Risk, Return, CAPM & WACC – Comprehensive Study Notes

Risk = uncertainty about returns; captures potential variability in outcomes and therefore shapes the cost of capital.
Two broad manifestations of risk in markets (illustrative real-world evidence):
- Bitcoin price path: from $\$123$ (Oct 2012) → $\$1{,}238$ (2013) → $\$20{,}000$ (2017) → $<\$4{,}000$ (early 2019) → $\$69{,}000$ (Nov 2021) → $<\$20{,}000$ (late 2022) → $>\$75{,}000$ (early 2024).
- Gold, NVIDIA, Nokia shares, Straits Times Index & S&P 500 charts—show long-term up-and-down swings; evidence that risk is omnipresent across assets, sectors, and time.

Total (or holding-period) rate of return over period $t$ :
- $rt = \dfrac{Ct + (Pt - P{t-1})}{P_{t-1}}$
- $C_t$ = cash flow received between $t-1$ and $t$ (e.g., dividend, coupon).
Example (DBS share):
- $P{0}=\$31$ , $C=\$2$ , $P{1}=\$36$ ⇒ $r = \dfrac{2 + (36-31)}{31}=0.1613 = 16.13\%\,\text{p.a.}$ (actual/historical return)

Based on probabilistic outcomes:
- $E(R)=\sum{i=1}^{n} pi R_i$
- Where $pi$ = probability of state $i$ , $Ri$ = return in state $i$ .
Coin-toss bet example:
- Heads $+\$1$ (50%), tails $-\$0.50$ (50%).
- $E(R)=1\times0.5 + (-0.5)\times0.5 = 0.25\ \text{(\$0.25 expected gain).}$

Variance: weighted average of squared deviations from expected return.
- $\sigma^{2}=\sum{i=1}^{n} pi (R_i-E(R))^{2}$
Standard deviation: $\sigma = \sqrt{\sigma^{2}}$ .
Higher $\sigma$ ⇒ higher risk.
Worked example (Stock A vs Stock B):
- State data (recession, neutral, boom) ⇒
- $E(RA)=11.25\%$ , $\sigmaA=19.8\%$
- $E(RB)=12.5\%$ , $\sigmaB=14.4\%$
- Interpretation: B offers slightly higher mean with lower volatility → preferable to risk-averse investor.

Weighted average of component expected returns:
- $E(Rp)=\sum{j=1}^{m} wj E(Rj)$
- $w_j$ = proportion of portfolio value in asset $j$ .
Example – single-scenario weights:
- Five-asset portfolio: $E(Rp)=10\%$ when $wA=30\%, w_B=17\%, \dots$ (see data sheet).
Example – multi-scenario:
- For each economy state compute portfolio return $R{p,i}=\sum wj R{j,i}$ , then apply probabilities to obtain $E(Rp)$ .

Not the weighted average of individual $\sigma$ ’s.
Steps:
1. Calculate $R_{p,i}$ per state.
2. Obtain overall $E(R_p)$ .
3. Compute deviations $DEVi = R{p,i} - E(R_p)$ .
4. Square, weight by $pi$ , sum ⇒ $\sigmap^{2}$ .
5. $\sigmap = \sqrt{\sigmap^{2}}$ .
Diversification benefit: portfolio $\sigmap < \sum wj \sigma_j$ unless all assets perfectly correlated.

Systematic (market) risk: economy-wide forces (inflation, rates, war, pandemics). Cannot be diversified away.
Unsystematic (unique) risk: firm or industry specific (strikes, regulation). Diversification can largely eliminate it.
Total risk decomposition: $\text{Total}\ \sigma^2 = \text{Systematic}\ \sigma^2 + \text{Unsystematic}\ \sigma^2$ .
Empirical diversification table: average portfolio $\sigma$ falls from $49.24\%$ (1 stock) to $\approx19.2\%$ (1,000 stocks) then flattens (non-diversifiable floor).

$\betaj = \dfrac{\%\,\text{change in }Rj}{\%\,\text{change in }R_m}$ .
- $\beta_m = 1$ for market portfolio.
Interpretation examples:
- $\beta=2$ ⇒ stock rises $20\%$ when market rises $10\%$ .
- $\beta=-2$ ⇒ stock falls $20\%$ when market rises $10\%$ .
Sample betas: Pfizer $0.63$ , Apple $1.27$ , Boeing $1.63$ etc.

Required/expected return for asset $j$ :
- $Rj = RF + \betaj (Rm - R_F)$
Components:
- $R_F$ = risk-free rate (e.g., treasury yield).
- $Rm - RF$ = market risk premium.
- $\betaj (Rm - R_F)$ = asset-specific risk premium.
AZ Corp valuation example:
- Data: $D1 = 0.05$ , $g = 0.03$ , $RF=2\%$ , $R_m=6\%$ , $\beta=1.30$ .
- Required return: $r = 0.02 + 1.30(0.06-0.02) = 0.072 = 7.2\%$ .
- Gordon model: $P_0 = \dfrac{0.05}{0.072-0.03}=1.19$ .

$\text{WACC} = (wi \times ri)(1-T) + (wp \times rp) + (ws \times rs)$ where
- $wi, wp, w_s$ = market value weights of debt, preferred, common.
- $r_i$ = before-tax cost of debt, adjusted by $(1-T)$ for taxes.
- $r_p$ = cost of preferred stock.
- $r_s$ = cost of common (retained or new equity).
Weights must be based on CURRENT market values, not book.

Interest tax-deductible → after-tax cost: $ri^{\text{after}} = ri^{\text{before}}(1-T)$ .
Company Z example: $ri^{\text{before}}=6\%$ , $T=17\%$ → $ri^{\text{after}} = 0.06(1-0.17)=4.98\%$ .

Two estimation methods:
1. Dividend Discount (constant-growth): $rs = \dfrac{D1}{P_0} + g$ .
2. CAPM: $rs = RF + \beta (Rm - RF)$ .
Company Z examples:
- Constant-growth: $r= \dfrac{4}{50}+0.05 = 13\%$ .
- CAPM: $r=0.02+2(0.08-0.02)=14\%$ .
- Average for planning: $(13\%+14\%)/2=13.5\%$ .
Retained earnings cost $rr = rs$ (opportunity cost to shareholders).

Capital structure: Debt $100$ , Preferred $50$ , Equity $150$ → weights $wi=0.3333, wp=0.1667, w_s=0.50$ .
Plug in:
- $\text{WACC}=0.3333\times6\%\times(1-0.17)+0.1667\times8\%+0.50\times13.5\% \approx 9.743\%$ .

Always distinguish between total, systematic, and unsystematic risk; only systematic commands a return premium.
Variance/standard deviation quantify volatility; higher $\sigma$ = greater risk.
Portfolio diversification reduces unsystematic risk; cannot remove systematic component.
Beta gauges relative systematic risk; feeds directly into CAPM for required return.
Total return formula is the foundation for historical performance measurement.
WACC blends component costs; correct weights = market-value proportions; debt component must be after-tax.
In valuation (e.g., Gordon model) ensure consistent units (decimals vs percentages) and growth g < r.
Ethical / practical implications: investors should demand compensation only for non-diversifiable risk; managers must hurdle projects above WACC to create value.