Financial Modelling « sudesca.org

Financial Modelling « sudesca.org

Jul 29th, 10 / 0 Comments / Share this Entry

FINANCIAL MODELING

Financial modeling is a process of forecasting performance of a certain asset, using relationships among operating, investing, and financing variables. The central aim of all financial modeling is valuation under uncertainty: how to estimate the value of a security when its future trajectory, or the trajectory of the other securities or economic variables it depends on, is unknown. Usually, financial modeling requires a great deal of spreadsheet work.

Financial Modeling Application

ü Business valuation, especially discounted cash flow
ü Cost of capital or WACC
ü Modeling the term structure of interest rate and credit spread
ü Option pricing
ü Real options
ü Risk modeling
ü Portfolio problems

Standard and Premise of Business Value

Before the value of a business can be measured, the valuation assignment must specify the reason for and circumstances surrounding the business valuation. These are formally known as the business value standard and premise of value.

Business valuation results can vary considerably depending upon the choice of both the standard and premise of value. For example, a business buyer and seller may bargain to establish the value of business assets that approaches the fair market value standard.

However, the value conclusions based on the going concern premise and that of assemblage of business assets may be quite different. One reason is that an operating business creates value by means of its ability to coordinate its capital, human and management resources to produce economic income. The same set of assets not currently used to produce income is generally worth less.

Reasons for Business Valuation

Business people may need to conduct business valuation for a number of reasons including sale, estate tax planning, estate tax valuation, divorce, business purchase price allocation, collateral documentation, litigation and documenting that a sales price is equitable.

Fair market value

“Fair market value”, a central standard of measuring business value, is defined as the price at which property would change hands between a willing buyer and a willing seller when the former is not under any compulsion to buy and the latter is not under any compulsion to sell, both parties having reasonable knowledge of relevant facts. See IRS Rev. Rul. 59-60, 1959-1, Cum. Bulletin 237, codified at 26 C.F.R. § 20.2031-1(b).

The fair market value standard incorporates certain assumptions, including the assumptions that the hypothetical purchaser is reasonably prudent and rational but is not motivated by any synergistic or strategic influences; that the business will continue as a going concern and not be liquidated; that the hypothetical transaction will be conducted in cash or equivalents; and that the parties are willing and able to consummate the transaction.

These assumptions might not, and probably do not, reflect the actual conditions of the market in which the subject business might be sold. However, these conditions are assumed because they yield a uniform standard of value, after applying generally-accepted valuation techniques, which allows meaningful comparison between businesses which are similarly situated.

Elements of business valuation

Economic conditions

A business valuation report generally begins with a description of national, regional and local economic conditions existing as of the valuation date, as well as the conditions of the industry in which the subject business operates. A common source of economic information for the first section of the business valuation report is the Federal Reserve Board’s Beige Book, published quarterly by the Federal Reserve Bank. State governments and industry associations often publish useful statistics describing regional and industry conditions.

Financial Analysis

The financial statement analysis generally involves common size analysis, ratio analysis (liquidity, turnover, profitability, etc.), trend analysis and industry comparative analysis. This permits the valuation analyst to compare the subject company to other businesses in the same or similar industry, and to discover trends affecting the company and/or the industry over time. By comparing a company’s financial statements in different time periods, the valuation expert can view growth or decline in revenues or expenses, changes in capital structure, or other financial trends. How the subject company compares to the industry will help with the risk assesment and ultimately help determine the discount rate and the selection of market multiples.

Normalization of financial statements

The most common normalization adjustments fall into the following four categories:

Comparability Adjustments. The valuator may adjust the subject company’s financial statements to facilitate a comparison between the subject company and other businesses in the same industry or geographic location. These adjustments are intended to eliminate differences between the way that published industry data is presented and the way that the subject company’s data is presented in its financial statements.

Non-operating Adjustments. It is reasonable to assume that if a business were sold in a hypothetical sales transaction (which is the underlying premise of the fair market value standard), the seller would retain any assets which were not related to the production of earnings or price those non-operating assets separately. For this reason, non-operating assets (such as excess cash) are usually eliminated from the balance sheet.

Non-recurring Adjustments. The subject company’s financial statements may be affected by events that are not expected to recur, such as the purchase or sale of assets, a lawsuit, or an unusually large revenue or expense. These non-recurring items are adjusted so that the financial statements will better reflect the management’s expectations of future performance.

Discretionary Adjustments. The owners of private companies may be paid at variance from the market level of compensation that similar executives in the industry might command. In order to determine fair market value, the owner’s compensation, benefits, perquisites and distributions must be adjusted to industry standards. Similarly, the rent paid by the subject business for the use of property owned by the company’s owners individually may be scrutinized.

