the market place is called the market portfolio.

2. Implications for Investors

If all investor in the market owns the identical market portfolio, how accurately do investors signal their risk aversion in their investments? In the capital asset pricing model, investors modify their risk preferences in their allocation choices, where they choose how much to invest in an asset with guaranteed returns - a riskless asset - and how much in risky assets (market portfolio). Investors who are risk averse might decide to invest much or even all of their fortune in the riskless investment. Investors who wish to take on added risk will invest the majority or even all of their fortune in the market portfolio. The investors who place all of their fortune in the market portfolio and are still eager to taking on more risk, could do so by borrowing at the riskless rate and investing in the same market portfolio as everyone else.

These outcomes are predicated on two other assumptions. One, a riskless asset is available. Two, investors are able to lend and borrow at this riskless rate to arrive at their best possible allocations. There are alternatives of the CAPM that permit these assumptions to be relaxed and still come to the same conclusions that are consistent with the general model.

3. What is the Market Risk of an Individual Asset?

The risk of any investment to an investor is the risk added on by that asset to the investor's overall portfolio. In the CAPM world, where all investors own the market portfolio, the risk of an added single investment to an investor will be the risk that this investment adds on to the market portfolio. Naturally, investments moving up and down with the market portfolio will generally be riskier than investments that move up or down less, since unrelated movements to the market portfolio will have little impact on the overall value of the portfolio when an investment is added on to the portfolio. Statistically, this added risk is measured by the covariance of the asset with the market portfolio.

The covariance is a non-standardized measure of market risk; knowing that the covariance of Granite Construction with the Market Portfolio is 55% does not provide a clue as to whether Granite Construction is riskier or safer than the average asset. We therefore standardize the risk measure by dividing the covariance of each asset with the market portfolio by the variance of the market portfolio. This yields the beta of the asset:

Beta of an asset i = Covariance of asset i with Market Portfolio/Variance of the Market Portfolio

Since the covariance of the market portfolio with itself is its variance, the beta of the market portfolio, and by extension, the average asset in it, is one. Investments that are riskier than average (using this measure of risk) will have betas that exceed one and assets that are safer than average will have betas that are lower than one. The riskless asset will have a beta of zero.

4. Computing Expected Returns

The fact that every investor maintains some combination of the riskless investment and the market portfolio leads to the next conclusion, which is that the expected return on an investment is linearly related to the beta of the asset. Specifically, the expected return on an investment can be shown as a function of the risk-free rate and the beta of that asset;

Expected Return on asset i

= Rf + βi [E(Rm) - Rf]

= Risk-free rate + Beta of asset i * (Risk premium on market portfolio) where,

E(Ri) = Expected Return on asset i Rf = Risk-free Rate

E(Rm) = Expected Return on market portfolio

βi = Beta of asset i

To use the capital asset pricing model, we need three inputs. While we will look at the estimation process in far more detail later, each of these inputs is estimated as follows:

• The riskless investment is defined to be an investment where the investor understands the expected return with certainty for the time period of the analysis. Therefore, the riskless rate used may change depending on whether the time period for the expected return is one year, five years or ten years.
• The risk premium is the premium required by investors for investing in the market portfolio, which includes all risky investments in the market, instead of investing in a riskless asset. So, it does not relate to any individual risky investment but to risky investment as a class.
• The beta, which we defined to be the covariance of the asset divided by the market portfolio, is the single company-specific contribution in this equation. In other words, the only reason two investments have different expected returns in the capital asset pricing model is because they have different betas.

In summary, in the capital asset pricing model all of the market risk is captured in one beta, measured relative to a market portfolio, which at least in theory should include all traded assets in the market place held in proportion to their market value.

## Borrowing Risk: Default Risk and the Cost of Debt

When a bank lends to an individual or a company, there is the chance that the borrower will not make interest and principal payments on the borrowing. This prospect of default is called the default risk. In general, borrowers with higher default risk ought to pay higher interest rates on their borrowing than those with lower default risk. This section looks at the extent of default risk, and the association of default risk to interest rates on borrowing.

In contrast to the general equity risk and return models, that measure the outcome of market risk on expected returns, default risk models measure the results of company-specific default risk on promised returns. Where diversification can be used to justify why company-specific risk will not be priced into expected returns for equities, the same basis cannot be applied to securities with limited upside possibilities and significantly larger downside possibilities from company-specific events. To see what we mean by limited upside possibilities, think about investing in the bond issued by a company. The coupons are fixed at the time of the issue, and these coupons denote the pledged cash flow on the bond. The best-case scenario for you as an investor is that you receive the pledged cash flows; you are not allowed to more than these cash flows even if the company is wildly successful. Every other development contains bad news, although in varying degrees, with the pledged cash flows being much lower than the promised cash flows.

