intermediate finance

Basic framework of finance

• Value creation framework: Free cash flow (FCF) discounted to present value using the Weighted Average Cost of Capital (WACC).

• WACC represents the cost of raising financing from both equity and debt sources; it is the expected return demanded by investors.

• Key intuition: WACC is tied to risk. Higher risk → higher cost of capital (financing risk, business risk, market risk).

• Core idea for the course: understand what affects WACC and how it interacts with investment decisions and risk.

Investment goals and risk basics

• Investors sacrifice current consumption to earn future returns; investment outcomes are not guaranteed.

• Expected return (denoted as the target or forecast) is not a guaranteed actual return.

• If you expect a return of R_expected, actual return can deviate due to risk; the difference reflects risk.

• Risk aversion: most people are risk-averse, meaning they require additional return to take additional risk.

• Risk vs reward trade-off: higher risk must be compensated with higher expected return for the investment to be attractive.

• Distinguishing dollar return from percentage return:

  • Dollar return example: investing $1,000 to receive $1,060 → $60 gain, but importance lies in the scale (percentage return).

  • Percentage return avoids scale issues: % return = (Gain / Initial investment) × 100.

• Positive vs negative returns: positive return occurs when the ending value exceeds the initial value; negative return when it ends lower.

Understanding return and risk through scenarios

• Investments often involve thinking in scenarios (discrete outcomes) rather than a single number.

• Discrete outcomes: a limited set of outcomes with assigned probabilities (e.g., rain/no rain; tariffs high/medium/low).

• Example of a four-scenario setup with probabilities and returns:

  • Probabilities: 0.10, 0.20, 0.30, 0.40 for scenarios 1–4.

  • Returns:

  • Scenario 1: -10% (r1 = -0.10)

  • Scenario 2: 0% (r2 = 0)

  • Scenario 3: 10% (r3 = 0.10)

  • Scenario 4: 30% (r4 = 0.30)

• Expected return calculation:

  • $E[R] = <br>\sum<em>{i} p</em>i r_i = 0.10(-0.10) + 0.20(0) + 0.30(0.10) + 0.40(0.30) = 0.14 = 14\%.$

• Measuring risk (stand-alone): how far actual outcomes deviate from the expected return.

• Deviation from expectation for each scenario:

  • Deviation values (r_i − E[R]):

  • Scenario 1: −0.10 − 0.14 = −0.24

  • Scenario 2: 0 − 0.14 = −0.14

  • Scenario 3: 0.10 − 0.14 = −0.04

  • Scenario 4: 0.30 − 0.14 = 0.16

• Squared deviations to remove sign and emphasize magnitude:

  • (−0.24)^2 = 0.0576

  • (−0.14)^2 = 0.0196

  • (−0.04)^2 = 0.0016

  • (0.16)^2 = 0.0256

• Weighted variance (discrete case):

  • $\mathrm{Var}(R) = \sum<em>i p</em>i (r_i - E[R])^2 = 0.10(0.0576) + 0.20(0.0196) + 0.30(0.0016) + 0.40(0.0256) = 0.0204.$

• Standard deviation (risk):

  • $\sigma(R) = \sqrt{\mathrm{Var}(R)} = \sqrt{0.0204} = 0.1428 = 14.28\%.$

• Interpretation: with these four scenarios, the stand-alone risk (standard deviation) is 14.28% (in percentage terms).

• For continuous distributions, returns are not limited to a few outcomes; real returns are often modeled as continuous (e.g., normal distribution).

Normal (continuous) distribution intuition

• If historical/observed returns approximate a normal distribution, then:

  • About 68% of outcomes lie within one standard deviation of the mean.

  • Example: If the expected return is $E[R] = 10\%$ and the standard deviation is $\sigma = 20\%$, then

  • Return range within one std: $[E[R] - \sigma,\, E[R] + \sigma] = [-10\%, 30\%].$

  • There is a 68% chance the realized return falls in this range; 32% outside (16% on either tail for a symmetric distribution).

• Visual intuition: tighter (narrow) distributions imply lower risk; wider distributions imply higher risk.

• Historical data for continuous modeling: use daily, weekly, or monthly returns to form an empirical distribution (smoothing toward normal over time).

• Caveat: historical data approximate but do not guarantee future results; probabilities are not assigned explicitly unless using a discrete model or a probabilistic forecast.