Income, Asset and Market Approaches

Three different approaches are commonly used in business valuation: the income approach, the asset-based approach, and the market approach. Within each of these approaches, there are various techniques for determining the fair market value of a business. Generally, the income approaches determine value by calculating the net present value of the benefit stream generated by the business (discounted cash flow); the asset-based approaches determine value by adding the sum of the parts of the business (net asset value); and the market approaches determine value by comparing the subject company to other companies in the same industry, of the same size, and/or within the same region.

In determining which of these approaches to use, the valuation professional must exercise discretion. Each technique has advantages and drawbacks, which must be considered when applying those techniques to a particular subject company. Most treatises and court decisions encourage the valuator to consider more than one technique, which must be reconciled with each other to arrive at a value conclusion. A measure of common sense and a good grasp of mathematics is helpful.

Income approaches

The income approaches determine fair market value by multiplying the benefit stream generated by the subject company times a discount or capitalization rate. The discount or capitalization rate converts the stream of benefits into present value. There are several different income approaches, including capitalization of earnings or cash flows, discounted future cash flows (“DCF”), and the excess earnings method (which is a hybrid of asset and income approaches). Most of the income approaches consider the subject company’s historical financial data; only the DCF method requires the subject company to provide projected financial data. Most of the income approaches look to the company’s adjusted historical financial data for a single period; only DCF requires data for multiple future periods. The discount or capitalization rate must be matched to the type of benefit stream to which it is applied. The result of a value calculation under the income approach is generally the fair market value of a controlling, marketable interest in the subject company, since the entire benefit stream of the subject company is most often valued, and the capitalization and discount rates are derived from statistics concerning public companies.

Discount or capitalization rates

A discount or capitalization rate is used to determine the present value of the expected returns of a business. The discount rate and capitalization rate are closely related to each other, but distinguishable. Generally speaking, the discount rate or capitalization rate may be defined as the yield necessary to attract investors to a particular investment, given the risks associated with that investment. The discount rate is applied only to discounted cash flow (DCF) valuations, which are based on projected business data over multiple periods of time. In DCF valuations, a series of projected cash flows is divided by the discount rate to derive the present value of the discounted cash flows. The sum of the discounted cash flows is added to a terminal value, which represents the present value of business cash flows into perpetuity. The sum of the discounted cash flows and the terminal value is the value of the business.

On the other hand, a capitalization rate is applied in methods of business valuation that are based on historical business data for a single period of time. The after-tax net cash flow capitalization rate is equal to the discount rate minus the long-term sustainable growth rate. The after-tax net cash flow of a business is divided by the capitalization rate to derive the present value. Capitalization rates may be modified so that they may be applied to after-tax net income or pre-tax cash flows or income. There are several different methods of determining the appropriate discount rates. The discount rate is composed of two elements: (1) the risk-free rate, which is the return that an investor would expect from a secure, practically risk-free investment, such as a government bond; plus (2) a risk premium that compensates an investor for the relative level of risk associated with a particular investment in excess of the risk-free rate. Most importantly, the selected discount or capitalization rate must be consistent with stream of benefits to which it is to be applied.

Build-Up Method

The Build-Up Method is a widely-recognized method of determining the after-tax net cash flow discount rate, which in turn yields the capitalization rate. The figures used in the Build-Up Method are derived from various sources. This method is called a “build-up” method because it is the sum of risks associated with various classes of assets. It is based on the principle that investors would require a greater return on classes of assets that are more risky. The first element of an Build-Up capitalization rate is the risk-free rate, which is the rate of return for long-term government bonds. Investors who buy large-cap equity stocks, which are inherently more risky than long-term government bonds, require a greater return, so the next element of the Build-Up method is the equity risk premium. In determining a company’s value, the long-horizon equity risk premium is used because the Company’s life is assumed to be infinite. The sum of the risk-free rate and the equity risk premium yields the long-term average market rate of return on large public company stocks.

Similarly, investors who invest in small cap stocks, which are riskier than blue-chip stocks, require a greater return, called the “size premium.” Size premium data is generally available from two sources: Morningstars’ (formerly Ibbotson & Associates’) Stocks, Bonds, Bills & Inflation and Duff & Phelps’ Risk Premium Report.

By adding the first three elements of a Build-Up discount rate, we can determine the rate of return that investors would require on their investments in small public company stocks. These three elements of the Build-Up discount rate are known collectively as the “systematic risks.”

In addition to systematic risks, the discount rate must include “unsystematic risks,” which fall into two categories. One of those categories is the “industry risk premium.” Morningstar’s yearbooks contain empirical data to quantify the risks associated with various industries, grouped by SIC industry code.

