Financial Calculations: Bond Valuation & Stock Risk Analysis

Bond Valuation and Interest Rate Premiums

Calculating Bond Rates and Premiums

1. Long-Term Treasury Bond Rate Calculation

You read in The Wall Street Journal that 30-day T-bills are currently yielding 2%. Your brother-in-law, a broker at Kyoto Securities, has provided the following estimates of current interest rate premiums on a 1-year bond:

• Liquidity Premium: 3%
• Maturity Risk Premium (MRP): 1.5%
• Default Risk Premium (DRP): 1.2%

Based on these data, what is the long-term Treasury bond rate?

For a Treasury bond, the Liquidity Premium and Default Risk Premium are typically not applicable. The 30-day T-bill yield (2%) can be considered the short-term risk-free rate plus inflation premium (r* + IP). To find the long-term Treasury bond rate, we add the Maturity Risk Premium.

rLTT = r* + IP + MRP

rLTT = 2% + 1.5% = 3.5%

2. Corporate Bond Default Risk Premium

A Treasury bond that matures in 10 years has a yield of 4.5%. A 10-year corporate bond has a yield of 6%. Assume that the liquidity premium on the corporate bond is 0.6%.

What is the default risk premium on the corporate bond?

The yield on a corporate bond includes the Treasury bond yield, plus liquidity and default risk premiums.

rLTC = rLTT + LP + DRP

Given:

• rLTC (10-year Corporate Bond Yield) = 6%
• rLTT (10-year Treasury Bond Yield) = 4.5%
• LP (Liquidity Premium) = 0.6%

Substituting the values:

6% = 4.5% + 0.6% + DRP

6% = 5.1% + DRP

DRP = 0.9%

3. Short-Term Corporate Bond Rate

You read in The Wall Street Journal that 30-day T-bills are currently yielding 0.5%. Your brother-in-law, a broker at Kyoto Securities, has provided the following estimates of current interest rate premiums on a 1-year bond:

• Liquidity Premium: 0.5%
• Maturity Risk Premium: 0.5%
• Default Risk Premium: 1%

Based on these data, what is the short-term corporate bond rate?

For a short-term corporate bond, the rate includes the short-term risk-free rate (T-bill yield), plus liquidity and default risk premiums. The maturity risk premium is often considered negligible for very short-term bonds or is not applied in this specific calculation context.

rSTC = r* + IP + LP + DRP

rSTC = 0.5% + 0.5% + 1% = 2%

4. Long-Term Corporate Bond Rate

You read in The Wall Street Journal that 30-day T-bills are currently yielding 1%. Your brother-in-law, a broker at Kyoto Securities, has provided the following estimates of current interest rate premiums on a 10-year bond:

• Liquidity Premium: 1%
• Maturity Risk Premium: 1%
• Default Risk Premium: 2%

Based on these data, what is the long-term corporate bond rate?

The long-term corporate bond rate includes the short-term risk-free rate (T-bill yield), plus maturity, liquidity, and default risk premiums.

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

rLTC = 1% + 1% + 1% + 2% = 5%

Bond Pricing and Yield Calculations

5. Price of an Annual Coupon Bond

The Carter Company's bonds mature in 6 years, have a par value of $1,000, and an annual coupon payment of $70. The yield to maturity (YTM) for the bonds is 9%. What is the price of these bonds?

Using a financial calculator or bond pricing formula:

• N (Number of periods) = 6
• I/Y (Yield to Maturity) = 9%
• PMT (Annual Coupon Payment) = $70
• FV (Face Value) = $1,000

PV (Present Value/Price) = $910.28

6. Price of a Semiannual Coupon Bond

The Carter Company's bonds mature in 8 years, have a par value of $1,000, and a semiannual coupon payment of $60. The yield to maturity (YTM) for the bonds is 7%. What is the price of these bonds?

For semiannual payments, adjust the number of periods, interest rate, and payment accordingly:

• N (Number of periods) = 8 years * 2 = 16
• I/Y (Semiannual YTM) = 7% / 2 = 3.5%
• PMT (Semiannual Coupon Payment) = $60
• FV (Face Value) = $1,000

PV (Present Value/Price) = $1,302.35

7. Yield to Maturity (YTM) for a Corporate Bond

A corporate bond has a face value of $1,000 and an 8% semiannual coupon. The bond matures in 8 years and sells at a price of $1,090. What is the bond’s yield to maturity?

