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Corporate Finance

Arguably, the role of a corporation's management is to increase the value of the firm to its shareholders while observing applicable laws and responsibilities. Corporate finance deals with the strategic financial issues associated with achieving this goal, such as how the corporation should raise and manage its capital, what investments the firm should make, what portion of profits should be returned to shareholders in the form of dividends, and whether it makes sense to merge with or acquire another firm.

Balance Sheet Approach to Valuation

If the role of management is to increase the shareholder value, then managers can make better decisions if they can predict the impact of those decisions on the firm's value. By observing the difference in the firm's equity value at different points in time, one can better evaluate the effectiveness of financial decisions. A rudimentary way of valuing the equity of a company is simply to take its balance sheet and subtract liabilities from assets to arrive at the equity value. However, this book value has little resemblance to the real value of the company. First, the assets are recorded at historical costs, which may be much greater than or much less their present market values. Second, assets such as patents, trademarks, loyal customers, and talented managers do not appear on the balance sheet but may have a significant impact on the firm's ability to generate future profits. So while the balance sheet method is simple, it is not accurate; there are better ways of accomplishing the task of valuation.

Cash vs. Profits

Another way to value the firm is to consider the future flow of cash. Since cash today is worth more than the same amount of cash tomorrow, a valuation model based on cash flow can discount the value of cash received in future years, thus providing a more accurate picture of the true impact of financial decisions.

Decisions about finances affect operations and vice versa; a company's finances and operations are interrelated. The firm's working capital flows in a cycle, beginning with cash that may be converted into equipment and raw materials. Additional cash is used to convert the raw materials into inventory, which then is converted into accounts receivable and eventually back to cash, completing the cycle. The goal is to have more cash at the end of the cycle than at the beginning.

The change in cash is different from accounting profits. A company can report consistent profits but still become insolvent. For example, if the firm extends customers increasingly longer periods of time to settle their accounts, even though the reported earnings do not change, the cash flow will decrease. As another example, take the case of a firm that produces more product than it sells, a situation that results in the accumulation of inventory. In such a situation, the inventory will appear as an asset on the balance sheet, but does not result in profit or loss. Even though the inventory was not sold, cash nonetheless was consumed in producing it.

Note also the distinction between cash and equity. Shareholders' equity is the sum of common stock at par value, additional paid-in capital, and retained earnings. Some people have been known to picture retained earnings as money sitting in a shoe box or bank account. But shareholders' equity is on the opposite side of the balance sheet from cash. In fact, retained earnings represent shareholders' claims on the assets of the firm, and do not represent cash that can be used if the cash balance gets too low. In this regard, one can say that retained earnings represent cash that already has been spent.

Shareholder equity changes due to three things:

• net income or losses
• payment of dividends
• share issuance or repurchase.

Changes in cash are reported by the cash flow statement, which organizes the sources and uses of cash into three categories: operating activities, investing activities, and financing activities.

Cash Cycle

The duration of the cash cycle is the time between the date the inventory (or raw materials) is paid for and the date the cash is collected from the sale of the inventory. A company's cash cycle is important because it affects the need for financing. The cash cycle is calculated as:

days in inventory + days in receivables - days in payables

Financing requirements will increase if either of the following occurs:

• Sales increase while the cash cycle remains fixed in duration. Increased sales increase the value of assets in the cycle.
• Sales remain flat but the cash cycle increases in duration.

While financially it makes sense to reduce the length of the cash cycle, such a reduction should not be done without considering the impact on operations. For example, one must consider the impact on customer and supplier relations as well as the impact on order fill rates.

Revenue, Expenses, and Inventory

A firm's income is calculated by subtracting its expenses from its revenue. However, not all costs are considered expenses; accounting standards and tax laws prohibit the expensing of costs incurred in the production of inventory. Rather, these costs must be allocated to inventory accounts and appear as assets on the balance sheet. Once the finished goods are drawn from inventory and sold, these costs are reported on the income statement as the cost of goods sold (COGS). If one wishes to know how much product the firm actually produced, the cost of goods produced in an accounting period is determined by adding the change in inventory to the COGS.

Assets

Assets can be classified as current assets and long-term assets. It is useful to know the number of days of certain assets and liabilities that a firm has on hand. These numbers are easily calculated from the financial statements as follows:

Accounts Receivable (A/R)

Number of days of A/R = ( accounts receivable / annual credit sales ) ( 365 ).
This also is known as the collection period.

Inventory

Number of days of inventory = ( inventory / annual COGS ) ( 365 ).
This also is known as the inventory period.

On the liabilities side:

Payables

Number of days of accounts payable = ( accounts payable / COGS ) ( 365 ), assuming that all accounts payable are for the production of goods. This also is known as the payables period.
Financial Ratios

A firm's performance can be evaluated using various financial ratios. Ratios are used to measure leverage, margins, turnover rates, return on assets, return on equity, and liquidity. Additional insight can be gained by comparing ratios among firms in the industry.
Bank Loans

Bank loans can be classified according to their durations. There are short-term loans (one year or less), long-term loans (also known as term loans), and revolving loans that allow one to borrow up to a specified credit level at any time over the duration of the loan. Some revolving loans automatically renew at maturity; these loans are said to be "evergreen."

Sources and Uses of Cash

It can be worthwhile to know where a firm's cash is originating and how it is being used. There are two sources of cash: reducing assets or increasing liabilities or equity. Similarly, a company uses cash either by increasing assets or decreasing liabilities or equity.

Sustainable Growth

A company's sustainable growth rate is calculated by multiplying the ROE by the earnings retention rate.

Firm Value, Equity Value, and Debt Value

The value of the firm is the value of its assets, or rather, the present value of the unlevered free cash flow resulting from the use of those assets. In the case of an all-equity financed firm, the equity value is equal to the firm value. When the firm has issued debt, the debt holders have a priority claim on their interest and principal, and the equity holders have a residual claim on what remains after the debt obligations are met. The sum of the value of the debt and the value of the equity then is equal to the value of the firm, ignoring the tax benefits from the interest paid on the debt. Considering taxes, the effective value of the firm will be higher since a levered firm has a tax benefit from the interest paid on the debt. If there is outstanding preferred stock, the firm value is the sum of the equity value, debt value, and preferred stock value, plus the value of the interest tax shield.

The debt holders and stock holders each have a claim on the cash flows of the firm. In a given time period, the debt holders have a claim equal to the interest payments during that period plus any principal payments that are due. The stock holders then have a claim equal to the unlevered free cash flow in that period plus the cash generated by the interest tax shield, minus the claims of the debt holders.

Capital Structure

The proportion of a firm's capital structure supplied by debt and by equity is reported as either the debt to equity ratio (D/E) or as the debt to value ratio (D/V), the latter of which is equal to the debt divided by the sum of the debt and the equity.

One can quickly convert between the D/E ratio and the D/V ratio by using the following relationships:

D / V = ( D / E ) / ( 1 + D / E )

D / E = ( D / V ) / ( 1 - D / V )

Risk Premiums

• Business risk is the risk associated with a firm's operations. It is the undiversifiable volatility in the operating earnings (EBIT). Business risk is affected by the firm's investment decisions. A measure for the business risk is the asset beta, also known the unlevered beta. In terms of the discount rate, the return on assets of a firm can be expressed as a function of the risk-free rate and the business risk premium (BRP):

r_A =   r_F +  BRP

• Financial risk is associated with the firm's capital structure. Financial risk magnifies the business risk of a firm. Financial risk is affected by the firm's financing decision.