The other category of unsystematic risk is referred to as “specific company risk.” Historically, no published data has been available to quantify specific company risks. However as of late 2006, new research has been able to quantify, or isolate, this risk for publicly-traded stocks through the use of Total Beta calculations. P. Butler and K. Pinkerton have outlined a procedure using a modified Capital Asset Pricing Model (CAPM) to calculate the company specific risk premium. The model uses an equality between the standard CAPM which relies on the total beta on one side of the equation; and the firm’s beta, size premium and company specific risk premium on the other. The equality is then solved for the company specific risk premium as the only unknown. While this is ground-breaking research, it has yet to be adopted and used by the valuation community at large.

It is important to understand why this capitalization rate for small, privately-held companies is significantly higher than the return that an investor might expect to receive from other common types of investments, such as money market accounts, mutual funds, or even real estate. Those investments involve substantially lower levels of risk than an investment in a closely-held company. Depository accounts are insured by the federal government (up to certain limits); mutual funds are composed of publicly-traded stocks, for which risk can be substantially minimized through portfolio diversification; and real estate almost invariably appreciates in value of long time horizons.

Closely-held companies, on the other hand, frequently fail for a variety of reasons too numerous to name. Examples of the risk can be witnessed in the storefronts on every Main Street in America. There are no federal guarantees. The risk of investing in a private company cannot be reduced through diversification, and most businesses do not own the type of hard assets that can ensure capital appreciation over time. This is why investors demand a much higher return on their investment in closely-held businesses; such investments are inherently much more risky.

Capital Asset Pricing Model (“CAP-M”)

The Capital Asset Pricing Model is another method of determining the appropriate discount rate in business valuations. The CAP-M method originated from the Nobel Prize winning studies of Harry Markowitz, James Tobin and William Sharpe. Like the Ibbotson Build-Up method, the CAP-M method derives the discount rate by adding a risk premium to the risk-free rate. In this instance, however, the risk premium is derived by multiplying the equity risk premium times “beta,” which is a measure of stock price volatility. Beta is published by various sources (including Ibbotson Associates, which was used in this valuation) for particular industries and companies. Beta is associated with the systematic risks of an investment.

One of the criticisms of the CAP-M method is that beta is derived from the volatility of prices of publicly-traded companies, which are likely to differ from private companies in their capital structures, diversification of products and markets, access to credit markets, size, management depth, and many other respects. Where private companies can be shown to be sufficiently similar to public companies, however, the CAP-M model may be appropriate.

Weighted Average Cost of Capital (“WACC”)

The weighted average cost of capital is the third major approach to determining a discount rate. The WACC method determines the subject company’s actual cost of capital by calculating the weighted average of the company’s cost of debt and cost of equity. The WACC capitalization rate must be applied to the subject company’s net cash flow to invested equity. One of the problems with this method is that the valuator may elect to calculate WACC according to the subject company’s existing capital structure, the average industry capital structure, or the optimal capital structure. Such discretion detracts from the objectivity of this approach, in the minds of some critics.

Once the capitalization or discount rate is determined, it must be applied to an appropriate economic income streams: pretax cash flow, aftertax cash flow, pretax net income, after tax net income, excess earnings, projected cash flow, etc. The result of this formula is the indicated value before discounts. Before moving on to calculate discounts, however, the valuation professional must consider the indicated value under the asset and market approaches.

Careful matching of the discount rate to the appropriate measure of economic income is critical to the accuracy of the business valuation results. Net cash flow is a frequent choice in professionally conducted business appraisals. The rationale behind this choice is that this earnings basis corresponds to the equity discount rate derived from the Build-Up or CAP-M models: the returns obtained from investments in publicly traded companies can easily be represented in terms of net cash flows. At the same time, the discount rates are generally also derived from the public capital markets data.

