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Chapter 7: Problem 1

Suppose you observe the following zero-coupon bond prices per \(\$ 1\) of maturity payment: \(0.96154(1-\text { year ) }, 0.91573 \text { (2-year), } 0.87630 \text { (3-year), } 0.82270(4-\text { year ) }\) 0.77611 (5-year). For cach maturity year compute the zero-coupon bond yields (effective annual and continuously compounded), the par coupon rate, and the I-ycar implied forward rate.

Short Answer

Expert verified
The solution involves computing the yield and coupon rate for each year, followed by calculating the implied forward rate.

Step by step solution

01

Compute Effective Annual Yield

This is computed by taking the reciprocal of the bond's price and raising it to the power of \(1/n\) (where n is the bond's term), then subtracting 1. i.e., \(Yield = (1/Price) ^ (1/n) - 1\) . Repeat this for each year with the provided bond prices.
02

Compute Continuously Compounded Yield

This is calculated by taking natural logarithm of the reciprocal of the bond's price divided by the term. i.e., \(Yield = ln(1/Price) / n\). Repeat this for each year with the provided bond prices.
03

Compute Par Coupon Rates

Par coupon rate is the yield that makes the present value of the bond's payments equal to 1. It can be found by setting the sum of the present value of unit coupon and present value of the face value (which is 1, here) equal to the bond's price, and then iterating to find the yield that satisfies the equation.
04

Compute 1-Year Implied Forward Rates

These can be calculated by using the relation between successive bond prices. An implied forward rate between years \(n-1\) and \(n\) can be computed by \((P_{n-1} / P_n) - 1 \), where \(P_n\) is the price of an \(n\)-year bond.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Effective Annual Yield
Understanding the effective annual yield of a zero-coupon bond is essential as it gives investors an idea of the return they can expect to earn when holding a bond for a year. It represents the actual interest rate an investor earns in a year after accounting for the effects of compounding.

For computing the effective annual yield, use the formula: \[ Y_{\text{effective}} = \bigg(\frac{1}{P}\bigg)^{\frac{1}{n}} - 1 \
Continuously Compounded Yield
The continuously compounded yield is a measure of return if the interest were compounded continuously rather than at discrete intervals. It provides an insight into the growth rate of an investment assuming that it is always earning interest on top of interest.

To find this yield for a zero-coupon bond, apply the natural logarithm to the reciprocal of the bond's price and then divide by \( n \), the term to maturity. This is expressed with the formula: \[ Y_{\text{cc}} = \frac{\ln(\frac{1}{P})}{n} \
Par Coupon Rate
The par coupon rate is the interest rate that will make the present value of a bond's cash flows (its coupon payments and return of principal at maturity) equal to its current market price. For a zero-coupon bond that pays no coupons, this rate would be the yield to maturity.

The concept can be a bit abstract, but essentially, the par coupon rate is the rate at which the invested funds would grow if they were invested at a consistent compounded rate. To find it, investors often use an iterative process where they guess a rate and check to see if it makes the sum of the discounted cash flows equal the bond's price, adjusting as needed. Mathematically, it requires solving for \( r \) in the equation \[ P = \frac{1}{(1+r)^{n}} \
1-Year Implied Forward Rate
The 1-year implied forward rate helps investors estimate the future yield of a bond. It's particularly relevant for forecasting interest rates and making decisions about future investments.

It is deduced from the current yields of several maturities by analyzing the relationship between successive bond prices. To calculate it, take the price of an \( n-1 \) year bond, divide it by the price of an \( n \) year bond, subtract one, and the result represents the 1-year forward rate beginning at \( n-1 \) years. The formula is \[ F_{n-1,n} = \bigg(\frac{P_{n-1}}{P_n}\bigg) - 1 \

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Most popular questions from this chapter

Use the following zero-coupon bond prices to answer the next three questions: $$\begin{array}{cl}\begin{array}{c}\text { Days to } \\\\\text { Maturity }\end{array} & \begin{array}{l}\text { Zero-Coupon } \\\\\text { Bond Price }\end{array} \\\\\hline 90 & 0.99009 \\\180 & 0.97943 \\\270 & 0.96525 \\\360 & 0.95238\end{array}$$ Suppose you are the counterparty for a lender who enters into an FRA to hedge the lending rate on \(\$ 10 \mathrm{m}\) for a 90 -day loan commencing on day \(270 .\) What positions in zero-coupon bonds would you use to hedge the risk on the FRA?

Suppose that in order to hedge interest rate risk on your borrowing, you enter into an FRA that will guarantee a \(6 \%\) effective annual interest rate for 1 year on \(\$ 500,000.00 .\) On the date you borrow the \(\$ 500,000.00,\) the actual interest rate is \(5 \% .\) Determine the dollar settlement of the FRA assuming a. Settlement occurs on the date the loan is initiated. b. Settlement occurs on the date the loan is repaid.

Suppose a 10-year zero coupon bond with a face value of \(\$ 100\) trades at \(\$ 69.20205\) a. What is the yield to maturity and modified duration of the zero-coupon bond? b. Calculate the approximate bond price change for a 50 basis point increase in the yield, based on the modified duration you calculated in parta). Also calculate the exact new bond price based on the new yield to maturity. c. Calculate the convexity of the 10 -year zero-coupon bond. d. Now use the formula (equation 7.15 ) that takes into account both duration and convexity to approximate the new bond price. Compare your result to that in part b).

Use the following zero-coupon bond prices to answer the next three questions: $$\begin{array}{cl}\begin{array}{c}\text { Days to } \\\\\text { Maturity }\end{array} & \begin{array}{l}\text { Zero-Coupon } \\\\\text { Bond Price }\end{array} \\\\\hline 90 & 0.99009 \\\180 & 0.97943 \\\270 & 0.96525 \\\360 & 0.95238\end{array}$$ What is the rate on a synthetic FRA for a 90 -day loan commencing on day \(90 ?\) A 180-day loan commencing on day 90? A 270-day loan commencing on day 90?

A lender plans to invest \(\$ 100 \mathrm{m}\) for 150 days, 60 days from today. (That is, if today is day \(0,\) the loan will be initiated on day 60 and will mature on day \(210 .\) ) The implied forward rate over 150 days, and hence the rate on a 150 -day \(\mathrm{FRA}\), is \(2.5 \% .\) The actual interest rate over that period could be either \(2.2 \%\) or \(2.8 \%\) a. If the interest rate on day 60 is \(2.8 \%\), how much will the lender have to pay if the FRA is settled on day \(60 ?\) How much if it is settled on day \(210 ?\) b. If the interest rate on day 60 is \(2.2 \%\), how much will the lender have to pay if the FRA is settled on day \(60 ?\) How much if it is settled on day \(210 ?\)
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