• Total corporate risk is the sum of the business and financial risks and is measured by the equity beta, also known as the levered beta. The business risk premium (BRP) and financial risk premium (FRP) are reflected in the levered (equity) beta, and the return on levered equity can be written as:

r_E =   r_F +  BRP  +  FRP

Debt beta is a measure of the risk of a firm's defaulting on its debt. The return on debt can be written as:

r_D =   r_F +  default risk premium

Cost of Capital

The cost of capital is the rate of return that must be realized in order to satisfy investors. The cost of debt capital is the return demanded by investors in the firm's debt; this return largely is related to the interest the firm pays on its debt. In the past some managers believed that equity capital had no cost if no dividends were paid; however, equity investors incur an opportunity cost in owning the equity of the firm and they therefore demand a rate of return comparable to what they could earn by investing in securities of comparable risk.

The return required by debt holders is found by applying the CAPM:

r_D =  r_F + beta_debt ( r_M - r_F )

The required rate of return on assets (that is, on unlevered equity) can be found using the CAPM:

r_A =  r_F + beta_unlevered ( r_M - r_F )

Using the CAPM, a firm's required return on equity is calculated as:

r_E =  r_F + beta_levered ( r_M - r_F )

Under the Modigliani-Miller assumptions of constant cash flows and constant debt level, the required return on equity is:

r_E =  r_A + (1-τ)(r_A - r_D)(D / E)

where  τ  is the corporate tax rate.

The overall cost of capital is a weighted-average of the cost of its equity capital and the after-tax cost of its debt capital. The weighted average cost of capital (WACC) then is given by:

WACC  =  r_E (E / V_L) + r_D (1-τ)(D / V_L)

Assuming perpetuities for the cash flows, the weighted average cost of capital can be calculated as:

WACC  =  r_A [ 1 - τ(D / V_L)]

Neglecting taxes, the WACC would be equal to the expected return on assets because the WACC is the return on a portfolio of all the firm's equity and all of its debt, and such a portfolio essentially has claim to all of the firm's assets.

For arbitrary cash flows, and under the assumption that the debt to value ratio is held constant, the following relationship derived by James A. Miles and John R. Ezzell is applicable:

WACC  =  r_A - τ r_D (D / V_L)(1+r_A) / (1+r_D)

Under the same assumptions, the cost of equity capital can be calculated from r_A and r_D using the following relationship from Miles and Ezzell:

r_E = r_A + [ 1 - τ r_D / (1+r_D)] [ r_A - r_D ] D/E

For low values of r_D,  [ 1 - τ r_D / (1+r_D)]  is approximately equal to one, and the expression can be simplified if high precision is not required.

If one cannot assume a constant debt to value ratio, then the APV method should be used.

Estimating Beta

In order to use the CAPM to calculate the return on assets or the return on equity, one needs to estimate the asset (unlevered) beta or the equity (levered) beta of the firm. The beta that often is reported for a stock is the levered beta for the firm. When estimating a beta for a particular line of business, it is better to use the beta of an existing firm in that exact line of business (a pure play) rather than an average beta of several firms in similar lines of business that are not exactly the same.

Expressing the levered beta, unlevered beta, and debt beta in terms of the covariance of their corresponding returns with that of the market, one can derive an expression relating the three betas. This relationship between the betas is:

beta_levered =  beta_unlevered­ [ 1 + (1 - τ) D/E ]  -  beta_debt(1- τ) D/E

beta_unlevered =  [ beta_levered + beta_debt(1- τ) D/E ]  /  [ 1 + (1 - τ) D/E ]
The debt beta can be estimated using CAPM given the risk-free rate, bond yield, and market risk premium.

Unlevered Free Cash Flows

To value the operations of the firm using a discounted cash flow model, the unlevered free cash flow is used. The unlevered free cash flow represents the cash generated by the firm's operations and is the cash that is free to be paid to stock and bond holders after all other operating cash outlays have been performed.
Terminal Value

The value of the firm at the end of the last year for which unique cash flows are projected is known as the terminal value. The terminal value is important because it can represent 50% or more of the total value of the firm.
Three Discounted Cash Flow Methods for Valuing Levered Assets

APV (Adjusted Present Value) Method
The APV approach first performs the valuation under an unlevered all-equity assumption, then adjusts this value for the effect of the interest tax shield. Using this approach,

V_L = V_U + PVITS

where V_L = value if levered
V_U = value if financed 100% with equity
PVITS = present value of interest tax shield

The unlevered value is found by discounting the unlevered free cash flow at the required return on assets. The present value of the interest tax shield is found by discounting the interest tax shield savings at the required return on debt, r_D.

The APV method is useful for valuing firms with a changing capital structure since the return on assets is independent of capital structure. For example, in a leveraged buyout, the debt to equity ratio gradually declines, so the required return on equity and the weighted average cost of capital change as the lenders are repaid. However, when calculating the terminal value it may be appropriate to assume a stable capital structure, so in calculating the terminal value in a leveraged buyout situation the WACC method may be a better approach.

Flows to Equity Method
The flows to equity method sums the NPV of the cash flows to equity and to debt.

Then, V_L = E + D

WACC Method
The WACC method discounts the unlevered free cash flow at the weighted average cost of capital to arrive at the levered value of the firm.

Cash Flows to Debt and Equity

When calculating the amount of cash flowing to debt and equity holders, it is not appropriate to use the unlevered free cash flows because these cash flows do not reflect the tax savings from the interest paid. Starting with the UFCF, add back the taxes saved to obtain the total amount of cash available to suppliers of capital.

Hurdle Price

At times a firm may wish to know at what price it would have to sell its product for a particular investment to have a positive net present value. A procedure for determining this price is as follows:

• Express the operating cash flow in terms of price. There may be multiple phases such as a short start-up period, a long operating period, and a final year in which the terminal value is calculated.

• Write out the expression for the NPV using the appropriate discount rate. For the longer operating period, one can calculate an annuity factor to multiply by the operating cash flow expression. Solve the expression for the cash flow that would result in an NPV of zero.

• Since the operating cash flow was written in terms of price, the price now can be found.

Debt Valuation

While debt may be issued at a particular face value and coupon rate, the debt value changes as market interest rates change. The debt can be valued by determining the present value of the cash flows, discounting the coupon payments at the market rate of interest for debt of the same duration and rating. The final period's cash flow will include the final coupon payment and the face value of the bond.

Investment Decision

If the unlevered NPV of a project is negative, aside from potential strategic benefits, the project is destroying value, even if the levered NPV is positive. The firm always could benefit from the tax shield of debt by borrowing money and putting it to other uses such as stock buybacks.

Optimal Capital Structure

The total value of a firm is the sum of the value of its equity and the value of its debt. The optimal capital structure is the amount of debt and equity that maximizes the value of the firm.

Share Buyback

If a firm has extra cash on hand it may choose to buy back some of its outstanding shares. One interesting aspect of such transactions is that they can be based on information that the firm has that the market does not have. Therefore, a share buyback could serve as a signal that the share price has potential to rise at above average rates.

Mergers and Acquisitions

Companies may combine for direct financial reasons or for non-financial ones such as expanding a product line. The target firm usually is acquired at a premium to its market value, with the hope that synergies from the merger will exceed the price premium. Mergers and acquisitions do not always achieve their goals, as promised syngeries may fail to materialize.
Appendix

Compounding and Discounting

Compound annual growth rate (CAGR):  ( FV/C )^1/T -  1

Continuous compounding:  FV_t =  C e^{r t}

Perpetuity:  PV = C / r

Growing perpetuity:  PV = C / ( r - g )

T-year annuity (T equally spaced payments):  PV = ( C / r ) [ 1 - 1/(1+r)^T ]

T-year growing annuity:  PV = [C / (r - g)] { 1 - [(1+g) / (1+r)]^T }

HP 19BII Calculator Tip

IRR Calculation:

• Press the yellow button then "EXIT" to reset the calculator.
• Press button under "FIN"
• Pess button under "CFLO"
• Press yellow button then INPUT to clear list
• Press button under "YES"
• Enter the initial cash inflow (negative number for outflow). Press "INPUT".
• For "FLOW(1)", enter the cash flow value for the end of year 1, then press "INPUT".
• Enter the number of periods for that value, then press "INPUT".
• For "FLOW(2)", enter the next cash flow value, then press "INPUT".
• The number of times will default to the previous number. Press "INPUT" to keep, or enter a new value.
• When the cash flow entries are complete, press the button under "CALC".
• Press the button under "IRR%" to calculate the IRR of the cash flow.