Asset-based approaches

The value of asset-based analysis a business is equal to the sum of its parts. That is the theory underlying the asset-based approaches to business valuation. The asset approach to business valuation is based on the principle of substitution: no rational investor will pay more for the business assets than the cost of procuring assets of similar economic utility. In contrast to the income-based approaches, which require the valuation professional to make subjective judgments about capitalization or discount rates, the adjusted net book value method is relatively objective. Pursuant to accounting convention, most assets are reported on the books of the subject company at their acquisition value, net of depreciation where applicable. These values must be adjusted to fair market value wherever possible. The value of a company’s intangible assets, such as goodwill, is generally impossible to determine apart from the company’s overall enterprise value. For this reason, the asset-based approach is not the most probative method of determining the value of going business concerns. In these cases, the asset-based approach yields a result that is probably lesser than the fair market value of the business. In considering an asset-based approach, the valuation professional must consider whether the shareholder whose interest is being valued would have any authority to access the value of the assets directly. Shareholders own shares in a corporation, but not its assets, which are owned by the corporation. A controlling shareholder may have the authority to direct the corporation to sell all or part of the assets it owns and to distribute the proceeds to the shareholder(s). The non-controlling shareholder, however, lacks this authority and cannot access the value of the assets. As a result, the value of a corporation’s assets is rarely the most relevant indicator of value to a shareholder who cannot avail himself of that value. Adjusted net book value may be the most relevant standard of value where liquidation is imminent or ongoing; where a company earnings or cash flow are nominal, negative or worth less than its assets; or where net book value is standard in the industry in which the company operates. None of these situations applies to the Company which is the subject of this valuation report. However, the adjusted net book value may be used as a “sanity check” when compared to other methods of valuation, such as the income and market approaches.

Market approaches

The market approach to business valuation is rooted in the economic principle of competition: that in a free market the supply and demand forces will drive the price of business assets to a certain equilibrium. Buyers would not pay more for the business, and the sellers will not accept less, than the price of a comparable business enterprise. It is similar in many respects to the “comparable sales” method that is commonly used in real estate appraisal. The market price of the stocks of publicly traded companies engaged in the same or a similar line of business, whose shares are actively traded in a free and open market, can be a valid indicator of value when the transactions in which stocks are traded are sufficiently similar to permit meaningful comparison.

The difficulty lies in identifying public companies that are sufficiently comparable to the subject company for this purpose. Also, as for a private company, the equity is less liquid (in other words its stocks are less easy to buy or sell) than for a public company, its value is considered to be slightly lower than such a market-based valuation would give

Guideline Public Company method

The Guideline Public Company method entails a comparison of the subject company to publicly traded companies. The comparison is generally based on published data regarding the public companies’ stock price and earnings, sales, or revenues, which is expressed as a fraction known as a “multiple.” If the guideline public companies are sufficiently similar to each other and the subject company to permit a meaningful comparison, then their multiples should be nearly equal. The public companies identified for comparison purposes should be similar to the subject company in terms of industry, product lines, market, growth, and risk.

Transaction Method or Direct Market Data Method

Using this method, the valuation analyst may determine market multiples by reviewing published data regarding actual transactions involving either minority or controlling interests in either publicly traded or closely held companies. In judging whether a reasonable basis for comparison exists, the valuation analysis must consider: (1) the similarity of qualitative and quantitative investment and investor characteristics; (2) the extent to which reliable data is known about the transactions in which interests in the guideline companies were bought and sold; and (3) whether or not the price paid for the guideline companies was in an arms-length transaction, or a forced or distressed sale.

The most widely used transactional databases include:

Institute of Business Appraisers (smaller companies)
BIZCOMPS® (smaller companies)
Pratt’s Stats® (smaller to mid-sized companies)
Public Stats™ (larger companies)
DoneDeals® (larger companies)
Alacra (larger companies)

Discounts and premiums

The valuation approaches yield the fair market value of the Company as a whole. In valuing a minority, non-controlling interest in a business, however, the valuation professional must consider the applicability of discounts that affect such interests. Discussions of discounts and premiums frequently begin with a review of the “levels of value.” There are three common levels of value: controlling interest, marketable minority, and non-marketable minority. The intermediate level, marketable minority interest, is lesser than the controlling interest level and higher than the non-marketable minority interest level. The marketable minority interest level represents the perceived value of equity interests that are freely traded without any restrictions. These interests are generally traded on the New York Stock Exchange, AMEX, NASDAQ, and other exchanges where there is a ready market for equity securities. These values represent a minority interest in the subject companies – small blocks of stock that represent less than 50% of the company’s equity, and usually much less than 50%. Controlling interest level is the value that an investor would be willing to pay to acquire more than 50% of a company’s stock, thereby gaining the attendant prerogatives of control. Some of the prerogatives of control include: electing directors, hiring and firing the company’s management and determining their compensation; declaring dividends and distributions, determining the company’s strategy and line of business, and acquiring, selling or liquidating the business. This level of value generally contains a control premium over the intermediate level of value, which typically ranges from 25% to 50%. An additional premium may be paid by strategic investors who are motivated by synergistic motives. Non-marketable, minority level is the lowest level on the chart, representing the level at which non-controlling equity interests in private companies are generally valued or traded. This level of value is discounted because no ready market exists in which to purchase or sell interests. Private companies are less “liquid” than publicly-traded companies, and transactions in private companies take longer and are more uncertain. Between the intermediate and lowest levels of the chart, there are restricted shares of publicly-traded companies. Despite a growing inclination of the IRS and Tax Courts to challenge valuation discounts , Shannon Pratt suggested in a scholarly presentation recently that valuation discounts are actually increasing as the differences between public and private companies is widening . Publicly-traded stocks have grown more liquid in the past decade due to rapid electronic trading, reduced commissions, and governmental deregulation. These developments have not improved the liquidity of interests in private companies, however. Valuation discounts are multiplicative, so they must be considered in order. Control premiums and their inverse, minority interest discounts, are considered before marketability discounts are applied.