Therefore, the expected return on a corporate bond is likely to reveal the company- specific default risk of the company issuing the bond.

## Understanding Default Risk

The default risk of a company is a function of its ability to produce cashflows from everyday operations and its financial responsibilities - including interest and principal payments.5 It is also a function of how liquid a company's assets are since companies with more liquid investments should have an easier time liquidating them, in a crisis, to meet debt requirements. Consequently, the following propositions relate to default risk:

• Companies that generate high cashflows relative to their financial debt requirements generally have lower default risk than do companies that generate low cashflows relative to their debt obligations. Thus, companies with large current investments that generate significant cashflows, will generally have lower default risk than will companies that do not.
• The more stable the cashflows, the lower is the default risk in the company. Companies that operate in conventional and predictable businesses will have lower default risk than will otherwise similar companies that operate in cyclical and/or unstable businesses, for the same level of indebtedness.
• The more liquid a company's assets, for any known amount of operating cashflows and financial obligations, the less default risk in the company.

For as long as there have been borrowers, lenders have had to evaluate default risk. Historically, measurements of default risk have been supported using financial ratios to compute the cashflow coverage (i.e., the degree of cashflows relative to debt repayment requirements) and control for industry effects, to sum up the unpredictability in cashflows and the liquidity of assets.

## Default Risk and Interest rates

When banks did much of the lending to companies, it made sense for banks to expend the resources to make their own assessments of default risk, and they still do for most lenders. The arrival of the corporate bond market produced a need for third party estimations of default risk on behalf of the bondholders. This demand came from the need for economies of scale, since few individual bondholders had the means to do the estimations themselves. In the United States, this led to the expansion of ratings agencies like Standard and Poor's and Moody's which formulated conclusions of the default risk of corporations, using a mix of private and public information, transformed these conclusions into measures of default risk (bond rating) and made these ratings public. Investors buying corporate bonds today use the bond ratings as a shorthand assessment of default risk.

### Bond Ratings

The method of rating a bond begins when a company asks for a rating from the ratings agency. This request is typically carried out by a company wanting to issue bonds. While ratings are not a legal pre-requisite for bond issues, it is unlikely that investors in the bond market will be willing to buy bonds issued by a company that is not well known and has shown itself to be unwilling to put itself through the rigor of a bond rating process. It is not surprising, therefore, that the largest number of rated companies are in the United States, which has the most active corporate bond markets, and that there are relatively few rated companies in Europe, where bank lending remains the norm for all but the largest companies.

The ratings agency then gathers information from both publicly available data, such as financial statements, and the company itself, and makes a judgment on the rating. If it disagrees with the rating, the company is given the chance to submit additional supporting documentation. This bond rating process is shown schematically for one ratings agency, Standard and Poor's (S&P), in Image 1.9:

Image 1.9: The Rating Process

The ratings assigned by these agencies are letter ratings. A rating of AAA from Standard and Poor's and Aaa from Moody's is the highest rating granted to companies deemed as having the lowest default risk. As the default risk increases, the ratings decrease toward D for companies in default (Standard and Poor's). Worksheet 1.1 provides a description of the bond ratings assigned by the two agencies.

Worksheet 1.1: List of Bond Ratings

 Standard and Poor's Moody's AAA The highest debt rating available. The borrower's ability to repay debt is really strong. Aaa Judged to be of the best quality with a low degree of risk. AA The ability to repay is solid and differs from the highest quality only by a trivial amount. Aa High quality but rated lower than Aaa because margin of protection may not be as large or because there may be other elements of long-term risk. A Has strong ability to repay; Borrower is subject to unfavorable results of changes in circumstances and economic conditions. A Bonds possess favorable investment attributes but may be subject to risk in the future. BBB Has adequate ability to repay, but unfavorable economic conditions or circumstances are more likely to lead to risk. Baa Neither highly protected nor poorly secured; adequate payment capacity. BB,B, Regarded as predominantly CCC, speculative, BB being the least CC speculative and CC the most. Ba Judged to have some speculative risk. B Generally lacking characteristics of a desirable investment; probability of payment small. D In default or with payments in arrears. Caa Poor standing and perhaps in default. Ca Very speculative; often in default. C Highly speculative; in default.