Recommended Reading

Higgens, Robert C., Analysis for Financial Management

Financial Ratios

A firm's performance can be evaluated using financial ratios. Referencing these ratios to those of other firms allows a comparison to be made. The following is a listing of some useful ratios.

Leverage : Assets / Shareholder's Equity

Gross Margin =   Gross Profit / Sales.
Gross margin measures the profitability considering only variable costs and is a measure of the percentage of revenue that goes to fixed costs and profit.

Net Profit Margin =   Net Income / Sales

Total Asset Turnover =   defined as Sales / Total Assets

Return on Assets (ROA) =   Net Income / Assets
ROA is a measure of the return on money provided by both owners and creditors, and is a measure of how efficiently all resources are managed.

Return on Equity (ROE) =   defined as Net Income / Equity
where the equity value is the shareholder's equity at the end of the period in which the income was earned. ROE is a measure of the return on money provided by the firm's owners.

ROE can be calculated indirectly as:

ROE  =  ( Net Income / Total Assets ) ( Total Assets / Equity )

ROE also can be calculated using DuPont analysis :

ROE  =  (Net Income / Sales)(Sales / Total Assets)(Total Assets / Equity)

This states that ROE is determined by multiplication of three levers:

ROE  =  (net profit margin) (total asset turnover) (leverage)

These levers are readily viewed on the company's financial statements. While ROE's may be similar among firms, the levers may differ significantly.
Liquidity

The term working capital is used to describe the current items of the balance sheet. Working capital includes current assets such as cash, accounts receivable, and inventory, and current liabilities such as accounts payable and other short term liabilities. Net working capital is defined as non-cash current operating assets minus non-debt current operating liabilities. Cash, short-term debt, and current portion of long-term debt are excluded from the net working capital calculation because they are related to financing and not to operations.

Two commonly used liquidity ratios are the current ratio and the quick ratio.

Current Ratio : defined as Current Assets / Current Liabilities.
The current ratio is a measure of the firm's ability to pay off current liabilities as they become due.

Quick Ratio : defined as Quick Assets / Current Liabilities.

The quick ratio also is known as the acid test. Quick assets are defined as cash, accounts receivable, and notes receivable - essentially current assets minus inventory.

Recommended Reading

Mulford, Charles W., Comiskey, Eugene E. The Financial Numbers Game: Detecting Creative Accounting Practices

Free Cash Flow

When valuing the operations of a firm using a discounted cash flow model, the operating cash flow is needed. This operating cash flow also is called the unlevered free cash flow (UFCF). The term "free cash flow" is used because this cash is free to be paid back to the suppliers of capital.

Calculating Free Cash Flow

For a particular year, the unlevered free cash flow is calculated as follows:

1.      Start with the annual sales and subtract cash costs and depreciation to calculate the earnings before interest and taxes (EBIT). The EBIT also is referred to as the operating income and represents the pre-tax earnings without regard to how the business is financed.

2.      Calculate the earnings before interest and after tax (EBIAT) by multiplying the EBIT by one minus the tax rate. Note that the EBIAT represents the after-tax earnings of the firm as if it were financed entirely with equity capital.

3.      To arrive at the UFCF, add the depreciation expense back to the EBIAT, and subtract capital expenditures (CAPEX) that were not charged against earnings and subtract any investments in net working capital (NWC).

The free cash flow calculation in equation form:

Operating Income (EBIT)   =   Revenues   -   Cash Costs   -   Depreciation Expense

EBIAT  =  EBIT  -  Taxes,    where Taxes = (tax rate)(EBIT)

UFCF  =  EBIAT  +  Depreciation Expense  -  CAPEX  -  Increase in NWC

Capital expenditures are calculated by solving for CAPEX in the following equation:

BV of Assets at Year End	=	BV of assets at Beginning of Year
		+  CAPEX
		-  Depreciation

An additional cash adjustment may be necessary for an increase in deferred taxes that would have a positive impact on cash flow.

Recommended Reading

Simon Z. Benninga and Oded Sarig, Corporate Finance: A Valuation Approach

Terminal Value

In a discounted cash flow valuation, the cash flow is projected for each year into the future for a certain number of years, after which unique annual cash flows cannot be forecasted with reasonable accuracy. At that point, rather than attempting to forecast the varying cash flow for each individual year, one uses a single value representing the discounted value of all subsequent cash flows. This single value is referred to as the terminal value.

The terminal value can represent a large portion of the valuation. The terminal value of a piece of manufacturing equipment at the end of its useful life is its salvage value, typically less than 10% of the present value. In contrast, the terminal value associated with a business often is more than 50% of the total present value. For this reason, the terminal value calculation often is critical in performing a valuation. The terminal value can be calculated either based on the value if liquidated or based on the value of the firm as an ongoing concern.

Terminal Value if Liquidated

If the firm is to be liquidated, the liquidation value can be based on book value, salvage value, or break-up value, but liquidation value usually understates the terminal value of a healthy business. One must make assumptions about the salvage value of the assets and net working capital. The net working capital may have a certain recovery rate since it might not be readily liquidated at balance sheet values. In the pro forma projections, one often may assume that net working capital will grow at the same rate as cash flow. The terminal value if the firm is liquidated then is the sum of the discounted value of the cash flow, the recovered net working capital, and the salvage value of the long-term assets, including any tax benefits.

Terminal Value of the Ongoing Firm

For an ongoing firm, the terminal value may be determined by either using discounted cash flow (DCF) estimates or by using multiples from comparable firms.

For the DCF method, if the unlevered free cash flow is growing at a rate of  g per year for a set number of years, the terminal value can be calculated by modeling the cash flow as a T-year growing perpetuity. At the end of T years, one can assume a different growth rate (possibly zero) or liquidation. If multiples from comparable firms are used, the price/earnings ratio, market/book values, or cash flow multiples are commonly used.

The unlevered terminal value is calculated using the return on assets (r_A) as the discount rate. The levered terminal value is calculated using the weighted average cost of capital (WACC) as the discount rate.

Terminal Value of the Debt

The terminal value of debt or preferred stock is simply the projected book value of the debt or preferred stock in the year that the terminal value is being calculated.

Terminal Value of the Common Stock

The terminal value of the common stock is the total levered terminal value less the terminal value of the debt, less the terminal value of the preferred stock (adding in the amount from any warrants that are exercised at their exercise price), plus the cash gained from the exercise of any common and preferred warrants.

Recommended Reading

Aswath Damodaran, Investment Valuation: Tools and Techniques for Determining the Value of Any Asset

Debt Valuation

In the enterprise model of valuation, the firm's equity value is calculated by subtracting the value of the firm's debt from the enterprise value. Debt valuation then becomes an important component of a valuation of the firm's equity.

A company's debt is valued by calculating the payoffs that debt holders can expect to receive, taking into account the risk of default. The default risk is addressed by considering the probability of default and the amount that could be recovered in that event. For modeling purposes, one may assume that the cash flow from the recovered amount is realized at the end of the year of default.

Debt valuation may take one of the following two approaches:

1. Discount the expected cash flow at the expected bond return; or
2. Discount the scheduled bond payments at the rating-adjusted yield-to-maturity.