Discount for lack of control

The first discount that must be considered is the discount for lack of control, which in this instance is also a minority interest discount. Minority interest discounts are the inverse of control premiums, to which the following mathematical relationship exists: MID = 1 – [ 1 / (1 + CP)] The most common source of data regarding control premiums is the Control Premium Study, published annually by Mergerstat since 1972. Mergerstat compiles data regarding publicly announced mergers, acquisitions and divestitures involving 10% or more of the equity interests in public companies, where the purchase price is $1 million or more and at least one of the parties to the transaction is a U.S. entity. Mergerstat defines the “control premium” as the percentage difference between the acquisition price and the share price of the freely-traded public shares five days prior to the announcement of the M&A transaction. While it is not without valid criticism, Mergerstat control premium data (and the minority interest discount derived therefrom) is widely accepted within the valuation profession.

Discount for lack of marketability

Another factor to be considered in valuing closely held companies is the marketability of an interest in such businesses. Marketability is defined as the ability to convert the business interest into cash quickly, with minimum transaction and administrative costs, and with a high degree of certainty as to the amount of net proceeds. There is usually a cost and a time lag associated with locating interested and capable buyers of interests in privately-held companies, because there is no established market of readily-available buyers and sellers. All other factors being equal, an interest in a publicly traded company is worth more because it is readily marketable. Conversely, an interest in a private-held company is worth less because no established market exists. The IRS Valuation Guide for Income, Estate and Gift Taxes, Valuation Training for Appeals Officers acknowledges the relationship between value and marketability, stating: “Investors prefer an asset which is easy to sell, that is, liquid.” The discount for lack of control is separate and distinguishable from the discount for lack of marketability. It is the valuation professional’s task to quantify the lack of marketability of an interest in a privately-held company. Because, in this case, the subject interest is not a controlling interest in the Company, and the owner of that interest cannot compel liquidation to convert the subject interest to cash quickly, and no established market exists on which that interest could be sold, the discount for lack of marketability is appropriate. Several empirical studies have been published that attempt to quantify the discount for lack of marketability. These studies include the restricted stock studies and the pre-IPO studies. The aggregate of these studies indicate average discounts of 35% and 50%, respectively. Some experts believe the Lack of Control and Marketabilty discounts can aggregate discounts for as much as ninety percent of a Company’s fair market value, specifically with family owned companies.

Restricted stock studies

Restricted stocks are equity securities of public companies that are similar in all respects to the freely traded stocks of those companies except that they carry a restriction that prevents them from being traded on the open market for a certain period of time, which is usually one year (two years prior to 1990). This restriction from active trading, which amounts to a lack of marketability, is the only distinction between the restricted stock and its freely-traded counterpart. Restricted stock can be traded in private transactions and usually do so at a discount. The restricted stock studies attempt to verify the difference in price at which the restricted shares trade versus the price at which the same unrestricted securities trade in the open market as of the same date. The underlying data by which these studies arrived at their conclusions has not been made public. Consequently, it is not possible when valuing a particular company to compare the characteristics of that company to the study data. Still, the existence of a marketability discount has been recognized by valuation professionals and the Courts, and the restricted stock studies are frequently cited as empirical evidence. Notably, the lowest average discount reported by these studies was 26% and the highest average discount was 45%.

Option pricing

In addition to the restricted stock studies, U.S. publicly traded companies are able to sell stock to offshore investors (SEC Regulation S, enacted in 1990) without registering the shares with the Securities and Exchange Commission. The offshore buyers may resell these shares in the United States, still without having to register the shares, after holding them for just 40 days. Typically, these shares are sold for 20% to 30% below the publicly traded share price. Some of these transactions have been reported with discounts of more than 30%, resulting from the lack of marketability. These discounts are similar to the marketability discounts inferred from the restricted and pre-IPO studies, despite the holding period being just 40 days. Studies based on the prices paid for options have also confirmed similar discounts. If one holds restricted stock and purchases an option to sell that stock at the market price (a put), the holder has, in effect, purchased marketability for the shares. The price of the put is equal to the marketability discount. The range of marketability discounts derived by this study was 32% to 49%.