While ratings can range from AAA (safest) to D (in default), a rating at or above BBB by Standard and Poor's (Baa for Moody's) is labeled as investment grade, suggesting the opinion of the ratings agency that there is comparatively modest default risk in investing in bonds issued by these companies. Bonds rated below BBB are generally identified as junk bonds or as high-yield bonds. While it is a subjective dividing line, it is a critical one for two reasons. First, different bond investment portfolios are constrained from investing in bonds below investment grade. So, the demand for investment grade bonds tends to be greater and deeper than that for bonds below that grade. Second, companies that are not rated investment grade have a tougher time when they try to raise new funding and they also pay much higher issuance costs when they do. In fact, until the early 1980s, companies below investment grade often could not issue new bonds.6 The view that they are exposed to default risk also creates a lot of other costs including tighter supplier credit and debt covenants.

### Bond Rating Coverage

The bond ratings given by ratings agencies are generally based upon publicly available information, although private information expressed by the company to the rating agency does play an important role. The rating that is delegated to a company's bonds will depend in large part on financial ratios that determine the ability of the company to make its debt payments and turn out steady and knowable cashflows. While a ton of financial ratios exist, Worksheet 1.2 sums up a few of the key ratios that are used to measure default risk:

Worksheet 1.2: Gauging Default Risk: Debt Ratios

 Ratio Description Pretax Interest Coverage = (Pretax Income from Continuing Operations + Interest Expense) / Gross Interest EBITDA Interest Coverage = EBITDA/ Gross Interest Funds from Operations / Total Debt =(Net Income from Continuing Operations + Depreciation) / Total Debt Free Operating Cashflow/ Total Debt = (Funds from Operations - Capital Expenditures - Change in Working Capital) / Total Debt Pretax Return on Permanent Capital = (Pretax Income from Continuing Operations + Interest Expense) / (Average of Beginning of the year and End of the year of long and short term debt, minority interest and Shareholders Equity) Operating Income/Sales (%) = (Sales - COGS (before depreciation) - Selling Expenses - Administrative Expenses - R&D Expenses) / Sales Long Term Debt/ Capital = Long Term Debt / (Long Term Debt + Equity) Total Debt/Capitalization = Total Debt / (Total Debt + Equity)

There is a solid correlation connecting the bond rating a company receives and its performance on these financial ratios. Worksheet 1.3 gives a review of the median ratios from 2006 to 2008 for different S&P ratings classes for manufacturing companies.

Worksheet 1.3: Bond Rating Ratios: 2019-2021

 AAA AA A BBB BB B CCC EBIT interest cov. (x) 17.5 10.8 6.8 3.9 2.3 1.0 0.2 EBITDA interest cov. 21.8 14.6 9.6 6.1 3.8 2.0 1.4 Funds flow/total debt 105.8 55.8 46.1 30.5 19.2 9.4 5.8 Free oper. cash flow/total debt (%) 55.4 24.6 15.6 6.6 1.9 -4.5 -14.0 Return on capital (%) 28.2 22.9 19.9 14.0 11.7 7.2 0.5 Oper.income/sales (%) 29.2 21.3 18.3 15.3 15.4 11.2 13.6 Long-term debt/capital (%) 15.2 26.4 32.5 41.0 55.8 70.7 80.3 Total Debt/ Capital (%) 26.9 35.6 40.1 47.4 61.3 74.6 89.4 Number of companies 10 34 150 234 276 240 23

Note that the pre-tax interest coverage ratio and the EBITDA interest coverage ratio are presented in terms of times interest earned, but the other ratios are presented in percentage terms.

Not unexpected, companies that produce income and cashflows that are drastically greater than debt payments, are profitable, with low debt ratios are expected to be highly rated than are companies that do not have these strengths. There will be individual companies whose ratings are not uniform with their financial ratios, yet, since the ratings agency do bring skewed conclusions into the final mix. So, a company that performs badly on financial ratios but is likely to further its performance spectacularly over the next period could get a higher rating than that reasoned by its current financials. For the majority of companies, however, the financial ratios ought to give a reasonable basis for guessing at the bond rating.

### Bond Ratings and Interest Rates

The interest rate on a corporate bond should be a measure of its default risk. If the rating is a good assessment of the default risk, higher rated bonds should be priced to yield lower interest rates than would lower rated bonds. The difference between the interest rate on a bond with default risk and a default-free government bond is called the default spread. This default spread will differ by maturity of the bond and may vary from time to time, depending on economic factors. Worksheet 1.4 summarizes default spreads in early 2021 for ten-year bonds in each ratings class (using S&P ratings) and the market interest rates on these bonds, based upon a treasury bond rate of 1.29%.