Debt Valuation - Method 1

Discount the expected cash flow at the expected bond return

Under this method, the value of the bond is the sum of the expected annual cash flows discounted at the expected bond return:

Value  =  the sum for each year  t of E(cash flow)_t / ( 1 + r_debt )^t

where  E(cash flow)_t =  expected cash flow in year  t.

For a one year bond:   Value  =  E(cash flow) / [1 + E(r_d)]

The expected bond return is the risk-adjusted discount rate, r_debt.

The expected cash flow is the cash flow considering the probability of default:

E(cash flow)  =  π ( 1 + C ) F  +  ( 1 - π ) λ F

	where	π  =  probability of no default
		λ  =  recovery rate in case of default, (percentage of face value)
		C  =  annual coupon rate of the bond
		F  =  face value of the bond

r_debt can be calculated using the CAPM:

r_debt =  r_f +  β_debtΠ_S&P500

where

	ΠS&P500 =  risk premium for the market portfolio
	βdebt =  covariance between rdebt and the market return;
	rf =   yield to maturity on a risk-free bond having the same maturity.

If  β_debt is not known, it can be found using ordinary least squares regression.

If  π  =  1 (no default risk), then  r_debt =  yield to maturity.

The difference in  r_debt and  YTM  reflects the default risk.

Debt Valuation - Method 2

Discount the scheduled bond payments at the rating-adjusted yield-to-maturity

For this method, estimate the rating-adjusted yield-to-maturity (RAYTM) by averaging the market yield-to-maturities (YTM) of bonds in the same group. The promised cash flows then are discounted at this rate that already has factored in the default risk.

Markov Chain Representation

A firm's debt rating can change over time, and the value of future cash flows should take into account the possibility of one or more rating changes. In this regard, bond valuation can be modeled as a Markov Chain problem in which a transition matrix is constructed for the probabilities of the firm's debt moving from one rating to another. For example, if there are five possible ratings: A, B, C, D, E, and F; and π_xy represents the probability of moving from state x to state y, then the transition matrix would look like the following:

πAA	πAB	πAC	πAD	πAE
πBA	πBB	πBC	πBD	πBE
πCA	πCB	πCC	πCD	πCE
πDA	πDB	πDC	πDD	πDE
πEA	πEB	πEC	πED	πEE

For multiple periods, the transition matrices for each period must be multiplied in order to calculate the multi-period probabilities. This multiplication easily can be performed by spreadsheet software.

Recommended Reading

Simon Z. Benninga and Oded Sarig, Corporate Finance : A Valuation Approach

Mergers and Acquisitions

A corporate merger is the combination of the assets and liabilities of two firms to form a single business entity. In everyday language, the term acquisition tends to be used when a larger firm absorbs a smaller firm, and merger tends to be used when the combination is portrayed to be between equals. In a merger of firms that are approximate equals, there often is an exchange of stock in which one firm issues new shares to the shareholders of the other firm at a certain ratio. For the sake of this discussion, the firm whose shares continue to exist (possibly under a different company name) will be referred to as the acquiring firm and the firm whose shares are being replaced by the acquiring firm will be referred to as the target firm.

Excluding any synergies resulting from the merger, the total post-merger value of the two firms is equal to the pre-merger value. However, the post-merger value of each individual firm likely will be different from the pre-merger value because the exchange ratio of the shares probably will not exactly reflect the firms' values with respect to one another. The exchange ratio is skewed because the target firm's shareholders are paid a premium for their shares.

Synergy takes the form of revenue enhancement and cost savings. When two companies in the same industry merge, such as two banks, combined revenue tends to decline to the extent that the businesses overlap in the same market and some customers become alienated. For the merger to benefit shareholders, there should be cost saving opportunities to offset the revenue decline; the synergies resulting from the merger must be more than the initial lost value.

To calculate the minimum value of synergies required so that the acquiring firm's shareholders do not lose value, an equation can be written to set the post-merger share price equal to the pre-merger share price of the acquiring firm as follows:

(pre-merger value of both firms  +  synergies)	=   pre-merger stock price
post-merger number of shares

The above equation then can be solved for the value of the minimum required synergies.

The success of a merger is measured by whether the value of the acquiring firm is enhanced by it. The practical aspects of mergers often prevent the forecasted benefits from being fully realized and the expected synergy may fall short of expectations.

Recommended Reading

Robert F. Bruner,  Applied Mergers and Acquisitions (Wiley Finance)

Investment Management

Investment management is about attaining investment objectives under specified constraints; for example, achieving the best possible return for a given level of risk. To meet these objectives, the investor may buy equity in an asset such a stock, a fund, or real estate, or buy debt issued by governments and corporations. By effectively managing such investments the investment manager can achieve a higher return for a specified acceptable level of risk. There are many tools for reaching this goal.

Expected Return and Portfolio Variance

The two basic metrics for an investment portfolio are the return and the variance.

In the case of an individual dividend-paying stock, the return is given by:

R_i =  [(P₁ + D₁) / P₀]  - 1,

where D₁ is the dividend paid at time t = 1.

The future return of a stock or a portfolio is not known with certainty; there are different probabilites for different return scenarios, one of which actually will unfold.

Given n possible return scenarios, each with its own probability p_i, the expected return is:

E(R) = Σ_i=1,n p_i R_i

The variance of such a stock or portfolio is given by:

σ² = Σ_i=1,n p_i [R_i - E(R)]²

Portfolios

A portfolio has certain advantages over a single security. The return of one security may tend to move in the same direction as the return of another security, but in the opposite direction of the return of a third security. Because of these tendencies, when securities are grouped into a portfolio, for a given expected return the variance of that return can be reduced. The joint tendencies between the returns can be measured by covariances.

The covariance in two securities' returns is given by:

Cov(R₁, R₂) = σ₁₂ = σ₁σ₂ ρ₁₂

The correlation coefficient between security i and the market is given by:

ρ_im = σ_im / σ_i σ_m

For two securities,

σ²_p = Σ_i=1,n Σ_j=1,n x_ix_j σ_ij

σ²_p = x²₁ σ²₁ + x²₂ σ²₂ + 2 x₁ x₂ σ₁₂

= x²₁ σ²₁ + x²₂ σ²₂ + 2 x₁ x₂ σ₁σ₂ ρ₁₂

where x₂ = 1 - x₁

Note that if T-bills that earn the risk-free rate are included, σ for R_F = 0.

Given two securities, many different portfolios can be constructed by varying the weighting of each security in the portfolio. To find the minimum variance portfolio,

set  dσ₁ / dx₁ = 0

=> x₁ = (σ²₂ - σ₁σ₂ ρ₁₂) / (σ²₁ + σ²₂ - 2 σ₁σ₂ ρ₁₂)

For an equally weighted portfolio with all standard deviations equal and all covariances equal to zero:

Var(R_p) = (1/N²) Σ_i=1,n Var(R_i)

= (1/N) Var(R_i)

= (1/N) σ²_i and

σ_p = (1/N^1/2) σ_i

Risk Adjusted Return

Different investors have different aversions to risk. When managing a portfolio for a particular investor, the goal is to maximize the portfolio return for the level of risk that the investor is willing to take. The following model can be used:

Maximize Z = E(R_p) - A Var(R_p)

where A = investor's aversion to risk as measured by the variance of the portfolio return.

To maximize the function assuming the investor's assets are only in the market portfolio and the riskfree asset,

first let w_m = the fraction of assets in the market portfolio. Then

E(R_p) = r_F + w_m (R_m - r_F)

and

Var(R_p) = w²_m σ²_m.

