# Calculate implied two year forward rates

Problem 1 (10
The following table depicts current market conditions (assume annual compounding):
Year Current spot rates () Implied 2-year forward rate (,)
3 2.00%
4 3.00%
5 3.80%
6 4.00% N/A
7 4.10% N/A

Calculate implied 2-year forward rates (3,2, 4,2, 5,2).

According to your analysis on the market condition, propose a strategy that goes long a zero coupon bond and goes short another zero coupon bond. For simplicity, use only zero coupon bonds.

According to (b), we will go long a m-year zero coupon bond and go short a n-year zero coupon bond. From mathematical analyses, we already know that the m-year zero coupon bond outperforms the n-year zero coupon bond whenever the forward rate between the m- and after 1 year. Here are our scenarios:

() ()
Current State % %
Steepening % + 0.5 %
Parallel Shift + 0.3 % + 0.3 %
Flattening + 0.3 % + 0.1 %

In which situation(s) do we make a profit from our strategy?

Problem 2 (24
Let us assume that there are 5 stocks and 5 bonds available in the market. The historical data are reported in the attached Excel files. You are a wealth manager, and you should propose a portfolio by using the given asset classes.

Your client has an asset of USD 1 million with a liability of USD 500,000 (the present value).
The maturity of the liability is 20 years with the yield to maturity of 4%.
The clients investment horizon is 3 years.
Regarding a bond portfolio, the client wants to hedge against a change in interest rates.
Regarding stock investment, your client wants to choose only one stock (because of, for example, huge transaction costs).

Our analysis is based on status quo financial analysis. Thus, there are several strong assumptions:
Assume that the is expected to be 3% constantly.
All the coupon payments are reinvested at 5%.
Coupon payment dates of all the bonds are 06/30/2002, 12/31/2002, 06/30/2003, 12/31/2003, 06/30/2004, and 12/31/2004.
Stock dividends and bond coupons are paid semi-annually. Note that the dividend and coupon rates are quoted on an annual basis: in other words, you should divide the coupon rate (dividend rate) by 2 when you compute the income.
An immunization portfolio is constructed on the offering dates of the bond (02 Jan
2002).
Tax is not considered here.

Using what we have learned, propose an asset allocation strategy. Specifically,
Propose the weight on each asset class
Calculate the return on the aggregate portfolio
Discuss risk embedded in the portfolio
Evaluate your portfolio performance using several measures

Problem 3 (16
In the bond market, available securities are provided as follows:
Maturity (years) Position name YTM (%) Coupon (%)
2 Short-wing 4.5 5
5 Body 5.5 5
10 Long-wing 6 5

Now, consider the following strategy
Maturity (years) Quantity (in USD Face Value)
2 X1
5 -10,000
10 X2

In other words, the strategy purchases \$ X1-par 2-year bond and \$ X2-par 10-year bond, while it sells short \$10,000-par 5-year bond.

A fifty-fifty butterfly strategy is to adjust the weights X1 and X2 so that the transaction has a zero dollar duration and the same dollar duration on each wing, short and long. The dollar duration of each wing then equals the fifty percent of the dollar duration of the body. Note that this strategy is, however, not cash neutral (i.e., you do not need to consider value matching). Determine X1 and X2 of this strategy.

Do you think this strategy will make a profit after a year from now (2022~2023)? Provide a reasonable scenario based on the real market situation nowadays, and approximate the profit.

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