Pre-IPO studies

Another approach to measure the marketability discount is to compare the prices of stock offered in initial public offerings (IPOs) to transactions in the same company’s stocks prior to the IPO. Companies that are going public are required to disclose all transactions in their stocks for a period of three years prior to the IPO. The pre-IPO studies are the leading alternative to the restricted stock stocks in quantifying the marketability discount. The pre-IPO studies are sometimes criticized because the sample size is relatively small, the pre-IPO transactions may not be arm’s length, and the financial structure and product lines of the studied companies may have changed during the three year pre-IPO window.

Applying the studies

The studies confirm what the marketplace knows intuitively: Investors covet liquidity and loathe obstacles that impair liquidity. Prudent investors buy illiquid investments only when there is a sufficient discount in the price to increase the rate of return to a level which brings risk-reward back into balance. The referenced studies establish a reasonable range of valuation discounts from the mid-30%s to the low 50%s. The more recent studies appeared to yield a more conservative range of discounts than older studies, which may have suffered from smaller sample sizes. Another method of quantifying the lack of marketability discount is the Quantifying Marketability Discounts Model (QMDM).

DISCOUNTED CASH FLOW

In finance, the discounted cash flow (or DCF) approach describes a method to value a project, company, or financial asset using the concepts of the time value of money. All future cash flows are estimated and discounted to give them a present value. The discount rate used is generally the appropriate cost of capital, and incorporates judgments of the uncertainty (riskiness) of the future cash flows.

FV=PV (1+i)n
DPV=FV/(1+i)n

COST OF CAPITAL

The cost of capital for a firm is a weighted sum of the cost of equity and the cost of debt (see Capital investment decisions). It is also known as the “Hurdle Rate” or “Discount Rate”.

Capital (money) used to fund a business should earn returns for the capital owner who risked his/her saved money. For an investment to be worthwhile the projected return on capital must be greater than the cost of capital. Otherwise stated, the risk-adjusted return on capital (that is, incorporating not just the projected returns, but the probabilities of those projections) must be higher than the cost of capital.

The cost of debt is relatively simple to calculate, as it is composed of the rate of interest paid. In practice, the interest-rate paid by the company will include the risk-free rate plus a risk component, which itself incorporates a probable rate of default (and amount of recovery given default). For companies with similar risk or credit ratings, the interest rate is largely exogenous.

Cost of equity is more challenging to calculate as equity does not pay a set return to its investors. Similar to the cost of debt, the cost of equity is broadly defined as the risk-weighted projected return required by investors, where the return is largely unknown. The cost of equity is therefore inferred by comparing the investment to other investments with similar risk profiles to determine the “market” cost of equity.

The cost of capital is often used as the discount rate, the rate at which projected cash flow will be discounted to give a present value or net present value.

Cost of debt

The cost of debt is computed by taking the rate on a non-defaulting bond whose duration matches the term structure of the corporate debt, then adding a default premium. This default premium will rise as the amount of debt increases (since the risk rises as the amount of debt rises). Since in most cases debt expense is a deductible expense, the cost of debt is computed as an after tax cost to make it comparable with the cost of equity (earnings are after-tax as well). Thus, for profitable firms, debt is discounted by the tax rate. Basically this is used for large corporations only.

Cost of equity

Cost of equity = Risk free rate of return + Premium expected for risk

Expected return

The expected return can be calculated as the “dividend capitalization model”, which is (dividend per share / price per share) + growth rate of dividends (that is, dividend yield + growth rate of dividends).

Capital asset pricing model

The capital asset pricing model (CAPM) is used in finance to determine a theoretically appropriate price of an asset such as a security. The expected return on equity according to the capital asset pricing model. The market risk is normally characterized by the ? parameter. Thus, the investors would expect (or demand) to receive:

WEIGHTED AVERAGE COST OF CAPITAL

The Weighted Average Cost of Capital (WACC) is used in finance to measure a firm’s cost of capital.

The total capital for a firm is the value of its equity (for a firm without outstanding warrants and options, this is the same as the company’s market capitalization) plus the cost of its debt (the cost of debt should be continually updated as the cost of debt changes as a result of interest rate changes). Notice that the “equity” in the debt to equity ratio is the market value of all equity, not the shareholders’ equity on the balance sheet.