Then

Z = r_F + w_m [E(R_m) - r_F] - 0.5 A w²_m σ²_m

and

dZ/dw_m = E(R_m) - r_F - A w_m σ²_m = 0

Solving for A,

A = [E(R_m) - r_F] / ( w_m σ²_m)

Beta

The risk of an individual security in a well diversified portfolio can be measured by its beta. Such risk is nondiversifiable.

Beta of an individual security with respect to the market is:

β_im = σ_im / σ²_m = Cov(R_i, R_SP500) / Var(R_SP500)

Beta of a risk-free asset with respect to the market = 0.

Betas determined using historical data are subject to estimation error. Merrill Lynch and some other firms adjust this value back towards the mean beta of the market (=1) or industry using

β_adjusted = w β_historical + (1-w) β_"true"

The lower the confidence in β_historical, the lower should be the value chosen for w.

Beta of a portfolio:

β_pm = Σ x_i β_im , where x_i is the weight.

One is willing to accept a lower return on a security or a portfolio having a negative beta since it can reduce the portfolio risk as part of a larger portfolio.

Efficient portfolios lie on the capital market line (CML). This CML is not a part of the CAPM. For this line to be used, there must be perfect correlation between the portfolio in question and the market portfolio. This implies that the line is only for those portfolios that are a combination of the tangential portfolio (usually the market portfolio) and the risk-free rate.

CML: E(R_p) = R_F + [ E(R_m) - R_F ] σ_p / σ_m

If borrowing is not permitted, the rational risk-averse investor will choose a portfolio along the capital market line up to the efficient frontier, and then follow the efficient frontier for levels of higher risk and return.

The variance and expected return of the market portfolio can be obtained by combining any two portfolios that lie on the efficient frontier and solving for the weights in the following expression:

E(R_m) = w₁ E(R₁) + (1-w₁) E(R₂)

The covariance between any two portfolios on the efficient frontier can be found by finding the weights needed to emulate the market portfolio and then solving for σ₁₂ in the following equation:

σ²_p = w²₁ σ²₁ + w²₂ σ²₂ + 2 w₁ w₂ σ₁₂

CAPM

The Sharpe-Lintner version of the capital asset pricing model implies that as a result of all investors holding the market portfolio, there is a linear relation between the expected return on a security and its β.

The following is the security market line - any security's expected return will lie on this line. This line applies to all securities, not just efficient portfolios.

E(R_i) = R_F + [ E(R_m) - R_F ] β_im Sharpe-Lintner

E(R_i) = R_z + [ E(R_m) - R_z ] β_im Black

Expected return of a portfolio using CAPM:

E(R_p) = R_F + [ E(R_m) - R_F ] β_pm

If the assumption of equal borrowing and lending rates is relaxed, investors no longer are required to hold the market portfolio; instead, they can hold a range of portfolios along the efficient frontier between the point of tangency of the lending line and the point of tangency of the borrowing line.

CAPM requires the measure of two unknown quantities - market risk premium and beta. However, attempts to estimate expected returns by using historical stock return data have resulted in std errors about double those of CAPM, because for CAPM the better precision in the estimate of the market risk premium more than offsets the additional estimation error in beta.

There have been many difficulties in testing CAPM. Roll argued that the CAPM must always hold for ex post data if the proxy chosen for the market is efficient. He also argued that it is impossible to measure the true market, so the CAPM cannot be tested. However, in 1982 Stambaugh found that adding other risky assets such as corporate bonds, real estate, and consumer durables to the market portfolio did not materially affect the tests.

Single Factor Model (Market Model)

R_it = a_i + β_iR_mt + e_it t = 1, ..., T

where e_it is the distance from the regression line at time t. The mean value of e_it = 0 and the covariance between R_m and e_i = 0. This is a regression model that characterizes the risk of a security over time by measuring its beta over a time interval. β_iis different from the β_im used in the CAPM in that β_im is more of a present-day beta rather than one taken over time. In the traditional approach of testing the CAPM, in the first step one uses this model to measure the beta of all securities (or portfolios). In the second step one estimates the CAPM itself by regressing the security returns on the estimated betas. When testing CAPM in this manner, one must question the validity of tests using ex post data to test the ex ante CAPM. Also, there is measurement error in individual security betas. Using portfolios instead in the first-pass regression helps.

Variance using the single-factor model:

Var(R_i) = β²_i Var(R_m) + Var(e_i)

where R_i, R_m, and e_i are random variables. The variance of the mean return a_i is zero by definition, so this term falls out. In a well-diversified portfolio, Var(e_i) = 0. In this equation, β²_i Var(R_m) is the variance explained by the market. The percent of variance explained by the market then is given by

β²_i Var(R_m) / σ²_i = R²

Note that (1-R²) is the idiosyncratic variance.

These expressions apply to portfolios as well by replacing i with p.

For two portfolios or securities in which their e_i's are uncorrelated, the covariance between them is given by:

σ_ij = β_i β_j σ²_m .

This is derived by finding the covariance between:

R_it = a_i + β_iR_mt + e_it and R_jt = a_i + β_jR_mt + e_jt

Cross Section of Common Stock Returns

Fama and French used a multi-factor model using additional risk factors related to size, price/book, etc.

They concluded that three "risk" factors were sufficient.

Gabriel Hawawini and Donald Keim's paper reports that stock returns depended size, E/P, CF/P, P/B, and prior returns. However, these factors were not due to risk.

The premia related to size and P/B are mainly due to the January effect. It is unlikely that the risk is higher in January. The size & P/B premia are uncorrelated across international markets. This is inconsistent with the notion of well-integrated international markets, in which similar risks should result in similar returns.

Market-Neutral Strategies

Market-neutral strategies balance the market risk by going long on some securities and short on others. Some people propose using the T-bill rate as a benchmark against which to compare the market return of a such a strategy. One can argue that even though the market-neutral strategy is risky, since it has zero beta it does not contribute to the risk of the market portfolio and therefore should not command a premium over the risk-free rate. On the other hand, if the expected returns on the long side are higher than those on the short, the benchmark return should exceed the risk-free rate.
Trading Costs

An important factor in the performance of high-turnover portfolios is the amount of the trading costs, including explicit costs such as commissions, fees, and taxes, the market maker spread, the impact of trading on market price, and the opportunity cost incurred during the delay between the time the decision is made and the time the trade is executed.

Trading costs can be reduced through passive fund management and electronic trading.

Long-Term Investing

The conventional wisdom is that over the long run, stock will generate returns superior to those of bonds. But while the variance of the geometric means of the returns declines as the time horizon increases, the variance of the terminal wealth increases. If a put option were purchased to insure a certain terminal wealth, the cost of that option would increase as the time horizon increases. To the extent that option prices are a measure of risk, the risk of stock investments then increases as the time horizon lengthens.

The optimal asset allocation is a function of the present wealth, target future wealth, risk tolerance, and time horizon. Long-term returns are difficult to analyze statistically because as the historical time horizon increases, the number of possible independent samples of returns decreases. In 1991, Butler and Domian illustrated a procedure that attempts to overcome this difficulty by first listing the monthly returns for the S&P 500 and the long-term bonds over a long historical time horizon. By randomly selecting data, returns over various long-term holding periods can be emulated by multiplying the appropriate number of random samples. An almost limitless number of samples for each holding period can be generated using this method. Performing such an analysis with data taken from the 792 months from 1926-1991 indicates that over a 10-year time period, there is an 11% chance that stocks will underperform bonds; over a 20-year time period this probability reduces to 5%.

