The Term Structure Of Risk-Free Interest Rates Is Shown

The Term Structure Of Risk Free Interest Rates Is As Shown

Suppose the term structure of risk-free interest rates is as shown below: Term 1 year 2 years 3 years 5 years 7 years 10 years 20 years Rate (EAR, %) 1....93

a. Calculate the present value of an investment that pays $1000 in two years and $2000 in five years for certain.

b. Calculate the present value of receiving $500 per year, with certainty, at the end of the next five years. To find the rates for the missing years in the table, linearly interpolate between the years for which you do know the rates. (For example, the rate in year 4 would be the average of the rate in year 3 and year 5.)

c. Calculate the present value of receiving $2300 per year, with certainty, for the next 20 years. Infer rates for the missing years using linear interpolation. (Hint: Use a spreadsheet.)

31. What is the shape of the yield curve given the term structure in Problem 29? What expectations are investors likely to have about future interest rates?

2. Assume that a bond will make payments every six months as shown on the following timeline (using six-month periods):

a. What is the maturity of the bond (in years)?

b. What is the coupon rate (in percent)?

c. What is the face value?

6. Suppose a 10-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading for a price of $1034.74.

a. What is the bond’s yield to maturity (expressed as an APR with semiannual compounding)?

b. If the bond’s yield to maturity changes to 9% APR, what will the bond’s price be?

7. Suppose a five-year, $1000 bond with annual coupons has a price of $900 and a yield to maturity of 6%. What is the bond’s coupon rate?

10. Suppose a seven-year, $1000 bond with an 8% coupon rate and semiannual coupons is trading with a yield to maturity of 6.75%.

a. Is this bond currently trading at a discount, at par, or at a premium? Explain.

b. If the yield to maturity of the bond rises to 7% (APR with semiannual compounding), what price will the bond trade for?

28. The following table summarizes the yields to maturity on several one-year, zero-coupon securities: Security Yield (%) Treasury 3.1 AAA corporate 3.2 BBB corporate 4.2 B corporate 4.9

a. What is the price (expressed as a percentage of the face value) of a one-year, zero-coupon corporate bond with a AAA rating?

b. What is the credit spread on AAA-rated corporate bonds?

c. What is the credit spread on B-rated corporate bonds?

d. How does the credit spread change with the bond rating? Why?

30. HMK Enterprises would like to raise $10 million to invest in capital expenditures. The company plans to issue five-year bonds with a face value of $1000 and a coupon rate of 6.5% (annual payments). The following table summarizes the yield to maturity for five-year (annual-pay) coupon corporate bonds of various ratings:

1. The figure below shows the one-year return distribution for RCS stock. Calculate:

a. The expected return.

b. The standard deviation of the return.

What does the beta of a stock measure?

35. Suppose the market risk premium is 5% and the risk-free interest rate is 4%. Using the data in Table 10.6, calculate the expected return of investing in:

a. Starbucks’ stock.

b. Hershey’s stock.

c. Autodesk’s stock.

37. Suppose the market risk premium is 6.5% and the risk-free interest rate is 5%. Calculate the cost of capital of investing in a project with a beta of 1.2.

11-2. You own three stocks: 600 shares of Apple Computer, 10,000 shares of Cisco Systems, and 5000 shares of Colgate-Palmolive. The current share prices and expected returns of Apple, Cisco, and Colgate-Palmolive are, respectively, $500, $20, $100 and 12%, 10%, 8%.:

a. What are the portfolio weights of the three stocks in your portfolio?

b. What is the expected return of your portfolio?

c. Suppose the price of Apple stock goes up by $25, Cisco rises by $5, and Colgate-Palmolive falls by $13. What are the new portfolio weights?

d. Assuming the stocks’ expected returns remain the same, what is the expected return of the portfolio at the new prices?

Paper For Above instruction

The term structure of risk-free interest rates is a fundamental concept in finance, reflecting the relationship between interest rates and different maturities. Understanding this structure aids in valuing securities, forecasting future interest rates, and assessing economic expectations. This paper explores the calculation of present values based on a given term structure, interpolation techniques for missing data, analysis of the yield curve, and implications for investor expectations. Additionally, the discussion extends to bond valuation, yield to maturity calculations, and risk premium considerations across varying credit qualities. The comprehensive analysis aims to elucidate the critical role of the term structure in financial decision-making and market predictions.