Calculation of WACC is an iterative procedure which requires estimation of the fair market value of equity capital

CAPITAL STRUCTURE

Because of tax advantages on debt issuance, it will be cheaper to issue debt rather than new equity (this is only true for profitable firms, tax breaks are available only to profitable firms). At some point, however, the cost of issuing new debt will be greater than the cost of issuing new equity. This is because adding debt increases the default risk – and thus the interest rate that the company must pay in order to borrow money. By utilizing too much debt in its capital structure, this increased default risk can also drive up the costs for other sources (such as retained earnings and preferred stock) as well. Management must identify the “optimal mix” of financing – the capital structure where the cost of capital is minimized so that the firms value can be maximized.

MODIGLIANI-MILLER THEOREM

If there were no tax advantages for issuing debt, and equity could be freely issued, Miller and Modigliani showed that the value of a leveraged firm and the value of an unleveraged firm should be the same.

INTEREST

Interest is a fee paid on borrowed capital. Assets lent include money, shares, consumer goods through hire purchase, major assets such as aircraft, and even entire factories in finance lease arrangements. The interest is calculated upon the value of the assets in the same manner as upon money. Interest can be thought of as “rent on money”.

The fee is compensation to the lender for foregoing other useful investments that could have been made with the loaned money. Instead of the lender using the assets directly, they are advanced to the borrower. The borrower then enjoys the benefit of using the assets ahead of the effort required to obtain them, while the lender enjoys the benefit of the fee paid by the borrower for the privilege. The amount lent, or the value of the assets lent, is called the principal. This principal value is held by the borrower on credit. Interest is therefore the price of credit, not the price of money as it is commonly – and mistakenly – believed to be. The percentage of the principal that is paid as a fee (the interest), over a certain period of time, is called the interest rate.

Interest rates and credit risk

It is increasingly recognized that the business cycle, interest rates and credit risk are tightly interrelated. The Jarrow-Turnbull model was the first model of credit risk which explicitly had random interest rates at its core. Lando (2004), Darrell Duffie and Singleton (2003), and van Deventer and Imai (2003) discuss interest rates when the issuer of the interest-bearing instrument can default.

Money and inflation

Loans, bonds, and shares have some of the characteristics of money and are included in the broad money supply.

By setting i*n, the government institution can affect the markets to alter the total of loans, bonds and shares issued. Generally speaking, a higher real interest rate reduces the broad money supply.

Open market operations in the United States

The Federal Reserve (often referred to as ‘The Fed’) implements monetary policy largely by targeting the federal funds rate. This is the rate that banks charge each other for overnight loans of federal funds. Federal funds are the reserves held by banks at the Fed.

Open market operations are one tool within monetary policy implemented by the Federal Reserve to steer short-term interest rates. Using the power to buy and sell treasury securities, the Open Market Desk at the Federal Reserve Bank of New York can supply the market with dollars by purchasing T-notes, hence increasing the nation’s money supply. By increasing the money supply or Aggregate Supply of Funding (ASF), interest rates will fall due to the excess of dollars banks will end up with in their reserves. Excess reserves may be lent in the Fed funds market to other banks, thus driving down rates.

Credit spread options: credit call spread is a “bearish” call spread, which has more premium on the short call. A credit put spread is a “bullish” put spread and has more premium on the short put.

Credit spread (bond): In finance, a credit spread is the difference in yield between different securities due to different credit quality. The credit spread reflects the additional net yield an investor can earn from a security with more credit risk relative to one with less credit risk. The credit spread of a particular security is often quoted in relation to the yield on a credit risk-free benchmark security or reference rate.

RISK MODELING

Risk modeling refers to the use of formal econometric techniques to determine the aggregate risk in a financial portfolio. Risk modeling is one of many subtasks within the broader area of financial modeling.

Risk modeling uses a variety of techniques including market risk, Value-at-Risk (VaR), Historical Simulation (HS), or Extreme Value Theory (EVT) in order to analyze a portfolio and make forecasts of the likely losses that would be incurred for a variety of risks. Such risks are typically grouped into credit risk, liquidity risk, interest rate risk, and operational risk categories.

Many large financial intermediary firms use risk modeling to help portfolio managers assess the amount of capital reserves to maintain, and to help guide their purchases and sales of various classes of financial assets.

Formal risk modeling is required under the Basel II proposal for all the major international banking institutions by the various national depository institution regulators.

Quantitative risk analysis and modeling have become important in the light of corporate scandals in the past few years (most notably, Enron), Basel II, the revised FAS 123R and the Sarbanes-Oxley Act. In the past, risk analysis was done qualitatively but now with the advent of powerful computing software, quantitative risk analysis can be done quickly and effortlessly.

