We observe the following annualized yields on four Treasury securities: (75%)
Maturity (years) Yield-to-maturity (%)
0.5 4.00
1 4.50
1.5 5.00
2 5.50
The par is $1000 for all the securities. The one with 0.5-year to mature is a zero coupon bond. Al other securities are coupon-bearing bonds selling at par. Note that, for par bonds, the coupon rate equals YTM. (20 points)
1. Calculate the spot rates for the maturities of 0.5, 1, 1.5, and 2 years.
2. What is the price of a 2-year bond with an 8% annual coupon rate (assume $1000 par)?
3. Suppose a 1-year zero-coupon bond with a par value of S1000 is selling at $900. Is there any arbitrage opportunity? If there is, construct an arbitrage portfolio and show the profit.
4. Calculate the one-period-ahead forward rates from 0 to 0.5, from 0.5 to 1, from 1 to 1.5, and from 1.5 to 2.
5. One year from now, you plan to purchase a then one-year bond with a 1000 par and an 8% annual coupon rate. What is the expected price of the bond? Assume the expectation hypothesis holds. Under the expectation hypothesis, the expected future spot rate equals the forward rate.

Answer:

Explanation:

1.

From the given information;

The spot rate for maturity at 0.5  year [tex](X_1) = 4\%/2 = 2\%[/tex]

The spot rate for maturity at 1 year is:

= [tex]\dfrac{22.5}{(1+X_1)}+ \dfrac{1000 + 22.5}{(1+X_2)^2}=1000[/tex]

= [tex]\dfrac{22.5}{(1+0.02)}+ \dfrac{1000 + 22.5}{(1+X_2)^2}=1000[/tex]

= [tex]\dfrac{22.5}{(1+0.02)}+ \dfrac{1022.5}{(1+X_2)^2}=1000[/tex]

By solving for [tex]X_2[/tex];

[tex]X_2[/tex] = 2.253%

The spot rate for maturity at 1.5 years is:

[tex]= \dfrac{25}{(1+X_1)}+ \dfrac{25}{(1+X_2)^2}+ \dfrac{1000 + 25}{(1+X_3)^3}=1000[/tex]

Solving for [tex]X_3[/tex]

[tex]X_3[/tex] = 2.510%

The spot rate for maturity at 2 years is:

[tex]= \dfrac{27.5}{(1+X_1)}+ \dfrac{27.5}{(1+X_2)^2}+ \dfrac{27.5}{(1+X_3)^3} +\dfrac{1000+27.5}{(1+X_4)^4} =1000[/tex]

By solving for [tex]X_4[/tex];

[tex]X_4[/tex] = 2.770%

Recall that:

Coupon rate = yield to maturity for par bond.

Thus, the annual coupon rates are 4%, 4.5%, 5%, and 5.5% for 0.5, 1, 1.5, 2 years respectively.

2.

For n years, the price of n-bond is:

[tex]= \dfrac{cash \ flow \ at \ year \ 1}{1+X_1}+ \dfrac{cash \ flow \ at \ year \ 2}{(1+X_2)^2}+... + \dfrac{cash \ flow \ at \ year \ b}{(1+X_n)^n}[/tex]

Thus, for 2 years bond implies 4 periods;

∴

[tex]= \dfrac{40}{1+0.02}+ \dfrac{40}{(1+0.02253)^2} + \dfrac{40}{(1+0.0252)^3}+ \dfrac{40}{(1+0.0277)^4}[/tex]

= $1047.024

3.

Suppose there exist no-arbitrage, then the price is:

[tex]= \dfrac{0}{(1+0.02)}+\dfrac{1000}{(1+0.02253)^2}[/tex]

= 956.4183

Since the market price < arbitrage price.

We then consider 0.5, 1-year bonds from the portfolio

Now;

weight 2 × 1000 + weight 2 × 22.5 = 1000

weight 2 × 1022.5 = 1000

weight 2 = 1022.5/1000

weight 2 = 0.976

weight 1 + weight 2 = 1

weight 1 = 1 - weight 2

weight 1 = 1 - 0.976

weight 1 =  0.022

The price of a 0.5-year bond will be:

[tex]= \dfrac{1000}{(1+0.02\%)} \\ \\ =\mathbf{980.39}[/tex]

The price of a 1-year bond will be = 1000

Market value on the bond portfolio = 0.022 × price of 0.5 bond + 0.978 × price 1-year bond = 956.42

= 0.022 × 980.39 + 0.978 ×  1000

= 956.42

So, to have arbitrage profit, the investor needs to purchase 1 unit of the 1-year zero-coupon bond as well as 0.022 units of the 0.5-year bond. Then sell 0.978 unit of the 1-year bond.

Then will he be able to have an arbitrage profit of $56.42

4.

The one-period ahead forward rates can be computed as follows:

Foward rate from 0 to 0.5 [tex]X_1[/tex] = 2%

Foward rate from 0.5 to 1

[tex](1+X_2)^2 = (1+X_1) \times (1+ Foward \ rate \ from \ 0.5 \ to \ 1 )[/tex]

[tex](1+0.0225)^2 = (1+0.02) \times (1+ Foward \ rate \ from \ 0.5 \ to \ 1 )[/tex]

Foward rate from 0.5 to 1 = 2.5%

Foward rate from 1 to 1.5

[tex](1+X_3)^3 = (1+X_2)^2 \times (1+ Foward \ rate \ from \ 1 \ to \ 1.5 )[/tex]

[tex](1+0.0251)^3 = (1+0.0225)^3 \times (1+ Foward \ rate \ from \ 1 \ to \ 1.5 )[/tex]

Foward rate from 1 to 1.5 =3.021%

Foward rate from 1.5 to 2

[tex](1+X_4)^4 = (1+X_3)^3 \times (1+ Foward \ rate \ from \ 1.5 \ to \ 2 )[/tex]

[tex](1+0.0277)^4 = (1+0.0251)^3 \times (1+ Foward \ rate \ from \ 1.5 \ to \ 2 )[/tex]

Foward rate from 1.5 to 2 =3.021%

5.

The expected price of the bond if the hypothesis hold :

= [tex]\dfrac{40}{1+ 0.03021}+ \dfrac{1000+40}{(1+0.03285)^2}[/tex]

[tex]= \dfrac{40}{(1.03021)}+ \dfrac{1040}{(1.03285)^2}}[/tex]

= 1013.724254

= 1013.72

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