The following table gives the prices of bonds: Bond principal ( Time to maturity (years)

(a) To calculate the zero rates, we can use the formula:u000Au000A$$u000Au005Ctext{Bond price} u003D u005Cfrac{u005Ctext{Cou

The following table gives the prices of bonds: Bond...

The following table gives the prices of bonds: $$ \begin{array}{cccc} \hline \text { Bond principal }(\$) & \text { Time to maturity (years) } & \text { Annual coupon }^*(\$) & \text { Bond price }(\$) \\ \hline 100 & 0.50 & 0.0 & 98 \\ 100 & 1.00 & 0.0 & 95 \\ 100 & 1.50 & 6.2 & 101 \\ 100 & 2.00 & 8.0 & 104 \\ \hline \end{array} $$ Half the stated coupon is assumed to be paid every six months. (a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months. (b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 months to 24 months? (c) What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual coupon payments? (d) Estimate the price and yield of a 2 -year bond providing a semiannual coupon of $7 \%$ per annum.

The following table gives the prices of bonds:
$$
\begin{array}{cccc}
\hline \text { Bond principal }(\$) & \text { Time to maturity (years) } & \text { Annual coupon }^*(\$) & \text { Bond price }(\$) \\
\hline 100 & 0.50 & 0.0 & 98 \\
100 & 1.00 & 0.0 & 95 \\
100 & 1.50 & 6.2 & 101 \\
100 & 2.00 & 8.0 & 104 \\
\hline
\end{array}
$$
Half the stated coupon is assumed to be paid every six months.
(a) Calculate zero rates for maturities of 6 months, 12 months, 18 months, and 24 months.
(b) What are the forward rates for the following periods: 6 months to 12 months, 12 months to 18 months, and 18 months to 24 months?
(c) What are the 6-month, 12-month, 18-month, and 24-month par yields for bonds that provide semiannual coupon payments?
(d) Estimate the price and yield of a 2 -year bond providing a semiannual coupon of $7 \%$ per annum.

Key Concepts

Zero Rates (Spot Rates)

Zero rates, or spot rates, are the yields on zero?coupon bonds, representing the interest rate for a single cash flow received at a specific future date. They are crucial in valuing any bond because they allow one to discount a cash flow received at a given time without any intermediate payments. The process of bootstrapping uses the prices of coupon and zero-coupon bonds to extract these rates for various maturities.

Coupon Bonds and Yield to Maturity (YTM)

Coupon bonds pay periodic interest payments in addition to the return of principal at maturity. The yield to maturity (YTM) is the internal rate of return on these bonds, equating the present value of all future cash flows (coupons and face value) to the current bond price. Understanding YTM is essential for comparing bonds with different coupon rates and maturities.

Forward Rates

Forward rates are the implied future short-term interest rates calculated from current zero rates. They represent the market's expectation of future interest rates over specific periods and are derived by comparing zero rates across different maturities. Forward rates are key to understanding the term structure of interest rates and for pricing interest rate derivatives.

Par Yields

A par yield is the coupon rate at which a bond is priced at its par value, meaning the bond's price equals its face value. It is determined by equating the present value of the coupon payments and the redemption amount to the bond’s par value using the current zero-coupon yield curve. Par yields are important for benchmarking and constructing the yield curve.

Discount Factors and Compounding Conventions

Discount factors convert future cash flows into their present values using an appropriate interest rate and compounding convention, such as semiannual compounding often used for bonds. They serve as the basic building blocks in bond valuation, linking bond prices with yields and facilitating the bootstrapping process of determining the zero-coupon yield curve.

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