Complete Problem Yield To Maturity For Annual Payments XYZ C
Complete Problem Yield To Maturity For Annual Paymentsxyz Corporation
Complete problem: Yield to Maturity for Annual Payments XYZ Corporation’s bonds have 14 years remaining to maturity. Interest is paid annually, the bonds have a $1,000 par value, and the coupon interest rate is 10%. The bonds sell at a price of $950. o What is their yield to maturity? Show your work. Complete problem: Required Rate of Return Suppose rRF = 5%, rM = 10%, and rA = 12%. Calculate Stock A's beta. If Stock A's beta were 2.0, then what would be A's new required rate of return? Show your work. Complete problem: Portfolio Beta You have a $2 million portfolio consisting of a $100,000 investment in each of 20 different stocks. The portfolio has a beta of 1.1. You are considering selling $100,000 worth of one stock with a beta of 0.9 and using the proceeds to purchase another stock with a beta of 1.4. o What will the portfolio’s new beta be after these transactions? Show your work. Prepare this Assignment as a Word® document. List each question, followed by your answer.
Paper For Above instruction
Introduction
Understanding key concepts in bond valuation, required rate of return, beta calculations, and portfolio management is fundamental in finance. This paper addresses three core problems: (1) calculating the yield to maturity (YTM) of a bond, (2) determining the required rate of return based on beta and other parameters, and (3) computing the impact on portfolio beta when adjusting holdings. Each problem is explored methodically, demonstrating essential financial formulas and reasoning.
Problem 1: Yield to Maturity (YTM) Calculation
XYZ Corporation’s bonds have 14 years remaining to maturity, a par value of $1,000, a coupon rate of 10%, and they are currently priced at $950. The goal is to find their YTM, which is the internal rate of return (IRR) of the bond’s cash flows.
The bond’s annual coupon payment (C) can be calculated as:
C = Par value × Coupon rate = $1,000 × 10% = $100
The bond pricing formula reflects the present value (PV) of future cash flows:
Price = \(\sum_{t=1}^{n} \frac{C}{(1 + y)^t} + \frac{FV}{(1 + y)^n}\)
where:
- \(PV = \$950\),
- \(C = \$100\),
- \(FV = \$1,000\),
- \(n = 14\),
- \(y =\) YTM (annual).
This formula is solved iteratively or via financial calculator or spreadsheet functions to find the YTM.
Using a financial calculator or Excel’s RATE function, the approximate YTM calculation is as follows:
- N = 14,
- PV = -950 (cash inflow but entered as negative),
- PMT = 100,
- FV = 1000.
In Excel:
=RATE(14,100,-950,1000)
This yields an approximate YTM of about 11.10%. This means the bond's annual return if held to maturity, given current price and cash flows, is approximately 11.10%.
Problem 2: Calculating Stock A's Beta and New Required Rate of Return
Given the risk-free rate \(r_{RF} = 5\%\), the market return \(r_{M} = 10\%\), and the asset's expected return \(r_A = 12\%\), the beta of Stock A can be derived using the Capital Asset Pricing Model (CAPM) formula:
\(r_A = r_{RF} + \beta (r_M - r_{RF})\)
Rearranged for beta:
\(\beta = \frac{r_A - r_{RF}}{r_M - r_{RF}}\)
Plugging in the values:
\(\beta = \frac{12\% - 5\%}{10\% - 5\%} = \frac{7\%}{5\%} = 1.4\)
Thus, Stock A’s beta is 1.4.
If Stock A’s beta were 2.0, its new required rate of return (\(r_{A,new}\)) would be:
\(r_{A,new} = r_{RF} + \beta_{new} (r_M - r_{RF})\)
Calculating:
\(r_{A,new} = 5\% + 2.0 \times (10\% - 5\%) = 5\% + 2.0 \times 5\% = 5\% + 10\% = 15\%\)
Therefore, with a beta of 2.0, the asset’s new required rate of return would be 15%.
Problem 3: Portfolio Beta Adjustment
The initial portfolio is valued at $2 million with equal investments of $100,000 in each of 20 stocks, and the total beta is 1.1.
The overall portfolio beta can be expressed as:
\(\beta_{portfolio} = \sum_{i=1}^{n} w_i \beta_i\)
where \(w_i\) are the weights of each stock in the portfolio.
Initially:
- Each stock weight: \(w = \frac{100,000}{2,000,000} = 0.05\),
- Total beta: 1.1.
Now, the investor considers selling $100,000 of a stock with beta 0.9 and purchasing an equivalent value of a stock with beta 1.4.
The new weights:
- Remaining investment in first stock: \(w_{original} = 0.05\),
- Sale reduces the total investment in that stock from $100,000 to $0,
- Purchase adds $100,000 in a stock with beta 1.4.
Calculating the new portfolio beta:
\(\beta_{new} = \frac{\text{Total Beta Dollars}}{\text{Total Portfolio Value}}\)
or, more precisely, the weighted sum:
\(\beta_{new} = \left(\text{Remaining stocks}\times \text{weight}\times \text{beta}\right) + \left(\text{New stock weight}\times \text{beta}\right)\)
The change affects the overall beta as:
- Removing $100,000 of beta contribution of stock with beta 0.9:
\(0.05 \times 0.9 = 0.045\),
- Adding $100,000 of stock with beta 1.4:
\(0.05 \times 1.4 = 0.07\),
- The net change:
New total beta = previous total beta \( - 0.045 + 0.07\).
Accounting for the full portfolio:
- Total portfolio value remains at $2 million,
- The proportional change yields:
\[
\beta_{new} = \frac{(1.1 \times 2,000,000) - 100,000 \times 0.9 + 100,000 \times 1.4}{2,000,000}
\]
Calculating numerator:
\[
(1.1 \times 2,000,000) = 2,200,000
\]
Subtracting the beta contribution of the sold stock:
\[
2,200,000 - 100,000 \times 0.9 = 2,200,000 - 90,000 = 2,110,000
\]
Adding the beta contribution of the purchased stock:
\[
2,110,000 + 100,000 \times 1.4 = 2,110,000 + 140,000 = 2,250,000
\]
Finally, dividing by total portfolio:
\[
\beta_{new} = \frac{2,250,000}{2,000,000} = 1.125
\]
Thus, after the transaction, the portfolio’s beta increases from 1.1 to approximately 1.125.
Conclusion
This analysis demonstrates the application of fundamental financial concepts—bond valuation via YTM calculation, CAPM-based risk and return estimation, and portfolio beta adjustments. Mastery of these tools enables financial analysts and investors to make informed decisions balancing risk and return across different investment instruments.
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