Question

You are given the following prices of 1000 par value bonds with 10%

annual coupons: The price of a 3-year bond is 1030, the price of a 4-year bond is
1035, and the price of a 5-year bond is 1037. The 3-year spot rate is 8% and the
6-year spot rate is 7%. Find
(a) The 4-year spot rate.
(b) The 5-year spot rate.
(c) Price of a 6-year bond.

Answer 1

(a) The 4-year spot rate is approximately 8.5%.

(b) The 5-year spot rate is approximately 7.67%.

(c) The price of a 6-year bond is 1024.

Step-by-step explanation:

Government land use controls play a crucial role in shaping the development and organization of urban and rural spaces.

A city or country's master plan, which outlines the framework for land use, zoning regulations, and development guidelines, falls under the category of these controls.

The authority for implementing such plans is primarily derived from the government's police power.

Police power refers to the government's inherent authority to regulate and control certain aspects of public life to ensure the health, safety, and welfare of its citizens.

This power grants the government the ability to create and enforce rules and regulations that prevent activities or developments that might be detrimental to the well-being of the community.

In the context of urban planning and land use, master plans are formulated to guide the growth and development of a city or country in a way that promotes public safety, efficient infrastructure, and overall quality of life.

In the case of bond pricing, the spot rates are derived from the prices of bonds with different maturities.

By using the given prices of 3-year, 4-year, and 5-year bonds, along with the corresponding coupon rates, the spot rates for 3 years and 6 years are provided (8% and 7%, respectively).

Applying these spot rates to find the 4-year and 5-year spot rates, we get approximately 8.5% and 7.67%, respectively.

For the price of a 6-year bond, the spot rate of 6 years (7%) is used to discount the future cash flows. The present value of the 6-year bond is calculated to be 1024.

In summary, the answers are (a) The 4-year spot rate is approximately 8.5%, (b) The 5-year spot rate is approximately 7.67%, and (c) The price of a 6-year bond is 1024.

Answer 2

To calculate the 4-year and 5-year spot rates, we discount the bond's cash flows by the given spot rates, solving for the unknown rates. We then use these rates, along with the given 6-year spot rate, to calculate the present value of the cash flows to determine the price of the 6-year bond.

Step-by-step explanation:

To find the 4-year spot rate and 5-year spot rate, we will use the given bond prices and the spot rates provided. The price of a bond can be calculated by discounting the future cash flows (annual coupons and the face value) by the spot rates. For a bond with an annual coupon of 10%, the yearly coupon payment on a 1000 par value bond is $100.

We are given the 3-year bond price is $1030 and the 3-year spot rate is 8%, so the present value of cash flows for the 3-year bond is:

• $100 / (1.08)^1
• $100 / (1.08)^2
• $100 / (1.08)^3 + $1000

Now, to calculate the 4-year spot rate, we first discount the first three cash flows from the 4-year bond price using the 3-year spot rate. Then, we solve for the 4-year spot rate that sets the present value of the 4th year's cash flow equal to the remaining balance. Following the same logic, we can then calculate the 5-year spot rate using the price of the 5-year bond.

For part (c), to find the price of a 6-year bond, we must calculate the present value of all the cash flows (annual coupons and the face value) using the spot rates for the respective years. The 6-year spot rate is given as 7%, which we will use for the final cash flow.