Question

5. Consider a 4-year zero-coupon bond priced such that its YTM is 7% per year. Assume the face value is $1000.

a. Determine the dollar price of the bond. (Enter the dollar price of the bond, such as 876.25 (without the dollar sign)). Do not enter the 32^nd conventional equivalent price.

b. Now assume that one year later interest rates have fallen and now the bond has a YTM of 5% (down from 7% one year earlier). What is the dollar price of the bond now?
HINT: The price is not equal to $822.70.

c. If you purchased the bond for price computed in Part A and sold it one year later for the price computed in Part B, then what is the holding period yield? i.e. what is the holding period yield from t = 0 to t = 1?
Enter your answer as a decimal, not as a percent.

d. With only one year remaining on the bond (i.e. at t = 3), the YTM on the bond is 10%. Compute the annualized effective holding period yield assuming you hold the bond from t = 1 (when it was, as described in Part B) to t = 3. Notice that this rate spans the two-year period from t = 1 to t = 3 but should be expressed as an effective annual value (in decimal form).

Enter the effective annualized holding period yield as a decimal, not as a percent, with at least 4 digits of precision.

Answer 1

a. Determine the dollar price of the bond.

Formula for calculation of YTM

YTM = (FV/MV)^1/n - 1

variable definitions:

• YTM = yield to maturity, as a decimal (multiply it by 100 to convert it to percent)
• FV = maturity value
• MV = price
• n = years until maturity

0.07 = (1000/MV)^1/4-1

1.07= (1000/MV)^1/4

(1.07)^{^4} = [(1000/MV)^1/4]^{^4}

1.3108 = 1000/MV

MV = 1000/1.3108

MV = 762.89

b. Now assume that one year later interest rates have fallen and now the bond has a YTM of 5% (down from 7% one year earlier). What is the dollar price of the bond now?

Formula for calculation of YTM

YTM = (FV/MV)^1/n - 1

variable definitions:

• YTM = yield to maturity, as a decimal (multiply it by 100 to convert it to percent)
• FV = maturity value
• MV = price
• n = years until maturity

0.05 = (1000/MV)^1/3-1

1.05= (1000/MV)^1/3

(1.05)^{^3} = [(1000/MV)^1/3]^{^3}

1.1576 = 1000/MV

MV = 1000/1.1576

MV = 863.86

c. If you purchased the bond for price computed in Part A and sold it one year later for the price computed in Part B, then what is the holding period yield? i.e. what is the holding period yield from t = 0 to t = 1?

Holding period return of Zero coupon bond

=   MV_{1 –} MV₀

          MV₀

= 863.86 - 762.89

         762.89

= 100.97

     762.89

= 0.1324

d. With only one year remaining on the bond (i.e. at t = 3), the YTM on the bond is 10%. Compute the annualized effective holding period yield assuming you hold the bond from t = 1 (when it was, as described in Part B) to t = 3. Notice that this rate spans the two-year period from t = 1 to t = 3 but should be expressed as an effective annual value (in decimal form).

Holding period return of Zero coupon bond for the t1 to t3 period

=   MV_{1 –} MV₀

          MV₀

= 909.09 - 863.86

         863.86

= 45.23

    863.86

= 0.0524

Annual effective holding period return

= ( 1 + total holding period return)^1/n - 1

= ( 1 + 0.0524)^1/2 – 1

= (1.0524) ^1/2 – 1

= 1.0258 – 1

= 0.0258