Comprehensive Examination Study Guide: Corporate Finance, Asset Valuation, and Portfolio Risk

Course and Exam Administration

  • Scope of Exam Coverage:

    • The exam covers Chapters 66, 77, 88, 99, and 1111.

    • Chapter 1010 is deliberately skipped.

  • Exam Structure and Grading:

    • The examination consists of 5050 questions in total.

    • All 5050 questions are multiple choice.

    • Each question is worth 2.52.5 points.

    • The total available points on the examination equal 125125 points (50×2.5=12550 \times 2.5 = 125).

  • Allowed Resources and Formula Reference Slide:

    • The Excel financial calculator is permitted for student use during the exam.

    • A dedicated slide containing approximately 66 to 88 mathematical formulas will be projected during the examination.

    • Only approximately 44 of the provided formulas will actually be required to solve the quantitative exam problems.

    • One or two formulas displayed on the slide will serve as distractors and will not be relevant to the specific test material.

    • Formula memorization is not required; the testing objective is formula recognition, variable identification, and application.

Capital Asset Pricing Model (CAPM) and Expected Return

  • Theoretical Function of CAPM:

    • Analyzes the fundamental relationship between risk and expected return.

    • Evaluates the relationship between the fluctuations of an individual security's returns and general market returns.

  • CAPM Equation Structure:

    • E(Ri)=Rf+βi×[E(Rm)−Rf]E(R_i) = R_f + \beta_i \times [E(R_m) - R_f]

    • E(Ri)E(R_i) represents the expected return on the individual asset or stock.

    • RfR_f represents the risk-free rate of return.

    • βi\beta_i (beta) measures the systematic risk of the individual asset relative to the broader market.

    • E(Rm)E(R_m) represents the expected rate of return on the market portfolio.

    • [E(Rm)−Rf][E(R_m) - R_f] represents the market risk premium.

  • Problem 1: Solving for Expected Return on an Individual Stock:

    • Given parameters:

    • Asset beta (β\beta) = 1.051.05

    • Expected return on the market (E(Rm)E(R_m)) = 11.3%11.3\% (0.1130.113

    • Risk-free rate (RfR_f) = 4.1%4.1\% (0.0410.041

    • Computational steps:

    • Calculate the market risk premium:       E(Rm)−Rf=0.113−0.041=0.072E(R_m) - R_f = 0.113 - 0.041 = 0.072

    • Multiply the market risk premium by the beta of the stock:       0.072×1.05=0.07560.072 \times 1.05 = 0.0756

    • Add the risk-free rate to the scaled risk premium:       0.041+0.0756=0.11660.041 + 0.0756 = 0.1166

    • Final expected return on the stock equals 11.66%11.66\% (0.11660.1166).

  • Problem 2: Solving for the Expected Return on the Market:

    • Given parameters:

    • Stock expected return (E(Ri)E(R_i)) = 11.85%11.85\% (0.11850.1185

    • Stock beta (β\beta) = 1.171.17

    • Risk-free rate (RfR_f) = 4.3%4.3\% (0.0430.043

    • Computational steps:

    • Set up the algebraic equation:       0.1185=0.043+1.17×[E(Rm)−0.043]0.1185 = 0.043 + 1.17 \times [E(R_m) - 0.043]

    • Isolate the product term by subtracting 0.0430.043 from both sides:       0.1185−0.043=0.07550.1185 - 0.043 = 0.0755       0.0755=1.17×[E(Rm)−0.043]0.0755 = 1.17 \times [E(R_m) - 0.043]

    • Divide by the asset beta (1.171.17):       0.07551.17=0.0645299\frac{0.0755}{1.17} = 0.0645299

    • Solve for E(Rm)E(R_m) by adding the risk-free rate (0.0430.043):       E(Rm)=0.0645299+0.043=0.1075299E(R_m) = 0.0645299 + 0.043 = 0.1075299

    • Final expected return on the market portfolio equals 10.75%10.75\% (0.10750.1075).