Given the specified term structure of risk-free interest rates, we start by filling in the missing rates through linear interpolation. If the rates for certain years are not provided, averaging known rates for neighboring maturities provides estimates. For example, the rate for year 4 would be the average of year 3 and year 5 rates. Once the complete set of interest rates is reconstructed, we can compute the present values of different cash flows.

For the first calculation, the present value of an investment paying $1000 in two years and $2000 in five years, with certainty, involves discounting each payoff with the appropriate rate corresponding to its maturity. Mathematically, the present value (PV) is computed as PV = FV / (1 + r)^n, where FV is the future value, r is the annualized interest rate, and n is the number of periods. For example, PV of $1000 received in two years would be $1000 / (1 + r_2)^2, where r_2 is the two-year rate.

Similarly, the valuation of an annuity of $500 per year for five years requires discounting each annual payment. The present value of an annuity is the sum of discounted payments, which can be simplified using the formula for the present value of an annuity if the rate is constant across years. However, since different rates may apply, individual discounting for each payment is necessary, especially when rates differ by year.

For the third case—receiving $2300 annually for 20 years—the process involves interpolating missing rates between known data points to estimate the appropriate discount rates for each year. This ensures a more precise valuation than assuming a constant rate. Using spreadsheet tools significantly simplifies computations, allowing the detailed discounting of each cash flow based on the interpolated rates.

The shape of the yield curve derived from the complete term structure reveals investors’ expectations. An upward-sloping curve suggests expectations of rising future interest rates, potentially due to economic growth or inflation expectations. Conversely, a flat or inverted curve signals expectations of stable or declining rates. The slope, therefore, serves as an indicator of market sentiment about future interest rates and economic conditions.

Bond valuation hinges on calculating the yield to maturity (YTM), which considers the present value of future coupon payments and face value against the current market price. For bonds with semiannual coupon payments, the YTM calculation involves solving for the interest rate in the bond pricing equation, often through iterative methods or calculator functions. Changes in market yields impact bond prices inversely; an increase in YTM leads to a decline in bond prices and vice versa.

Credit spreads reflect the additional yield demanded by investors for bearing credit risk over risk-free securities. These spreads vary with credit ratings, with lower-rated bonds commanding higher spreads due to increased default risk. The spread is calculated as the difference between the yield on a corporate bond and a comparable risk-free treasury security. Empirical data show that spreads tend to widen as credit quality deteriorates, highlighting investor risk perceptions.

When a company like HMK Enterprises considers issuing bonds, it evaluates current yield curves, credit ratings, and bond features to determine optimal pricing. The yields on similar-rated bonds inform the coupon rate and issuance price, influencing the company's cost of capital. A higher YTM increases the cost of borrowing, making it vital for companies to balance desired capital raise against market conditions.

Market risk and stock returns are analyzed through expected return, variance, and beta. The expected return encapsulates the mean of the return distribution, while the standard deviation measures volatility. Beta quantifies systematic risk relative to the overall market; a beta above one indicates higher sensitivity to market movements, while a lower beta suggests less volatility. These measures assist investors in portfolio risk management and performance assessment.

Finally, the capital asset pricing model (CAPM) connects stock expected returns to market risk premium, risk-free rate, and beta. It posits that the expected return on a stock equals the risk-free rate plus the product of the market risk premium and the stock’s beta. For example, with a market risk premium of 5% and a risk-free rate of 4%, a stock with a beta of 1.2 would have an expected return of 4% + 1.2 * 5% = 10%. Such calculations guide investment decisions and portfolio optimization.

In summary, the term structure of risk-free rates is crucial in valuing financial instruments, assessing market expectations, and understanding interest rate dynamics. Interpolating missing data enhances valuation precision. Bond pricing, yield calculations, and credit risk analysis leverage the yield curve and credit spreads, providing market insights. Stock return analyses, beta, and CAPM underscore the importance of systematic risk in investment portfolios. Collectively, these concepts underpin prudent financial decision-making and contribute to efficient capital markets.

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