Worksheet 1.4: Bond Default Spreads: July 2021

 Rating is Default Spread Interest Rate on Bond A1/A+ 0.62% 1.90% A2/A 0.52% 1.80% A3/A- 0.55% 1.83% Aa2/AA 0.58% 1.86% Aaa/AAA 0.81% 2.09% B1/B+ 1.03% 2.31% B2/B 1.50% 2.78% B3/B- 1.96% 3.24% Ba1/BB+ 2.62% 3.90% Ba2/BB 3.28% 4.56% Baa2/BBB 4.42% 5.70% C2/C 5.55% 6.83% Ca2/CC 5.95% 7.23% Caa/CCC 10.88% 12.16% D2/D 13.55% 14.83%

Source: https://fred.stlouisfed.org

Worksheet 1.4 shows default spreads at a point in time, but default spreads not only fluctuate over time but they can change for bonds with an identical rating but different maturities. For the bonds with higher ratings, the default spread usually widens for the longer maturities. For bonds with lower ratings, the spreads can decrease as we go to longer maturities, suggesting the truth that near term default risk is greater than long term default risk. Historically, default spreads for all ratings categories have increased during recessions and decreased during economic booms. In Image 1.10, we take a look at the changes of default spreads for different bond rating classes through 2021:

Image 1.10: Default Spreads on Rating Classes

Note how much default spreads changed through 2021. This suggests that default spreads for bonds must be re-estimated at regular intervals, particularly if the economy shifts from low to high growth or vice versa.

A final note is that everything that has been mentioned about the relationship between interest rates and bond ratings could be stated generally about interest rates and default risk. The availability of bond ratings is a convenience that makes the evaluation of default risk fairly easier for us when examining companies. In its absence, we would still have to evaluate default risk on our own and come up with estimates of the default spread we would charge if we were lending to a company.

Industry-Specific Risk: These are unexpected outcomes on project cashflows of industry-wide changes in technology, amendments in laws or in the price of a commodity.

Market Risk: Market risk is the unanticipated adjustments in project cashflows produced by changes in interest rates, inflation rates and the economy that impact all companies, though to differing degrees.

Competitive Risk: This is the unanticipated effect on the cashflows in a project of competitor actions - these effects can be positive or negative.

International Risk: This is the additional uncertainty created in cashflows of projects by unanticipated changes in exchange rates and by political risk in foreign markets.

Riskless Asset: A riskless asset is one, where the actual return is always equal to the expected return.

Risk is in the eyes of the beholder: A privately owned company will usually end up with a higher discount rate for a project than would an otherwise similar publicly traded company with diversified investors.

Diversification: This is the practice of owning many investments in a portfolio, either amongst the same asset class (e.g. stocks) or across many asset classes (real estate, bonds etc.

Project-Specific: This is risk that influences just the project under consideration, and might come from things exclusive to the project or estimation error.

Beta: The beta of any investment in the CAPM is a standardized measure of the risk that it adds to the market portfolio.

1A utility function is a way of summarizing investor preferences into a generic term called 'utility' on the basis of some choice variables. In this case, for instance, investor utility or satisfaction is stated as a function of wealth. By doing so, we effectively can answer questions such as - Will an investor be twice as happy if he has twice as much wealth? Does each marginal increase in wealth lead to less additional utility than the prior marginal increase? In one specific form of this function, the quadratic utility function, the entire utility of an investor can be compressed into the expected wealth measure and the standard deviation in that wealth, which provides a justification for the use of a framework where only the expected return (mean) and its standard deviation (variance) matter.

2The variance is percent squared. In other words, if you stated the standard deviation of 9.96% in decimal terms, it would be 0.0996 but the variance of 99.15% would be 0.009915 in decimal terms.

3Companies could conceivably diversify away competitive risk by acquiring their existing competitors. Doing so would expose them to attacks under the anti-trust law, however and would not eliminate the risk from as yet unannounced competitors.

4If investments are not held in proportion to their market value, investors are still losing some diversification benefits. Since there is no gain from over weighting some sectors and under weighting others in a market place where the odds are random of finding undervalued and overvalued assets, investors will not do so.

5Financial obligation refers to any payment that the company has legally obligated itself to make, such as interest and principal payments. It does not include discretionary cashflows, such as dividend payments or new capital expenditures, which can be deferred or delayed, without legal consequences, though there may be economic consequences.

6In the early 1980s, Michael Milken and Drexel Burnham that created the junk bond market, allowing for original issuance of junk bonds. They did this primarily to facilitate hostile takeovers by the raiders of the era.

Excerpts (reworded) from Aswath Damodarn, Corporate Finance 2009.

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