Finance: Present Value, Yield Curves, Bonds, and CAPM

Present Value

Present value decreases because higher rates reduce current worth.

Effective Annual Rate (EAR)

More compounding periods per year → higher EAR.

Formula for EAR

EAR=(1+r/m)^m - 1

Comparing Loans

Use the effective annual rate (EAR).

Effective Rate from Nominal Rate

It's slightly higher than 8% due to monthly compounding.

Nominal Interest Rate Formula

r=r* + IP + DRP + LP + MRP

Yield Curve and Inflation Expectations

It becomes upward sloping as long-term rates increase.

Upward-Sloping Yield Curve

Investors expect higher inflation and/or higher real rates in the future.

Lowest Risk Premium Bonds

Short-term U.S. Treasury securities (T-bills).

Default and Liquidity Premiums

They raise yields above comparable Treasury bonds.

Interest Rates and Bond Prices

Inverse — as rates rise, prices fall.

Reinvestment-Rate Risk

Risk that coupon payments must be reinvested at lower rates.

Callable Bonds and Falling Rates

They are more likely to be called by the issuer.

Bond Price and Discount Rate

Price increases.

Bonds Selling at Premium

When coupon rate > current yield or YTM.

Bonds Selling at Discount

When coupon rate < current yield or YTM.

Bond Prices and Par

Because market interest rates are constantly changing.

Call Provision

Issuer can repurchase bonds early, usually when interest rates drop.

Dirty Price vs. Clean Price

Dirty = Clean + Accrued Interest.

Total (Holding-Period) Return

Formula for total (holding-period) return?

Standard deviation (σ)

Total volatility or stand-alone risk.

Coefficient of variation (CV) formula

CV=σ/r̄

Beta (β)

Measures systematic (market) risk; sensitivity to market returns.

CAPM equation

r=r_F + β(r_M - r_F)

Effect of increased risk aversion on SML

Slope increases because market-risk premium rises.

Risk eliminated by diversification

Unsystematic (firm-specific) risk.

Portfolio beta formula

β_p = ∑w_i β_i

Required return calculation

If β = 1.7, r_F = 5.5%, RPM = 8.2%, required return = 5.5 + 1.7 × 8.2 = 19.44.

Effect of risk aversion on market-risk premium

Higher risk aversion → larger market-risk premium.

Relationship between β and required return

Higher β → higher required return.

Risk comparison in diversified portfolio

β = 1.2 is riskier than 0.9.

Effect of RPM increase on required return

It rises in proportion to each stock's β.

Portfolio beta calculation

When 75% β = 0.9 and 25% β = 1.2, β_p = 0.975.

Portfolio required return calculation

If β_p = 0.975, r_F = 6%, RPM = 5%, then required return = 6 + 0.975 × 5 = 10.875.

Percentage in T-bills for desired β

If the portfolio β = 1.5 and you want β_p = 1, then 33% in T-bills, 67% in the stock.

Constant-growth (Gordon) model

P_0 = D_1 / (r - g)

Calculation of D₁

If D₀ = $1.50 and g = 4%, then D₁ = 1.50 × 1.04 = 1.56.

Price calculation for stock

If r = 14.1% and g = 4%, find P₀ for D₁ = 1.56: P₀ = 1.56 / (0.141 - 0.04) = 15.45.

Stock pricing when expected return > CAPM required return

Underpriced → buy signal.

Stock pricing when expected return < CAPM required return

Overpriced → sell signal.

Effect of g increase on price

Price rises; future dividends grow faster.

Valuation sensitivity when g ≈ r

Price becomes extremely sensitive to small changes in r or g.