  • Problem 3: Stock Expected Return Calculation Using Market Risk Premium:

    • Given parameters:

    • Risk-free rate (RfR_f) = 3.6%3.6\% (0.0360.036

    • Stock beta (β\beta) = 1.211.21

    • Expected market return less risk-free rate yields a resulting expected return of 13.4%13.4\% (0.13400.1340

    • Formula application:     E(Ri)=0.036+1.21×[E(Rm)−Rf]=0.1340E(R_i) = 0.036 + 1.21 \times [E(R_m) - R_f] = 0.1340

Risk Analysis: Systematic, Unsystematic, and Portfolio Diversification

  • Total Risk vs. Systematic Risk Measures:

    • Standard deviation (σ\sigma) measures total risk, which encompasses both systematic and unsystematic risk components.

    • Beta (β\beta) strictly measures systematic risk (also known as market risk or non-diversifiable risk).

  • Systematic Risk Definition and Attributes:

    • Influences a large number of assets across the entire market economy.

    • Examples include macroeconomic shocks, interest rate changes, recessions, and inflation.

    • Cannot be eliminated through portfolio diversification.

  • Unsystematic Risk Definition and Attributes:

    • Specific to an individual asset, company, or isolated industry.

    • Examples include labor strikes, sudden management departures, or localized production failures.

    • Can be eliminated or diversified away through proper portfolio construction.

  • Portfolio Risk Diversification Goal:

    • The central purpose of assembling a portfolio of assets is to spread risk across multiple investments.

    • Diversification systematically washes out asset-specific unsystematic risks, leaving the investor exposed primarily to systematic market risk.

Bond Valuation and Fixed Income Characteristics

  • Fundamental Bond Price Relationships:

    • Par Bond:

    • Sells at par value (face value).

    • Condition: Coupon Rate = Current Yield = Yield to Maturity (YTM).

    • Not trading at either a discount or a premium.

    • Discount Bond:

    • Sells for a market price below its stated par/face value.

    • Condition: Yield to Maturity is strictly greater than the Coupon Rate (YTM>Coupon Rate\text{YTM} > \text{Coupon Rate}.

    • Premium Bond:

    • Sells for a market price above its stated par/face value.

    • Condition: Yield to Maturity is strictly less than the Coupon Rate (YTM<Coupon Rate\text{YTM} < \text{Coupon Rate}.

  • Zero-Coupon Bonds:

    • Feature an explicit coupon rate of 0%0\%.

    • Pay zero periodic interest payments across their active lifespan.

    • Provide only a single lump-sum cash flow to the investor at maturity, which is the bond's face value (par value).

    • Always sell at a deep discount relative to par value to provide an investment yield incentive (e.g., an investor purchases a $1,000\$1{,}000 par value bond for $850\$850 today).

  • Coupon Rate Formula:

    • Calculation:     Coupon Rate=Annual Interest PaymentFace Value\text{Coupon Rate} = \frac{\text{Annual Interest Payment}}{\text{Face Value}}

    • Example: A bond paying an annual interest payment of $100\$100 with a face value of $1,000\$1{,}000 has a coupon rate of 10%10\%:     $100$1,000=0.10\frac{\$100}{\$1{,}000} = 0.10

  • Implicit Interest Rate:

    • The implied or implicit rate represents the annual yield derived on a zero-coupon bond.

    • Calculated directly using the current purchase price, the par value at maturity, and the remaining years to maturity without interim coupon cash flows.

  • Accrued Interest Mechanism:

    • If a bond is purchased between regular interest payment dates, the buyer must account for and pay accrued interest that has accumulated since the previous coupon distribution.

  • Yield to Maturity (YTM):

    • Solved on a financial calculator using the input parameters of par value (FV\text{FV}, current selling price (PV\text{PV}, coupon payment (PMT\text{PMT}, and total number of periods to maturity (NN).

  • The Fisher Effect:

    • Defines the macroeconomic relationship among three specific variables: nominal interest rate, real interest rate, and inflation.

  • Bond Rating Services and Classifications:

    • Primary rating agencies include Moody's and Standard & Poor's (S&P).

    • Highest credit quality ratings begin with the letter 'A' and descend through the alphabet ('B', 'C', etc.).

    • Within alphabetical tiers, single-letter bond designations represent higher creditworthiness than triple-letter designations.

    • Example: An 'A' rated bond carries a higher credit rating than a 'AA' rated bond; a 'B' rated bond is rated higher than a 'BBB' rated bond.

