Business, 22.02.2021 19:40 ReeceLehua01

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.

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Answer from: dayanara72

Explanation:

From the given information;

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

The spot rate for maturity at 1 year is:

= $\dfrac{22.5}{(1+X_1)}+ \dfrac{1000 + 22.5}{(1+X_2)^2}=1000$

= $\dfrac{22.5}{(1+0.02)}+ \dfrac{1000 + 22.5}{(1+X_2)^2}=1000$

= $\dfrac{22.5}{(1+0.02)}+ \dfrac{1022.5}{(1+X_2)^2}=1000$

By solving for X_2 ;

X_2 = 2.253%

The spot rate for maturity at 1.5 years is:

$= \dfrac{25}{(1+X_1)}+ \dfrac{25}{(1+X_2)^2}+ \dfrac{1000 + 25}{(1+X_3)^3}=1000$

Solving for X_3

X_3 = 2.510%

The spot rate for maturity at 2 years is:

$= \dfrac{27.5}{(1+X_1)}+ \dfrac{27.5}{(1+X_2)^2}+ \dfrac{27.5}{(1+X_3)^3} +\dfrac{1000+27.5}{(1+X_4)^4} =1000$

By solving for X_4 ;

X_4 = 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.

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

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

Thus, for 2 years bond implies 4 periods;

∴

$= \dfrac{40}{1+0.02}+ \dfrac{40}{(1+0.02253)^2} + \dfrac{40}{(1+0.0252)^3}+ \dfrac{40}{(1+0.0277)^4}$

= $1047.024

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

$= \dfrac{0}{(1+0.02)}+\dfrac{1000}{(1+0.02253)^2}$

= 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:

$= \dfrac{1000}{(1+0.02\%)} \\ \\ =\mathbf{980.39}$

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

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

Foward rate from 0 to 0.5 X_1 = 2%

Foward rate from 0.5 to 1

$(1+X_2)^2 = (1+X_1) \times (1+ Foward \ rate \ from \ 0.5 \ to \ 1 )$

$(1+0.0225)^2 = (1+0.02) \times (1+ Foward \ rate \ from \ 0.5 \ to \ 1 )$

Foward rate from 0.5 to 1 = 2.5%

Foward rate from 1 to 1.5

$(1+X_3)^3 = (1+X_2)^2 \times (1+ Foward \ rate \ from \ 1 \ to \ 1.5 )$

$(1+0.0251)^3 = (1+0.0225)^3 \times (1+ Foward \ rate \ from \ 1 \ to \ 1.5 )$

Foward rate from 1 to 1.5 =3.021%

Foward rate from 1.5 to 2

$(1+X_4)^4 = (1+X_3)^3 \times (1+ Foward \ rate \ from \ 1.5 \ to \ 2 )$

$(1+0.0277)^4 = (1+0.0251)^3 \times (1+ Foward \ rate \ from \ 1.5 \ to \ 2 )$

Foward rate from 1.5 to 2 =3.021%

The expected price of the bond if the hypothesis hold :

= $\dfrac{40}{1+ 0.03021}+ \dfrac{1000+40}{(1+0.03285)^2}$

$= \dfrac{40}{(1.03021)}+ \dfrac{1040}{(1.03285)^2}}$

= 1013.724254

= 1013.72

Answer from: Quest

there are so many examples for that in different areas, like biology experiment carried out in our lab recently.

here's one link :

Answer from: Quest

The federal reserve took an expansionary approach during the crisis. this was done by expanding the money supply and boosting liquidity. this can be seen in the fed's actions of lending to banks, purchasing securities, and lowering the federal funds rate in order to lower overall interest rates. the fed's goal was to increase consumer spending and overall liquidity within the system, and they pursued this by expanding the supply of liquid money.

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