Financial Calculations: Yields, Index Returns, and Security Pricing

Municipal Bond Yield Calculations

Equivalent Taxable Yield (ETY)

A municipal bond carries a coupon rate of 6.00% and is trading at par.

Required Calculation 1: ETY for a 38% Tax Bracket

What would be the equivalent taxable yield of this bond to a taxpayer in a 38% combined tax bracket?

The formula for Equivalent Taxable Yield ($r$) is:

$$r = r_m / (1 - t)$$

• $r$: Equivalent Taxable Yield
• $r_m$: Municipal Bond Yield (6.00% or 0.06)
• $t$: Tax Rate (38% or 0.38)

Calculation:

$$0.06 / (1 - 0.38) = 0.096774 \rightarrow \mathbf{9.68\%}$$

Required Calculation 2: Municipal Yield Preference

An investor is in a 40% combined federal plus state tax bracket. If corporate bonds offer 7.75% yields, what yield must municipals offer for the investor to prefer them to corporate bonds?

The required municipal yield ($r_m$) is calculated as:

$$r_m = r \times (1 - t)$$

Calculation:

$$r_m = 0.0775 \times (1 - 0.4) = 0.0465 \rightarrow \mathbf{4.65\%}$$

Required Calculation 3: ETY Across Different Tax Brackets

Find the equivalent taxable yield of the municipal bond for tax brackets of zero, 10%, 20%, and 30%, if it offers a yield of 3.20%.

• 0% Tax Bracket: $0.032 / (1 - 0) = \mathbf{3.2\%}$
• 10% Tax Bracket: $0.032 / (1 - 0.1) \approx 0.0356 \rightarrow \mathbf{3.56\%}$
• 20% Tax Bracket: $0.032 / (1 - 0.2) = \mathbf{4.0\%}$
• 30% Tax Bracket: $0.032 / (1 - 0.3) \approx 0.0457 \rightarrow \mathbf{4.57\%}$

Price-Weighted Index Analysis

Consider the three stocks in the following table. $P_t$ represents price at time t, and $Q_t$ represents shares outstanding at time t. Stock C splits two-for-one in the last period ($t=2$).

Price-Weighted Index Requirements

1. Calculate the Rate of Return for Period 1 (t=0 to t=1)

   Sum of Prices at $t=0$ ($P_0$): $82 + 42 + 84 = 208$

   Sum of Prices at $t=1$ ($P_1$): $87 + 37 + 94 = 218$

   Rate of Return:

   $$\frac{P_1 - P_0}{P_0} = \frac{218 - 208}{208} \approx 0.0480769 \rightarrow \mathbf{4.81\%}$$

2. Determine the Divisor for the Price-Weighted Index in Year 2

   Stock C splits two-for-one at $t=2$. The price of C adjusts from $94$ to $47$. We must find the new divisor ($D_2$) such that the index value remains constant immediately before and after the split (at $t=1$).

   Sum of Prices before split (using $P_1$): $218$ (Divisor $D_1 = 3$)

   Sum of Prices adjusted for split: $87 + 37 + 47 = 171$

   Index Value Equality:

   $$\frac{218}{3} = \frac{171}{D_2}$$

   $$D_2 = \frac{171 \times 3}{218} \approx \mathbf{2.3532}$$

3. Calculate the Rate of Return for Period 2 (t=1 to t=2)

   Index Value at $t=1$ (Index 1): $218 / 3 \approx 72.6667$

   Index Value at $t=2$ (Index 2): $171 / 2.3532 \approx 72.6667$

   Rate of Return:

   $$\frac{\text{Index}_2 - \text{Index}_1}{\text{Index}_1} = \frac{72.6667 - 72.6667}{72.6667} = \mathbf{0\%}$$

Market-Weighted and Equally Weighted Index Returns

Consider the three stocks in the following table. $P_t$ represents price at time t, and $Q_t$ represents shares outstanding at time t. Stock C splits two-for-one in the last period.

Required: Calculate First-Period Rates of Return (t=0 to t=1)

1. Market Value–Weighted Index

   Market Value at $t=0$ ($MV_0$):

   • Stock A: $88 \times 100 = 8,800$
   • Stock B: $48 \times 200 = 9,600$
   • Stock C: $96 \times 200 = 19,200$

   Total $MV_0 = 8,800 + 9,600 + 19,200 = \mathbf{37,600}$

   Market Value at $t=1$ ($MV_1$):

   • Stock A: $93 \times 100 = 9,300$
   • Stock B: $43 \times 200 = 8,600$
   • Stock C: $106 \times 200 = 21,200$

   Total $MV_1 = 9,300 + 8,600 + 21,200 = \mathbf{39,100}$

   Return (0 $\rightarrow$ 1):

   $$\frac{39,100 - 37,600}{37,600} \approx 0.039893 \rightarrow \mathbf{3.99\%}$$

2. Equally Weighted Index

   Calculate the return for each stock individually:

   • Stock A Return: $(93 - 88) / 88 \approx 0.056818$
   • Stock B Return: $(43 - 48) / 48 \approx -0.104167$
   • Stock C Return: $(106 - 96) / 96 \approx 0.104167$

   Equally Weighted Return (Average of Returns):

   $$\frac{1}{3} \times (0.056818 + (-0.104167) + 0.104167) \approx 0.018939 \rightarrow \mathbf{1.89\%}$$

Treasury Bill Pricing and Yield

A T-bill has a face value of $10,000 and 85 days to maturity, selling at a bank discount ask yield ($Y_d$) of 3.2%.

1. What is the Price of the Bill?

   First, calculate the dollar discount ($D$):

   $$D = \text{Face Value} \times Y_d \times \frac{\text{Days to Maturity}}{360}$$

   $$D = 10,000 \times 0.032 \times \frac{85}{360} \approx \$75.56$$

   Price is Face Value minus Discount:

   $$\text{Price} = 10,000 - 75.5555 = \mathbf{\$9,924.44}$$

2. What is its Bond Equivalent Yield (BEY)?

   The BEY formula uses the actual price and a 365-day year convention:

   $$\text{BEY} = \frac{\text{Face Value} - \text{Price}}{\text{Price}} \times \frac{365}{\text{Days to Maturity}}$$

   $$\text{BEY} = \frac{10,000 - 9,924.4444}{9,924.4444} \times \frac{365}{85} \approx 0.0326955 \rightarrow \mathbf{3.27\%}$$

Security Pricing Comparisons

Determine which security should sell at a greater price, assuming all other relevant features are identical.

1. Treasury Bond Coupon Rate Comparison

   An 8-year Treasury bond with a 10.25% coupon rate or an 8-year Treasury bond with a 11.25% coupon?

   Answer: An 8-year Treasury bond with a 11.25% coupon.

   Reason: A bond that offers a higher periodic coupon payment generates a greater stream of cash flows, resulting in a higher present value (price).

2. Call Option Exercise Price Comparison

   A four-month expiration call option with an exercise price of $45 or a four-month call on the same stock with an exercise price of $40?

   Answer: A four-month call on the same stock with an exercise price of $40.

   Reason: A call option grants the holder the right to buy the stock. A lower exercise price allows the holder to acquire the stock for less money, making the option inherently more valuable.

3. Put Option Stock Price Comparison

   A put option on a stock selling at $55 or a put option on another stock selling at $65?

   Answer: A put option on a stock selling at $55.

   Reason: A put option grants the holder the right to sell the stock at the strike price. A put option becomes more valuable as the underlying stock price drops (assuming the strike price is identical). Therefore, the option on the stock currently trading at the lower price ($55) is more valuable.