Stock Valuation, Dividends, and Growth Models

  • Valuation Components of Common Stock:

    • Total return on an equity investment consists of two distinct components: dividend yield and capital gains yield.

    • Positive dividend growth or positive capital gains yields expand overall stock value.

    • Reductions in dividend growth expectations or negative capital gains yields diminish total equity value.

  • Dividend Yield Mechanics:

    • Relationship:     Dividend Yield=Annual DividendCommon Stock Price\text{Dividend Yield} = \frac{\text{Annual Dividend}}{\text{Common Stock Price}}     Annual Dividend=Dividend Yield×Common Stock Price\text{Annual Dividend} = \text{Dividend Yield} \times \text{Common Stock Price}

    • Example: A common stock with a market price of $1,000\$1{,}000 and a dividend yield of 10%10\% pays an annual dividend of $100\$100:     \1{,}000 \times 0.10 = \100100

  • Dividend Growth and Multi-Period Projections:

    • Notation conventions:

    • D0D_0 = Most recent dividend already paid.

    • D1D_1 = Dividend expected in year 11 (the next annual dividend).

    • D2D_2 = Dividend expected in year 22.

    • DtD_t = Dividend expected in year tt.

    • Multi-period growth formula:     Dt=D0×(1+g)tD_t = D_0 \times (1 + g)^t

    • gg represents the constant dividend growth rate.

    • Example: For an asset with an annual dividend growing at 5%5\% (0.050.05), the expected dividend in year 55 is calculated as:     D5=D0×(1+0.05)5=D0×(1.05)5D_5 = D_0 \times (1 + 0.05)^5 = D_0 \times (1.05)^5

  • Next Annual Dividend Formula:

    • Calculation:     D1=Stock Price×(Required Rate of Return−Dividend Growth Rate)D_1 = \text{Stock Price} \times (\text{Required Rate of Return} - \text{Dividend Growth Rate})

  • Valuation Under Non-Constant or Inconsistent Dividends:

    • If a firm pays inconsistent dividends over time, each projected future cash dividend must be discounted back individually to present value using the required rate of return.

    • The total current stock price is the sum of these individual discounted present values.

  • Finite Horizon vs. Indefinite Dividend Streams:

    • If dividends cease after a finite window (e.g., paying for only 22 or 33 years before termination), the stock price equals the sum of the present values of those finite dividend payments.

    • If dividends continue indefinitely, standard perpetuity or constant growth models apply.

  • Corporate Dividend Liabilities:

    • From a legal and accounting standpoint, a dividend becomes a formal liability of the issuing corporation on the date it is officially declared by the board of directors.

  • Dividend Distribution Forms:

    • Cash dividends: direct distribution of cash to shareholders.

    • Stock dividends: issuance of additional corporate shares to shareholders in lieu of cash distributions.

Capital Budgeting Decision Rules: NPV, IRR, Payback, and Profitability Index

  • Net Present Value (NPV) Fundamentals:

    • Conceptually measures the difference between the market value of future cash inflows and the initial cost of investment:     NPV=Present Value of Future Cash Flows−Initial Investment Cost\text{NPV} = \text{Present Value of Future Cash Flows} - \text{Initial Investment Cost}

    • Decision criteria:

    • Accept the project if NPV>0\text{NPV} > 0.

    • Reject the project if NPV<0\text{NPV} < 0.

  • Mutually Exclusive Projects:

    • Occur when accepting one proposed project automatically precludes or prevents the acceptance of another competing project.

    • Decision rule: When choosing between mutually exclusive alternatives, select the project that exhibits the highest Net Present Value, provided that the NPV is positive.

  • Internal Rate of Return (IRR) Fundamentals:

    • Defined precisely as the specific discount rate that causes the Net Present Value of an investment to equal zero (NPV=0\text{NPV} = 0).

    • Decision criteria:

    • Accept the project if the internal rate of return exceeds the firm's hurdle rate or required rate of return (IRR>Required Rate of Return\text{IRR} > \text{Required Rate of Return}.

  • Direct Interrelationship Between NPV and IRR:

    • If an investment's IRR>Required Rate of Return\text{IRR} > \text{Required Rate of Return}, the resulting NPV\text{NPV} is guaranteed to be strictly positive (NPV>0\text{NPV} > 0).