Defined-Benefit Pension Plans

In a defined-benefit plan, the plan sponsor (usually an employer) guarantees a level of future benefits to the plan participants, taking responsibility for any shortfall in the investment performance of the plan. FASB 87 requires that any unfunded liability in the present value of the benefits appear on the balance sheet of the employer. One alternative for the plan sponsor is to place the present value of the plan liability into government bonds of the same duration as the liability, in which case there is no chance of shortfall and the liability is fully immunized. Furthermore, because pension plans are not taxed, the incentive to hold equity in order to take advantage of lower taxes on capital gains is diminished. For a given level of risk, tax implications increase the return most for investments such as bonds, which have a large spread between pre-tax and after-tax return. An alternative to bonds is for the pension fund to place its money into riskier assets such as common stocks. Under this latter alternative, there exists both the chance of a shortfall and the chance of a surplus. However, FASB 87 does not permit a surplus to be reported as an asset on the sponsor's balance sheet, and the surplus often gets allocated to the plan participants. Nonetheless, for many reasons it is common for firms to hold equity in their pension funds. The Pension Benefit Guarantee Corporation (a federal agency) guarantees the benefits, and the sponsor's premiums are independent of the risk level of the pension fund's investments. For employers in financial distress, the pension guarantee from PBGC effectively is a put option. Given that put options increase in value as risk increases, there is an incentive for some firms to invest the pension fund in risky assets.

In defined-benefit plans there exists the opportunity for tax arbitrage. The plan sponsor can issue debt in order to buy equity in the pension plan. The pension plan then can invest the funds in bonds. Because of the tax status of the pension fund, the taxes on the pension plan's bond interest will be deferred, and the sponsor will enjoy the interest tax shield from its debt issuance. The sponsor then realizes an arbitrage profit equal to the interest rate multiplied by the corporate tax rate, with no increase in the firm's overall risk.

Arbitrage Pricing Theory

In 1976, Steve Ross presented the arbitrage pricing theory (APT) as an alternative to the CAPM that requires fewer assumptions. The APT is an equilibrium theory, which differs from a factor model in that it specifies relationships between expected returns across securities and attributes that influence those securities. A factor model allows the first term in the model, the expected return, to differ across securities and therefore can represent either and efficient or an inefficient market. Ross assumed that returns have the first term in common, and the other terms depend on several different systematic factors, as opposed to the single market risk premium factor of the CAPM. The model takes the form:

R_pt = E(R_p) + β_p1I_1t + β_p2I_2t + ... + β_pKI_Kt + e_pt

where I_i = value of the i^th factor, β_pi = sensitivity of the return to the i^th factor, k = number of factors, and e_pt equals the idiosyncratic variation in the return. Assuming an efficient market in equilibrium, the first term to the right of the equal sign is the same for all securities and is approximately equal to the risk-free rate. Examples of factors that could be included in the model are monthly industrial production, changes in expected inflation, unexpected inflation, unexpected changes in the risk premium, and unexpected shifts in the term structure of interest rates. Such variables likely affect most or all stocks.

Market-Neutral Strategies Revisited

Given that "alpha" is the return above the market return, by constructing a portfolio long on positive alpha stocks and short on negative alpha stocks, one can cancel the effect of the market. This market-neutral strategy sometimes is referred to as a "double alpha, no beta" strategy. Because such a strategy is uncorrelated with the market, the volatility depends on non-market factors. If the market-neutral portfolio is well-diversified across many types of industries, the volatility can be low. If the portfolio is concentrated in a smaller number of stocks and industries, the volatility can be high. Furthermore, if the portfolio is not balanced among stocks of different size or value/growth measures, there could be higher volatility as a result of these non-market risk factors.

Mutual Funds

Mutual funds charge fees to their investors. Transaction fees called loads sometimes are charged for fund purchases or redemptions. Such fees are deducted directly from the investor's account and represent a charge for the broker's service of providing information and fund selection advice. Operating expenses are fees that are deducted from the fund earnings before distribution to investors, and typically average slightly more than 1% per year. Two components of operating expenses are management fees and 12b-1 fees. The 12b-1 fees represent a reimbursement for the fund's marketing expenses.

Style Analysis

Mutual funds can be characterized according to investment style, such as value or growth. However, the actual fund composition may not correspond closely to its stated investment style, and reports of portfolio holdings may not be very representative since they are only snapshots taken at one point in time. This limitation makes the public's view of the holdings subject to distortions such as "window-dressing," in which the portfolio manager buys stocks that have performed well so that investors will see those stocks in the portfolio holdings (cost basis is not reported) and perceive the manager to be capable of selecting the top performers. Style analysis is a method of characterizing the true style of a fund based on its behavior, not on its stated objectives and holdings.

Style analysis is performed by first selecting a set of indices that correspond to particular styles, such as small-cap value, small-cap growth, large-cap value, large-cap growth, and cash. Using a weighted combination of these indices, one can construct a passive benchmark portfolio that tracks the return of the portfolio being analyzed as closely as possible. Assume that there are five indices available with which to compose the benchmark. The following steps are used to analyze the style:

1. Define the benchmark return for period t to be:

R_Benchmark,t = w₁R_1,t + w₂R_2,t + w₃R_3,t + w₄R_4,t + w₅R_5,t

1. Define the tracking error to be:

e_t = R_Fund,t - R_Benchmark,t

1. Solve for the weights by minimizing the standard deviation of the mean tracking error over the entire time period being analyzed under the constraint that the weights sum to one and are each greater than or equal to zero (unless net short positions are permitted in the fund). The standard deviation of the tracking error is given by:

σ(e) = [ ( 1 / T-1 ) Σ_t=1,T (e_t - e_mean)² ]^1/2

Evaluating Fund Performance

When the popular press publishes mutual fund performance rankings, it usually does not consider the risk that the portfolio manager took to achieve that return. Such rankings do not necessarily reflect the skill of the manager. To adjust for risk, one should consider the ratio of excess returns to risk, or consider risk-adjusted differential returns. For the risk, one can use standard deviations or betas.

The "Sharpe Measure": [ E(R_p) - E(R_F) ] / σ_p

The "Treynor Measure": [ E(R_p) - E(R_F) ] / β_p

Std dev. differential measure: [ E(R_p) - E(R_F) ] - [ E(R_B) - E(R_F) ] σ_p / σ_B

"Jensen Measure": [ E(R_p) - E(R_F) ] - [ E(R_B) - E(R_F) ] β_p

The Jensen measure is perhaps the most widely used measure of fund performance.

In the above measures, R_p is the return of the portfolio under test, R_B is the return of a passive benchmark portfolio, R_F is the risk-free rate, and E(R) represents the mean historical returns.

In determining which measure to use, one should consider the purpose of the measurement.

For portfolios that represent a large portion of its investors' assets, a method that uses standard deviation should be used; the Sharpe Measure and the other Std. deviation differential measure are more appropriate.

For ranking fund performance, the ratio of excess return to risk should be measured; the Sharpe Measure or the Treynor Measure are more appropriate.

Market Efficiency

There is some evidence of some autocorrelation in stock prices. Small amounts of both positive autocorrelation, in which stock returns tend to move in the direction of the previous period, and negative autocorrelation, in which returns tend to move in a direction opposite to that of the previous period, have been observed. In situations of positive autocorrelation, momentum investing strategies should be employed, and in situations of negative autocorrelation, contrarian strategies should be used. However, for shorter term trading, any advantage from these techniques is neutralized by trading costs, and for longer terms there is not yet enough data to confirm or deny any net advantage.

Timing the Market

Some investors have attempted to time the market to increase their returns, increasing their stake in equities when they predict an up market and decreasing it when they predict a down market.

QuickMBA's market timing page covers this topic in more detail.

Bonds

Coupon bearing notes and bonds typically make fixed interest payments two times per year. Zero coupon bonds are sold at a discount and pay off their face values at maturity. Zero coupon treasury securities are issued by commercial institutions who separate the interest and principal payments. These zero coupon bonds are known as CAT's, TIGR's, and STRIP's.