Using historical data and Excel for risk measures

• Historical approach to estimating mean and risk:

  • Compute average return (sample mean) from historical observations: e.g., four-year strip: $\bar{r} = \frac{\sum r_t}{T}.$

  • Compute standard deviation from the data using either population or sample formula; for a sample, use $\mathrm{SD} = \sqrt{\frac{\sum (r_t - \bar{r})^2}{T-1}}.$

• Excel usage (practical notes):

  • Average: $=AVERAGE(range)$

  • Standard deviation (sample): $=STDEV.S(range)$

  • Standard deviation (population): $=STDEV.P(range)$

  • Correlation between two series: $=CORREL(range1, range2)$

• Example with two stocks (B and C):

  • Returns (e.g., over 10 years): mean returns: $\bar{r}<em>B = 6.4\%,$ $\bar{r}</em>C = 9.2\%.$

  • Standard deviations: $\sigma<em>B = 25\%,$ $\sigma</em>C = 38\%.$

  • Correlation: $\rho_{BC} = 0.11.$

• Portfolio construction intuition: diversify to reduce risk; combine assets with weights to form a portfolio.

Portfolio returns and risks with two assets

• Portfolio return with weights: if you invest $wB$ in stock B and $wC$ in stock C (with wB + wC = 1), then

  • $R<em>p = w</em>B R<em>B + w</em>C R_C.$

  • Example: with weights 75% in B and 25% in C, and year where B returns 26% and C returns 47%,

  • $R_p = 0.75(0.26) + 0.25(0.47) = 0.3125 = 31.25\%.$

• Portfolio risk (standard deviation) for two assets with correlation ρ:

  • General formula: $\sigma<em>p^2 = w</em>B^2 \sigma<em>B^2 + w</em>C^2 \sigma<em>C^2 + 2 w</em>B w<em>C \sigma</em>B \sigma<em>C \rho</em>{BC}.$

  • Using the numbers from the example: wB = 0.75, wC = 0.25, $\sigma<em>B = 0.25, \ \sigma</em>C = 0.38, \ \rho_{BC} = 0.11.$

  • Compute components:

  • $w<em>B^2 \sigma</em>B^2 = (0.75)^2 (0.25)^2 = 0.5625 \times 0.0625 = 0.03515625.$

  • $w<em>C^2 \sigma</em>C^2 = (0.25)^2 (0.38)^2 = 0.0625 \times 0.1444 = 0.009025.$

  • $2 w<em>B w</em>C \sigma<em>B \sigma</em>C \rho_{BC} = 2(0.75)(0.25)(0.25)(0.38)(0.11) = 0.00391875.$

  • Sum to get variance: $\sigma_p^2 = 0.03515625 + 0.009025 + 0.00391875 \approx 0.0481.$

  • Thus, $\sigma_p = \sqrt{0.0481} \approx 0.2195 = 21.95\%.$

• Diversification insight: portfolio risk can be lower than the risk of individual assets when there is imperfect (positive but not perfect) correlation between assets.

• Two-stocks correlation intuition:

  • Correlation ranges from −1 to 1.

  • Negative correlation (ρ < 0) can reduce portfolio risk; perfect negative correlation (ρ = −1) with appropriate weights can, in theory, produce zero portfolio risk.

  • In practice, perfectly negative correlation is rare; typical stock correlations in the US might be around 0.2–0.3, enabling meaningful diversification benefits but not perfect elimination of risk.

• Diversification and the limits of risk reduction:

  • Adding more stocks generally reduces unsystematic (diversifiable) risk.

  • With many assets, portfolio risk bottoms out toward the market (systematic) risk component, which cannot be diversified away.

  • A common teaching point: for a broad stock portfolio, diversifiable risk can be reduced substantially with enough independent assets, but economy-wide risk remains.

• Conceptual split of risk:

  • Diversifiable (unsystematic) risk: company-specific events (e.g., product failure, management changes) that can be mitigated through diversification.

  • Non-diversifiable (systematic) risk: economy-wide factors (e.g., recession, inflation, war) affecting almost all assets; cannot be eliminated by diversification.

• Practical takeaway: the goal of diversification is to reduce the portfolio’s unsystematic risk, leaving the systematic risk to be priced by the market (e.g., CAPM framework to come).

Real-world relevance and takeaways

• WACC and risk pricing are central to capital budgeting and corporate finance decisions; higher risk requires higher expected returns.

• Investors evaluate risk using a mix of discrete scenarios, continuous distributions, and historical data to form expectations and quantify risk.

• The quantitative tools cover expected return, variance, standard deviation, and correlations, all of which underpin portfolio construction and risk management.

• Understanding diversification and correlation helps explain why investors seek broad, partially uncorrelated assets to reduce risk without sacrificing expected return.

• These concepts set up the next parts of the course on CAPM, market efficiency, and behavioral finance, tying risk, return, and pricing to broader financial theory.