The following is a list of prices today for zero coupon bonds with different mat

Question

The following is a list of prices today for zero coupon bonds with different maturities and par value of $1,000.

Maturity (years) Price

1                     $917.43

2                     $826.51

  3                     $737.96

  4                     $653.06

(a) What is the yield to maturity on the three year zero coupon bond?

(b) According to the expectations theory, what would be the market's expectation of the one year interest rate in the third year?

(c) Assume that you purchased a four year maturity coupon bearing bond with a 10% coupon rate paid annually, with a par value of $1,000. Calculate the price of the bond one year from now if the implied forward rates remain the same.

Explanation / Answer

Sigh, you've already asked this exact same question and you've already received the answer but clearly you do not understand the answer. The issue is interpolating on the bond yield curve. Basically interest rates change over time and in general are expected to increase so bonds are priced based upon what the expected interest rates are believed to be. If we express the interest rates as an annual growth factor i.e.: 1 + rate/100 such that 10% per annum would be R=1.10 then your four prices would mean:

$917.43 * R1 = $1,000
$826.51 * R1 * R2 = $1,000
$737.96 * R1 * R2 * R3 = $1,000
$653.06 * R1 * R2 * R3 * R4 = $1,000

For a), the question is with the three year case, what value of R1, R2 and R3 if they are al equal to each other would satisfy the equation $737.96 * R1 * R2 * R3 = $1,000 and the answer is the third root of $1,000 / $737.96 or 1.1066 which is 10.66%

For b), you want to know what R3 is. This is simple algebra and with the information at hand can be calculated from the two year bond and the three year bond. The three year bond says:

$737.96 * R1 * R2 * R3 = $1,000
.:
R3 = $1,000 / ( R1 * R2 * $737.96 )

The two year bond says

$826.51 * R1 * R2 = $1,000
.:
R1 * R2 = $1,000 / $826.51

Plugging in R1 and R2 from the two year bond into the three year bond and we have:

R3 = $1000 / ( $1,000 / $826.51 * $737.96 )
.:
R3 = 1.1200 or 12.00%

For C, you want to know what price X satisfies the equation:

X * R2 * R3 * R4 = $1,000

From the 4 year bond you know that:

$653.06 * R1 * R2 * R3 * R4 = $1,000
.:
X = $653.06 * R1

From the 1 year bond you know that:

$917.43 * R1 = $1,000
.:
R1 = $1,000 / $917.43

Plugging R1 from the 1 year bond into the four year bond and you get:

X = $653.06 * $1,000 / $917.43
.:
X = $711.84

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