Bond prices often are quoted in the format x:y, where x is the integer dollar amount and y is the fractional amount in 32^nd's of a dollar.

The spot rate is the rate that would correspond to a single cash flow at maturity for a bond purchased today, as is the case with a zero coupon bond. A notation used for spot rates is r_n, where n is the number of periods (e.g. years) into the future when a loan made today is to mature. The forward rate is the rate at which a future loan is made today. A notation used for forward rates is f_m,n, where m is the number of periods from the present when the loan is to commence, and n is the number of periods into the future when the loan is to end. Forward rates can be expressed in terms of spot rates:

1 + f_m,n = ( 1 + r_n ) / ( 1 + r_m )

The ask price of a U.S. Treasury bill is calculated from the "asked" rate (not asked yield) as follows:

Ask Price = 10,000 [ 1 - asked rate ( N / 360 ) ]

where N = the number of days until maturity.

The implied rate (spot rate) is ( 10,000 / Ask Price - 1 ). This implied rate does not represent an annualized basis. The annualized rate is found by raising the implied rate to the 365/N power:

Annualized Rate = ( Implied Rate )^{365 / N}

The bond equivalent yield is the yield to maturity y that satisfies the following equation:

P = Σ_n=1,N C_n / ( 1 + y/2 )ⁿ

where P = price, C_n = cash flow at the end of each period, N = number of periods.

For a zero coupon bond there is only one cash flow at maturity.

The value of a coupon bond can be modeled as a portfolio of zero-coupon bonds having face values and maturity dates that correspond to the coupon payments and dates. Summing the prices of the zero-coupon bonds then would give the value of the coupon bond, and any difference would represent an arbitrage opportunity.

Forward rates can be calculated using the prices and returns of bills, notes, or bonds provided they cover the proper time periods. For example, given the six month spot rate r_0.5, one can calculate the one year spot rate r_1.0 by using the data for a one year note the following equation:

Price = coupon1 / (1+r_0.5) + (coupon2 + face value) / (1+r_1.0)

Once the spot rates are known, the forward rate can be calculated as already illustrated.

The spot rate is not quoted on an annualized basis. To annualize it:

Annualized Yield = (Spot Rate)^x/y

where x is the number of periods in one year, and y is the number of periods included in the spot rate.

The duration of a bond often is thought of in terms of time until maturity. However, in addition to the payoff of the face value at maturity, there are the coupon payment cash flows that influence effective duration. Two bond with equal yield-to-maturities and maturity dates will have different effective durations if their coupon rates are different. Frederick Macaulay suggested the following method of determining duration:

Effective Duration = Σ_t=1,T t { [ C_t / ( 1+y/2 )^t ]  /  [Σ_t=1,T C_t / ( 1+y/2 )^t] }

where T = life of the bond in semiannual periods,

C_t = cash flow at end of t^th semiannual period,

y = yield to maturity, expressed as a bond-equivalent yield.

A zero-coupon bond has no coupon payments and therefore its effective duration always is equal to the time until maturity and does not change as yield-to-maturity changes. Duration essentially measures the sensitivity of a bond's price to movements in interest rates. By this definition, duration is defined as

D » ( D P / P ) / [ D ( 1 + r ) / ( 1 + r ) ]

If one plots the price of a non-callable bond as a function of its yield, the plot will be concave up (convex down) rather than linear. This curvature is called convexity, and in this case, positive convexity. Convexity is due to the fact that effective duration increases as interest rates decrease.

Because of the effectively shorter duration, the coupon bond yield curve will be below that of the zero coupon bond when forward rates are rising with time, and above it when they are dropping. Zero coupon rates often are more useful for capital budgeting purposes.

Research has found that diversified portfolios of junk bonds have lower variance than those of high-grade bonds. There are several contributing factors to this initially surprising result. First, while individual junk bonds are risky, much of this risk can be diversified in a portfolio. Second, because of the higher coupon rate, junk bonds effectively have a shorter duration than do higher grade bonds and therefore a lower sensitivity to interest rate movements. Third, junk bonds are more likely to be called than are higher-grade bonds, since there is a strong incentive to refinance at lower rates if the issuer's credit improves. This characteristic reduces the effective duration resulting in less volatility.

Recommended Reading

David F. Swensen, Pioneering Portfolio Management : An Unconventional Approach to Institutional Investment

Trading Costs

The cost associated with trading securities can have a non-negligible impact on portfolio return. Trading costs include the following:

• Explicit costs - commissions, fees, and taxes.
• Market maker spread - difference between the bid and ask prices that the specialist sets for a stock; the specialist keeps the difference as compensation for providing immediacy. For less liquid stocks, the specialist has greater exposure to adverse price movements and likely will make the spread larger.
• Market impact - results when high volume trades influence the market price. Market impact can be broken into two components - a temporary one and a permanent one. The temporary component is due to the need for liquidity to fill the order. The permanent impact is due to the change in the market's perception of the security as a result of the block trade.
• Opportunity cost - the effective cost of price movements that occur before the trade executes.

NYSE specialists sometimes may appear to have a monopoly on trading their respective securities, creating a larger than necessary spread between bid and ask. However, there is more competition than is initially obvious. First, there is competition for the specialist positions, providing the specialist incentive to price fairly. Furthermore, there are other specialists on the floor who may be willing to trade within the spread if it is too wide.

The total trading cost of a buy transaction is calculated by taking the percentage increase of the average purchase price as compared to the price when the buy decision was made, and adding the commissions, fees, and taxes as a percentage of the price when the buy decision was made.

Active portfolio managers attempt to outperform passive benchmarks, but trading costs reduce any realized advantages. Typical trading commissions run 0.20% of the transaction amount, and the typical cost due to bid-ask spread and market impact is 0.55%. The total cost of a trade then is 0.75% of the trade amount. If a fund has a portfolio turnover rate of 80%, and for every sell transaction the stock is replaced via a buy transaction, a total of 160% of the portfolio value will be transacted each year. For trading costs of 0.75% per transaction, the annual trading costs amount to (1.6)(0.75%) = 1.20% of the portfolio value. If one adds a 0.3% management fee to this amount, the total becomes 1.50%.

Reducing Trading Costs: Passively Traded Funds

Passive portfolios have lower transaction costs and overall trading costs. The transaction cost is typically 0.25% of the transaction value, since a passive portfolio does not have to trade as quickly and can be more patient with each transaction. A typical turnover rate for a passive portfolio is about 4% per year, and assuming replacement 8% of the portfolio value will be transacted each year for annual trading costs of only (0.08)(0.25%) = 0.02% of the portfolio value. Passive portfolios have lower management fees, for example, 0.10%, so the total of trading costs and management fees is only 0.12%, compared to 1.50% for a typical actively managed fund.

Passively managed funds that track an index often have returns less than that of the index because of trading costs, especially for small-cap indices in which the securities are less liquid. These trading costs can be reduced if the weights of the securities in the fund are allowed to deviate somewhat from the index, since both trading volume and the need for immediacy are reduced. The correlation with the index still can remain quite high under the relaxed weights.

In 1982 Dimensional Fund Advisors (DFA) introduced a passive small-cap "9-10" fund composed of the lower two deciles of NYSE market capitalization. The fund sacrificed tracking accuracy by allowing the weights to deviate in order to minimize trading costs. The result was higher performance than other small-cap funds. The 9-10 fund even outperformed the stocks in the lower two market capitalization deciles of the NYSE, partly due to the following strategies:

1.      The 8^th decile is treated as a hold range, not a sell range,

2.      The DFA waits a minimum of one year before buying IPO's,

3.      The fund does not buy stocks selling for less than $2 or having less than $10 million in market capitalization,

4.      The fund does not buy NASDAQ stocks having fewer than four market makers,

5.      The fund does not buy bankrupt stocks, and

6.      The fund is passive, not rigidly indexed.

Note that using the 8^th decile as a hold range effectively increases the average market cap of the portfolio and increases returns in periods in which large caps outperform small caps, such as in the 1980's.

