Concept Guide: Risk, Bond Pricing, and Stock Valuation

Expected return is the average return investors hope to earn, while risk measures uncertainty around that average. Riskier assets usually must offer a higher expected return to attract investors.

Standard deviation uses all possible outcomes and measures dispersion around the mean, so it captures total volatility for one asset by itself.

Expected return and variance are weighted averages, so the probabilities must describe all possible states completely.

Each asset contributes to the portfolio in proportion to its portfolio weight, so the expected return is linear in the weights.

Portfolio variance depends on how assets move together, not just on each asset's own volatility.

When returns do not move together, bad performance in one asset can be offset by better performance in another.

Firm-specific shocks can be offset across many holdings, but economy-wide shocks affect most assets at the same time.

Beta focuses on how much an asset moves relative to the market, so it captures market-related risk rather than total risk.

CAPM assumes diversified investors can eliminate firm-specific risk, so only non-diversifiable market risk earns a premium.

The SML shows the return required for a given beta. Above the line means expected return exceeds required return; below means the opposite.

Beta scales market exposure. A beta of 1 matches market sensitivity, while values above or below 1 amplify or dampen that sensitivity.

CAPM splits required return into a safe base return and a premium for market exposure.

Shifting every outcome upward raises the average return, while risk depends on spread, not just on level.

If assets move exactly together, combining them does not smooth out fluctuations much.

If an asset does not co-move with market swings, its market sensitivity is essentially zero.

The SML is a theoretical pricing line from CAPM, based on required returns, not a line fit to past realized returns.

Diversification removes most firm-specific noise, leaving market risk as the main priced risk.

Total volatility includes both market and firm-specific movements, while beta isolates only the market-related part.

Weights determine how much each asset contributes to the portfolio average return and to the portfolio covariance structure.

CAPM is a pricing model for required return, so it gives a benchmark, not a promise about future realized performance.

A bond pays coupons and principal in the future, so today’s value is the discounted value of those promised payments.

Yield is the discount rate. A higher discount rate lowers present value, and a lower discount rate raises it.

When the bond’s coupon rate matches the required market return, the present value of the cash flows equals par value.

Its coupon payments are more attractive than current market rates, so investors pay more than par.

Its coupon payments are less attractive than current market rates, so the bond must sell below par to offer the required return.

These three measures use different bases, so they answer different questions about the bond.

The investor buys below par and earns the return from the bond moving up to face value by maturity.

The bond pays twice per year, so both the cash flow timing and the discounting frequency must be adjusted.

Cash flows that arrive later are more sensitive to discount-rate changes.

With lower coupons, more of the bond’s value is tied to the distant principal payment, which is more rate-sensitive.

As time passes, there are fewer remaining cash flows, so the price converges to the amount paid at maturity.

Coupon rate tells the annual percentage, and coupon payment converts that rate into an actual dollar cash flow.

Current yield uses only annual coupon divided by current price, while YTM includes both coupon income and price convergence to par.

The same coupon dollars are divided by a price above par for a premium bond and below par for a discount bond.

Since there are no interim coupon cash flows, there is nothing to reinvest before maturity.

Fair pricing comes from discounting promised cash flows at the market-required rate for comparable risk.

Once the annual inputs are converted to period inputs correctly, the problem is just a standard present value calculation.

Longer maturity stretches cash flows farther out, making present value more responsive to rate changes.

The promised cash flows stay the same, but the rate used to value them changes in the market.

At maturity the issuer repays face value, so temporary price deviations disappear by the final payment date.

Valuation always links price today to the present value of future cash paid to shareholders.

The model values future cash flows, so it must start with the next dividend to be received.

That perpetual constant-growth assumption is what lets the infinite dividend stream collapse into a simple formula.

The denominator is r − g, so required return must exceed growth for the present value series to converge.

Higher expected growth increases future dividends and therefore increases present value.

A higher required return means heavier discounting, which lowers present value.

In the constant-growth model, total expected return is the cash yield plus expected price growth.

If the same dividend continues forever, the present value formula is the perpetuity formula.

The DDM works best when dividends are meaningful, predictable, and tied to long-run shareholder payouts.

Multi-stage valuation separates the unusual early years from the long-run steady stage.

In the Gordon model the dividend is in the numerator, so a bigger next dividend directly raises price.

Higher required return means a larger discount rate, which reduces present value.

When the denominator becomes very small, even tiny input changes create large valuation swings.

Equity holders own the whole future stream of payouts, so value must reflect the entire expected stream.

Value can still come from future dividends or other eventual shareholder payouts even if current dividends are zero.

If dividends and price are expected to grow at the same constant rate, the expected price appreciation rate is g.

It is the discount rate appropriate for the uncertainty of that stock’s future cash flows.

A high dividend yield may reflect low growth or higher perceived risk, so it is only one piece of valuation.

The formula uses next period’s dividend, so the timing of the dividend input matters.

Growth affects both the numerator and denominator in constant-growth valuation, making estimates very sensitive.