PORTFOLIO PROBLEMS

In finance, a portfolio is an appropriate mix of or collection of investments held by an institution or a private individual. In building up an investment portfolio a financial institution will typically conduct its own investment analysis, whilst a private individual may make use of the services of a financial advisor or a financial institution which offers portfolio management services. Holding a portfolio is part of an investment and risk-limiting strategy called diversification. By owning several assets, certain types of risk (in particular specific risk) can be reduced. The assets in the portfolio could include stocks, bonds, options, warrants, gold certificates, real estate, futures contracts, production facilities, or any other item that is expected to retain its value.

Portfolio management involves deciding what assets to include in the portfolio, given the goals of the portfolio owner and changing economic conditions. Selection involves deciding what assets to purchase, how many to purchase, when to purchase them, and what assets to divest. These decisions always involve some sort of performance measurement, most typically expected return on the portfolio, and the risk associated with this return (i.e. the standard deviation of the return). Typically the expected return from portfolios of different asset bundles are compared.

Porfolio formation

Many strategies have been developed to form a portfolio.

Ø equally-weighted portfolio
Ø capitalization-weighted portfolio
Ø price-weighted portfolio
Ø optimal portfolio (for which the Sharpe ratio is highest)

VALUATION OF OPTIONS

Black–Scholes:

The term Black–Scholes refers to three closely related concepts:

Ø The Black–Scholes model is a mathematical model of the market for an equity, in which the equity’s price is a stochastic process.
Ø The Black–Scholes PDE is a partial differential equation which (in the model) must be satisfied by the price of a derivative on the equity.
Ø The Black–Scholes formula is the result obtained by solving the Black-Scholes PDE for European put and call options.

Binomial options pricing model: In finance, the binomial options pricing model (BOPM) provides a generalisable numerical method for the valuation of options. The binomial model was first proposed by Cox, Ross and Rubinstein (1979). Essentially, the model uses a “discrete-time” model of the varying price over time of the underlying financial instrument. Option valuation is then computed via application of the risk neutrality assumption over the life of the option, as the price of the underlying instrument evolves.

Monte Carlo option model: In mathematical finance, a Monte Carlo option model uses Monte Carlo methods to calculate the value of an option with multiple sources of uncertainty or with complicated features.

REAL OPTIONS ANALYSIS

In corporate finance, real options analysis or ROA applies put option and call option valuation techniques to capital budgeting decisions.[1]

A real option is the right, but not the obligation, to undertake some business decision, typically the option to make a capital investment. For example, the opportunity to invest in the expansion of a firm’s factory is a real option. In contrast to financial options, a real option is not often tradeable—e.g. the factory owner cannot sell the right to extend his factory to another party, only he can make this decision; however, some real options can be sold, e.g., ownership of a vacant lot of land is a real option to develop that land in the future. Some real options are proprietory (owned or exercisable by a single individual or a company); others are shared (can be exercised by many parties). Therefore, a project may have a portfolio of embedded real options; some of them can be mutually exclusive.

The terminology “real option” is relatively new, whereas business operators have been making capital investment decisions for centuries. However, the description of such opportunities as real options has occurred at the same time as thinking about such decisions in new, more analytically-based, ways. As such, the terminology “real option” is closely tied to these new methods. The term “real option” was coined by Professor Stewart Myers at the MIT Sloan School of Management; this happened most likely around 1977.

The concept of real options was popularized by Michael J. Mauboussin, the chief U.S. investment strategist for Credit Suisse First Boston and an adjunct professor of finance at the Columbia School of Business. Mauboussin uses real options in part to explain the gap between how the stock market prices some businesses and the “intrinsic value” for those businesses as calculated by traditional financial analysis, specifically discounted cash flows.

Additionally, with real option analysis, uncertainty inherent in investment projects is usually accounted for by risk-adjusting probabilities (a technique known as the equivalent martingale approach). Cash flows can then be discounted at the risk-free rate. With regular DCF analysis, on the other hand, this uncertainty is accounted for by adjusting the discount rate, using e.g. the cost of capital) or the cash flows (using certainty equivalents). These methods normally do not properly account for changes in risk over a project’s lifecycle and fail to appropriately adapt the risk adjustment. More importantly, the real options approach forces decision makers to be more explicit about the assumptions underlying their projections.

Generally, the most widely used methods are: Closed form solutions, partial differential equations, and the binomial lattices. In business strategy, real options have been advanced by the construction of option space, where volatility is compared with value-to-cost, NPVq. Latest advances in real option valuation are models that incorporate fuzzy logic and option valuation in fuzzy real option valuation models.

Real options are a field of academic research, and at the present one of the leading names in academic real options is Professor Lenos Trigeorgis (University of Cyprus). An academic conference on real options is organized yearly (Annual International Conference on Real Options).