Reducing Trading Costs: Electronic Trading

Electronic crossing networks have lower trading costs than do exchanges because of lower commissions, no bid-ask spread, and elimination of market impact. By matching the natural buyers and sellers of a security at some predetermined price, for example, the NYSE closing price, electronic crossing networks eliminate the need for a market maker to provide liquidity. However, crossing networks require buyers and sellers to participate in order for there to be liquidity. Furthermore, there are the disadvantages of potentially limited liquidity and no inherent price discovery mechanism.

Electronic communications networks are computerized bulletin boards for matching trades. Because the traders can remain anonymous, price impact is diminished.

Another electronic trading mechanism is the single-price call auction in which buyers and sellers simply place limit orders. The market clearing price is set at the intersection of the supply and demand curves.

Recommended Reading

Gregory Baer and Gary Gensler, The Great Mutual Fund Trap : An Investment Recovery Plan

Market Timing

Some investment managers and individual investors attempt to improve their performance by timing the market and adjusting their portfolio according to predictions about the market or specific sectors. Examples of market timing include switching among sectors, switching among different countries' securities, switching between stocks and bonds, or switching between stocks and risk-free treasury bills. The effect of correctly timing the market would be to increase the portfolio beta in up markets and decrease it in down markets. For the purpose of this discussion, an up market is one in which the market return exceeds the risk-free rate, and a down market is one in which the market return is less than the risk-free rate.

Proponents of market timing may argue that the market timer does not have to be correct 100% of the time in order to benefit from timing. Some even may argue that for market timing to be worthwhile, the timer simply must be right more often than wrong.

Opponents to market timing may argue that the financial markets are fairly efficient, and therefore there is little to be gained from attempting to time them. Furthermore, there are transaction costs and tax implications associated with buying and selling stocks, both of which create an inherent disadvantage for the market timer. Finally, opponents of market timing may argue that no market timer can be correct 100% of the time, and the lost opportunity caused by missing a bull market or the significant losses of getting caught in a bear market require much more than 50% of a market timer's predictions to be correct in order to benefit from the strategy.

One can test this argument by creating a model to determine how accurate the market timer's predictive ability must be in order to benefit from the strategy. William Sharpe provided such a framework for evaluating the potential of market timing in his 1975 publication "Likely Gains from Market Timing". The potential gains from market timing can be modeled by considering an investor who switches between 100% equity and 100% cash equivalents invested at the risk-free rate. The goal is to determine what the probability of correctly predicting up or down markets must be in order to make timing worthwhile. Define:

π_up =  probability of an up market

π_down =  probability of a down market

p_correct =  probability of correctly predicting an up or down market

where an up market is defined as the situation in which stock returns exceed the risk-free rate in the period under consideration. Historically,

π_up =  67%   and   π_down =  33%

One then can draw a tree that leads to four outcomes:

1. Up market, predicted up. [ probability  =  π_upp_correct ]
2. Up market, predicted down. [ probability  =  π_up(1 - p_correct) ]
3. Down market, predicted up. [ probability  =  π_down(1 - p_correct) ]
4. Down market, predicted down. [ probability  =  π_downp_correct ]

Using historical market data from 1934 to 1972 and analyzing returns assuming various levels of predictive ability, the result is that in order to perform better than simply remaining fully invested in stocks, one must be able to predict the market with at least 83% accuracy, a predictive ability that would be extremely difficult for even the best market timer to sustain.

However, this comparison has not considered risk - staying fully invested at all times results in more portfolio variance. The market timer is not invested in stocks 100% of the time, and therefore experiences less variability in portfolio return. To make a fair comparison, one must adjust for the differences in risk. If one compares the market timer's return to that of a portfolio of stocks and cash weighted to have the same standard deviation as the market timer's portfolio, the result is that the market timer must be correct 74% of the time in order to perform better than the passive portfolio of the same risk. So even after adjusting for risk, a significant predictive ability still is required.

One can evaluate the success or failure of a portfolio manager's market timing strategy by performing the following regression:

R_pt - R_Ft =    a  +  b(R_mt - R_Ft)  +  c(R_mt - R_Ft)² +  e_pt

where R_p is the portfolio return, R_F is the risk-free rate, R_m is the market return. If the value of c is greater than zero, than some ability to time the market has been demonstrated. An alternative method is to perform the following regression:

R_pt - R_Ft =    a  +  b(R_mt - R_Ft)  +  c[(R_mt - R_Ft)D_t]  +  u_pt

In this regression, D_t = 1  if  R_mt > R_Ft,  0 otherwise. If the value of c is greater than zero, than some ability to time the market has been demonstrated. Using this equation, b is the beta in down markets, b+c is the beta in up markets, and c is the difference in the up market and down market betas.

Recommended Reading

Roger Lowenstein, When Genius Failed : The Rise and Fall of Long-Term Capital Management

Black-Scholes Option Pricing Formula

In their 1973 paper, The Pricing of Options and Corporate Liabilities, Fischer Black and Myron Scholes published an option valuation formula that today is known as the Black-Scholes model. It has become the standard method of pricing options.

The Black-Scholes formula calculates the price of a call option to be:

C  =  S N(d₁)  -  X e^-rT N(d₂)

where

	C  =  price of the call option
	S  =  price of the underlying stock
	X  =  option exercise price
	r  =  risk-free interest rate
	T  =  current time until expiration
	N()  =  area under the normal curve
	d1 =  [ ln(S/X) + (r + σ2/2) T ] / σ T1/2
	d2 =  d1 - σ T1/2

Put-call parity requires that:

P  =  C  -  S + Xe^-rT

Then the price of a put option is:

P  =  Xe^-rT N(-d₂)  -  S N(-d₁)

Assumptions

The Black-Scholes model assumes that the option can be exercised only at expiration. It requires that both the risk-free rate and the volatility of the underlying stock price remain constant over the period of analysis. The model also assumes that the underlying stock does not pay dividends; adjustments can be made to correct for such distributions. For example, the present value of estimated dividends can be deducted from the stock price in the model.

Warrant Pricing

Warrants are call options issued by a corporation. They tend to have longer durations than do exchange-traded call options. Warrants can be valued by the Black-Scholes model, but some modifications must be made to the parameters.

When warrants are exercised, the company typically issues new shares at the exercise price to fill the order. The resulting increase in shares outstanding dilutes the share value. If there were  n shares outstanding, and  m warrants are exercised,  α represents the percentage of the value of the firm that is represented by the warrants, where

α  =  m / ( m + n )

When using the Black-Scholes model to value the warrants, it is worthwhile to use total amounts instead of per share amounts in order to better account for the dilution. The current share price  S becomes the enterprise value (less debt) to be acquired by the warrant holders. The exercise price is the total warrant exercise amount, adjusted for the fact that in paying cash to the firm to exercise the warrants, the warrant holders in effect are paying a portion of the cash, α, to themselves.

The inputs to the Black-Scholes model for both option pricing and warrant pricing are outlined in the following table.

Black-Scholes Parameters for Pricing Options and Warrants

Input Parameter	Option Pricing	Warrant Pricing
S	current share price	α V, where V is enterprise value minus debt.
X	exercise price per share	total warrant exercise amount multiplied by (1 - α).
T	current time to expiration	average T for warrants
r	interest rate	interest rate
σ	standard deviation of stock return	standard deviation for returns on enterprise value, including warrants

Recommended Reading

Jerry Marlow, Option Pricing: Black-Scholes Made Easy