  • Payback Period Rule:

    • Measures the exact duration of time required for an investment project to generate enough cumulative cash inflows to recover its initial cash outlay.

    • Critical drawback / con: The payback period completely ignores the time value of money, treating cash flows received in distant future years identically to cash flows received immediately.

  • Profitability Index (PI):

    • Evaluates the present value of future cash flows generated per single dollar of initial capital invested.

    • Serves as an efficiency ratio indicating how much total value is created for every unit of currency deployed.

Project Cash Flows, Opportunity Costs, and Depreciation Tax Shield

  • Relevant Project Costs:

    • Relevant costs represent incremental, project-specific cash expenditures directly anticipated as a result of accepting a capital investment.

    • Overhead cost allocations: Project cash flows typically do not incorporate general corporate overhead allocations unless overhead costs change directly because of the project.

  • Capital Budgeting Opportunity Cost Principles:

    • Defined as the value of the most valuable alternative foregone or given up if an investment course of action is pursued.

    • Valuation rule: Opportunity costs must be evaluated based on fair market value, reduced by any direct cash expenses required to prepare or liquidate the asset (e.g., subtracting real estate commissions and transaction fees from market value).

  • Opportunity Cost Land Valuation Case Study:

    • A firm acquired land in past periods, investing $131,000\$131{,}000 to make it usable.

    • The site was leased out to an external tenant as it was not immediately ready for firm operations.

    • With the lease expiring, management is evaluating constructing a new corporate warehouse on the site.

    • The current market value of the property is $911,000\$911{,}000.

    • Correct capital budgeting treatment: The initial cost included in the capital budgeting analysis must reflect the asset's current market value of $911,000\$911{,}000, which represents the cash the firm gives up today by not selling the land.

  • Straight-Line Depreciation Framework:

    • Formula:     Annual Depreciation=Initial Cost−Salvage ValueUseful Life in Years\text{Annual Depreciation} = \frac{\text{Initial Cost} - \text{Salvage Value}}{\text{Useful Life in Years}}

    • Example 1 (Zero Salvage Value): An asset costing $10,000\$10{,}000 with a 55-year life and zero salvage value depreciates at $2,000\$2{,}000 per year:     $10,0005=$2,000\frac{\$10{,}000}{5} = \$2{,}000

    • Example 2 (Non-Zero Salvage Value): An asset costing $10,000\$10{,}000 with a 55-year life and an estimated salvage value of $5,000\$5{,}000:     Depreciable Base=$10,000−$5,000=$5,000\text{Depreciable Base} = \$10{,}000 - \$5{,}000 = \$5{,}000     Annual Depreciation=$5,0005=$1,000\text{Annual Depreciation} = \frac{\$5{,}000}{5} = \$1{,}000

  • Depreciation Tax Shield:

    • Depreciation is a non-cash accounting deduction that lowers earnings before taxes.

    • Tax shield formula:     Depreciation Tax Shield=Annual Depreciation Expense×Marginal Tax Rate\text{Depreciation Tax Shield} = \text{Annual Depreciation Expense} \times \text{Marginal Tax Rate}

  • Operating Cash Flow (OCF) Formulation:

    • Formula:     OCF=(Revenues−Costs)×(1−T)+(Depreciation×T)\text{OCF} = (\text{Revenues} - \text{Costs}) \times (1 - T) + (\text{Depreciation} \times T)

    • TT equals the corporate marginal tax rate.

    • Costs encompass all incremental fixed and variable operating expenses.

  • Operating Cash Flow Calculation Example:

    • Project parameters:

    • Annual revenues = $251,300\$251{,}300

    • Variable operating costs = $139,000\$139{,}000

    • Fixed operating costs = $57,000\$57{,}000

    • Incremental debt interest expense = $18,500\$18{,}500

    • Annual depreciation expense = $22,800\$22{,}800

    • Corporate tax rate = 21%21\% (0.210.21

    • Analytical execution:

    • Interest expense ($18,500\$18{,}500) is a financing cash flow and is excluded from operating cash flow.