Using a financial calculator:

• N (Number of periods) = 8 years * 2 = 16
• PV (Present Value/Price) = -$1,090 (negative as it's an outflow)
• PMT (Semiannual Coupon Payment) = ($1,000 * 0.08) / 2 = $40
• FV (Face Value) = $1,000

I/Y (Semiannual Yield) = 3.27%

YTM (Annual Yield to Maturity) = 3.27% * 2 = 6.54%

8. Yield to Call (YTC) for a Corporate Bond

Consider the same bond from the previous question (Question 7). The bond can be called in 4 years at a call price of $1,040. What is the bond’s yield to call?

Using a financial calculator, with adjusted call period and call price:

• N (Number of periods to call) = 4 years * 2 = 8
• PV (Present Value/Price) = -$1,090
• PMT (Semiannual Coupon Payment) = $40
• FV (Call Price) = $1,040

I/Y (Semiannual Yield to Call) = 3.16%

YTC (Annual Yield to Call) = 3.16% * 2 = 6.32%

Stock Valuation and Portfolio Risk Analysis

Stock Return and Risk Metrics

1. Expected Return, Standard Deviation, and Coefficient of Variation

Roenfeld Corp believes the following probability distribution exists for its stock. What is the expected return, standard deviation, and coefficient of variation on the company's stock?

Expected Return (E(r)) Calculation:

E(r) = Σ (Probability * Return)

E(r) = 0.40(30%) + 0.50(12%) + 0.10(-10%)

E(r) = 12% + 6% - 1% = 17%

Variance (σ²) and Standard Deviation (σ) Calculation:

σ2 = Σ [Probability * (Return - E(r))2]

σ2 = 0.40(30% - 17%)2 + 0.50(12% - 17%)2 + 0.10(-10% - 17%)2

σ2 = 0.40(13%)2 + 0.50(-5%)2 + 0.10(-27%)2

σ2 = 0.40(169) + 0.50(25) + 0.10(729)

σ2 = 67.6 + 12.5 + 72.9 = 153

σ = √153 ≈ 12.37%

Coefficient of Variation (CV) Calculation:

CV = σ / E(r)

CV = 12.37% / 17% ≈ 0.73

Capital Asset Pricing Model (CAPM) and Portfolio Beta

2. Return on Market and Required Rate of Return (CAPM)

Cooley Company's stock has a beta of 1.20, the risk-free rate is 3%, and the market risk premium is 6%. What is the return on the market? What is the firm's required rate of return?

Return on Market (r_m) Calculation:

Market Risk Premium (MRP) = rm - rf

6% = rm - 3%

rm = 9%

Required Rate of Return (r) Calculation:

r = rf + Beta * MRP

r = 3% + 1.20(6%) = 3% + 7.2% = 10.2%

3. Portfolio Beta Calculation

Mike Flannery holds the following portfolio:

What is his portfolio’s beta?

First, calculate the total investment and the weight of each stock:

• Total Investment = $75,000 + $30,000 = $105,000
• Weight of Stock A (w_A) = $75,000 / $105,000 ≈ 0.7143
• Weight of Stock B (w_B) = $30,000 / $105,000 ≈ 0.2857

Portfolio Beta (Bp) = (wA * BA) + (wB * BB)

Bp = (75,000 / 105,000)(1.40) + (30,000 / 105,000)(0.80)

Bp = 1.00 + 0.23 = 1.23

4. Required Rate of Return with CAPM

Assume that the risk-free rate is 2% and the expected return on the market is 12%. What is the required rate of return on a stock with a beta of 1.1?

r = rf + Beta * (rm - rf)

r = 2% + 1.1(12% - 2%)

r = 2% + 1.1(10%)

r = 2% + 11% = 13%

5. Stock Beta and Impact of Market Risk Premium Change

A stock has a required return of 11%, the risk-free rate is 2.5%, and the market risk premium is 8%. What is the stock’s beta? If the market risk premium increased to 10%, what would happen to the stock’s required rate of return? Assume that the risk-free rate and the beta remain unchanged.

Calculate Stock's Beta:

Using the CAPM formula:

r = rf + Beta * MRP

11% = 2.5% + Beta * 8%

8.5% = Beta * 8%

Beta = 8.5% / 8% = 1.0625

Beta ≈ 1.06

Calculate New Required Rate of Return:

If the market risk premium increases to 10% (with rf = 2.5% and Beta = 1.06):

rnew = rf + Beta * MRPnew

rnew = 2.5% + 1.06 * 10%

rnew = 2.5% + 10.6% = 13.1%