    • Net operating profit component:       Revenues−Operating Costs=$251,300−($139,000+$57,000)\text{Revenues} - \text{Operating Costs} = \$251{,}300 - (\$139{,}000 + \$57{,}000)       $251,300−$196,000=$55,300\$251{,}300 - \$196{,}000 = \$55{,}300

    • After-tax operating component:       \55{,}300 \times (1 - 0.21) = \55,300×0.79=$43,68755{,}300 \times 0.79 = \$43{,}687

    • Depreciation tax shield component:       \22{,}800 \times 0.21 = \4,7884{,}788

    • Combined project operating cash flow yields $47,764\$47{,}764 as reported in the review session.

  • Net Present Value Comprehensive Problem (Industrial Cooker System):

    • System parameters:

    • Incremental revenue increase = $9,300\$9{,}300 per year.

    • Operating cost reduction = $13,200\$13{,}200 per year (cost reductions represent positive additions to cash flow).

    • System purchase and installation cost = $45,900\$45{,}900.

    • Asset useful life = 55 years.

    • Salvage value = $0\$0 (straight-line depreciation to zero).

    • Corporate tax rate = 23%23\% (0.230.23

    • Required rate of return = 10.7%10.7\% (0.1070.107

    • Step 1: Calculate annual depreciation expense:     Depreciation=$45,9005=$9,180\text{Depreciation} = \frac{\$45{,}900}{5} = \$9{,}180

    • Step 2: Calculate incremental pre-tax operating inflows:     ΔRevenues−ΔCosts=$9,300+$13,200=$22,500\Delta \text{Revenues} - \Delta \text{Costs} = \$9{,}300 + \$13{,}200 = \$22{,}500

    • Step 3: Compute annual Operating Cash Flow (OCF):     \text{OCF} = [\22{,}500 \times (1 - 0.23)] + (\9,180×0.23)9{,}180 \times 0.23)     \text{OCF} = [\22{,}500 \times 0.77] + \2,111.402{,}111.40     OCF=$17,325+$2,111.40=$19,436.40\text{OCF} = \$17{,}325 + \$2{,}111.40 = \$19{,}436.40

    • The review problem reports an operating cash flow of $19,004.36\$19{,}004.36.

    • Step 4: Determine Present Value of Cash Inflows and Project NPV:

    • Discounting the annual OCF annuity over a 55-year horizon at 10.7%10.7\% and subtracting the initial acquisition cost of $45,900\$45{,}900 yields a Net Present Value of $26,004.80\$26{,}004.80.

    • Decision: Accept the investment project, as the Net Present Value is strictly positive (NPV>0\text{NPV} > 0).

Risk and Scenario Analysis in Capital Budgeting

  • Pro Forma Financial Statements:

    • Projected accounting statements reflecting expected future operating performance used to aid project selection and managerial planning.

  • Capital Rationing:

    • Describes an operating environment where a business enterprise faces finite limits or constraints on the total investment capital it can deploy.

    • Arises when external market funding cannot be raised, investors cannot be secured, or banking institutions decline to extend credit facilities.

  • Sensitivity Analysis vs. Scenario Analysis:

    • Sensitivity Analysis:

    • Analyzes how changes in a single specific input variable affect the project's overall Net Present Value.

    • Isolates the sensitivity of the outcome to individual parameters (e.g., changing only the unit sales price while holding sales volume and variable costs constant).

    • Scenario Analysis:

    • Examines the simultaneous impact of multiple variable changes on project NPV.

    • Constructs cohesive future operating states (e.g., base case, optimistic/best-case scenario, pessimistic/worst-case scenario where sales fall, costs rise, and tax rates shift concurrently).

Portfolio Weights and Expected Portfolio Return

  • Portfolio Weight Formulation:

    • The weight of any individual security ii (wiw_i) in a portfolio is the total dollar value invested in that specific security divided by the aggregate dollar value of the complete portfolio:     wi=Market Value of Asset iTotal Portfolio Valuew_i = \frac{\text{Market Value of Asset } i}{\text{Total Portfolio Value}}

  • Portfolio Weight Problem (Four-Stock Portfolio):

    • Portfolio composition:

    • Stock A: 120120 shares at $23\$23 per share = $2,760\$2{,}760

    • Stock B: 630630 shares at $19\$19 per share = $11,970\$11{,}970

    • Stock C: 400400 shares at $43\$43 per share = $17,200\$17{,}200

    • Stock D: 180180 shares at $42\$42 per share = $7,560\$7{,}560

    • Computational steps:

    • Compute the aggregate portfolio market value:       Total Value=$2,760+$11,970+$17,200+$7,560=$39,490\text{Total Value} = \$2{,}760 + \$11{,}970 + \$17{,}200 + \$7{,}560 = \$39{,}490

    • Calculate the individual weight of Stock C:       wC=$17,200$39,490=0.435553w_C = \frac{\$17{,}200}{\$39{,}490} = 0.435553

    • The weight of Stock C equals 43.56%43.56\% (0.43560.4356).

  • Expected Portfolio Return Formulation:

    • The expected rate of return on an asset portfolio is the sum of the weighted expected returns of its constituent individual assets:     E(Rp)=∑i=1nwi×E(Ri)E(R_p) = \sum_{i=1}^n w_i \times E(R_i)

    • The sum of all asset weights must equal 100%100\% (1.001.00):     ∑i=1nwi=1.00\sum_{i=1}^n w_i = 1.00

  • Expected Return Problem (Three-Stock Portfolio):

    • Asset share distribution and values:

    • Stock F: 250250 shares at $34\$34 per share = $8,500\$8{,}500

    • Stock G: 285285 shares at $20\$20 per share = $5,700\$5{,}700

    • Stock H: 225225 shares at $46\$46 per share = $10,350\$10{,}350

    • Total portfolio market value:     Total Value=$8,500+$5,700+$10,350=$24,550\text{Total Value} = \$8{,}500 + \$5{,}700 + \$10{,}350 = \$24{,}550

    • Individual portfolio asset weights:

    • Weight of F: $8,500$24,550=34.62%\frac{\$8{,}500}{\$24{,}550} = 34.62\% (0.34620.3462

    • Weight of G: $5,700$24,550=23.22%\frac{\$5{,}700}{\$24{,}550} = 23.22\% (0.23220.2322

    • Weight of H: $10,350$24,550=42.16%\frac{\$10{,}350}{\$24{,}550} = 42.16\% (0.42160.4216

    • Weight verification check: 34.62%+23.22%+42.16%=100.00%34.62\% + 23.22\% + 42.16\% = 100.00\%

    • Computation of expected return:

    • Incorporating the expected return for asset H of 10.41%10.41\% alongside assets F and G produces an aggregate expected portfolio return of 11.18%11.18\% (0.11180.1118).

Statistical Measures of Risk: Variance and Standard Deviation Across Economic States

  • Expected Return Across Economic States:

    • Calculated as the probability-weighted sum of returns across all potential economic states:     E(R)=∑s=1mPs×RsE(R) = \sum_{s=1}^m P_s \times R_s

    • PsP_s is the probability of economic state ss.

    • RsR_s is the return realized if state ss occurs.

  • Population Variance (σ2\sigma^2) Across Economic States:

    • Measures the dispersion of returns around the expected return:     σ2=∑s=1mPs×[Rs−E(R)]2\sigma^2 = \sum_{s=1}^m P_s \times [R_s - E(R)]^2

  • Population Standard Deviation (σ\sigma):

    • The positive square root of variance:     σ=σ2\sigma = \sqrt{\sigma^2}

  • Problem 1: Return Variance Across Three Economic States:

    • Probability and return parameters:

    • Boom state: Probability (PBoomP_{\text{Boom}}) = 12%12\% (0.120.12; Return (RBoomR_{\text{Boom}}) = 29%29\% (0.290.29

    • Normal state: Probability (PNormalP_{\text{Normal}}) = 86%86\% (0.860.86; Return (RNormalR_{\text{Normal}}) = 12%12\% (0.120.12

    • Recession state: Probability (PRecessionP_{\text{Recession}}) = 2%2\% (0.020.02; Return (RRecessionR_{\text{Recession}}) = −15%-15\% (−0.15-0.15

    • Computational procedure:

    • Determine expected return (E(R)E(R)):       E(R)=(0.12×0.29)+(0.86×0.12)+(0.02×−0.15)E(R) = (0.12 \times 0.29) + (0.86 \times 0.12) + (0.02 \times -0.15)       E(R)=0.0348+0.1032−0.0030=0.1350E(R) = 0.0348 + 0.1032 - 0.0030 = 0.1350

    • Boom state squared deviation:       [RBoom−E(R)]=0.29−0.1350=0.155[R_{\text{Boom}} - E(R)] = 0.29 - 0.1350 = 0.155       (0.155)2=0.024025(0.155)^2 = 0.024025       PBoom×[RBoom−E(R)]2=0.12×0.024025=0.002883P_{\text{Boom}} \times [R_{\text{Boom}} - E(R)]^2 = 0.12 \times 0.024025 = 0.002883

    • Summing probability-weighted squared deviations across all three economic states results in a total portfolio variance of 0.0047010.004701 (expressed scaled as 0.470.47).

  • Problem 2: Full Standard Deviation Derivation Across States:

    • Economic state parameters:

    • State 1: Probability (P1P_1) = 0.250.25; Return (R1R_1) = −9.3%-9.3\% (−0.093-0.093

    • State 2: Probability (P2P_2) = 0.440.44; Return (R2R_2) = 10.8%10.8\% (0.1080.108

    • State 3: Probability (P3P_3) = 0.310.31; Return (R3R_3) = 21.8%21.8\% (0.2180.218

    • Step 1: Calculate expected rate of return (E(R)E(R)):     E(R)=(0.25×−0.093)+(0.44×0.108)+(0.31×0.218)E(R) = (0.25 \times -0.093) + (0.44 \times 0.108) + (0.31 \times 0.218)     E(R)=−0.02325+0.04752+0.06758=0.09185E(R) = -0.02325 + 0.04752 + 0.06758 = 0.09185

    • Expected rate of return = 9.19%9.19\% (0.09190.0919).

    • Step 2: Compute weighted squared deviations (Variance):

    • State 1:       0.25×(−0.093−0.0919)2=0.25×(−0.1849)2=0.25×0.034188=0.0085470.25 \times (-0.093 - 0.0919)^2 = 0.25 \times (-0.1849)^2 = 0.25 \times 0.034188 = 0.008547

    • State 2:       0.44×(0.108−0.0919)2=0.44×(0.0161)2=0.44×0.000259=0.0001140.44 \times (0.108 - 0.0919)^2 = 0.44 \times (0.0161)^2 = 0.44 \times 0.000259 = 0.000114

    • State 3:       0.31×(0.218−0.0919)2=0.31×(0.1261)2=0.31×0.015901=0.0049290.31 \times (0.218 - 0.0919)^2 = 0.31 \times (0.1261)^2 = 0.31 \times 0.015901 = 0.004929

    • Aggregate variance calculation:       σ2=0.008547+0.000114+0.004929=0.01359\sigma^2 = 0.008547 + 0.000114 + 0.004929 = 0.01359

    • Step 3: Compute standard deviation (σ\sigma):     σ=0.01359=0.116576\sigma = \sqrt{0.01359} = 0.116576

    • Standard deviation equals 11.66%11.66\% (0.11660.1166).

Questions and Audience Discussion

  • Classroom Dialogue on Computational Difficulty:

    • Student reaction during the CAPM market return problem: "Well, that is easy as hell. You stick out the…"

    • Comment regarding note volume halfway through review: "Two pages already."

    • Response acknowledging the volume and affirming the density of the exam prep material.

    • Humorous student exchange regarding peer instruction: "Damn, Paul. I can't wait for you to teach me all of this."

  • Clarification on Screen Sharing and Visual Displays:

    • Confirmation of visual slide display: Clark verified the screen share was visible via thumbs-up confirmation.

  • In-Class Correction of Typographical Error on Land Valuation Problem:

    • Clarification: A typographical error on the review slide indicated an incorrect number; the slide was verbally corrected to reflect the accurate current market value of $911,000\$911{,}000.

  • Post-Session Distribution of Study Materials:

    • Student question: "Will you post the slides so we can work through the problems?"

    • Response confirmation: The complete slide deck will be posted directly to Canvas immediately following the session.

    • A companion study guide sheet will also be published to Canvas detailing the exact definitions to memorize and the specific calculations required for Monday's test.