INSTRUCTIONS:

❼ The total marks for this assignment are 200.

❼ Please work on this assignment in a group of at most 4 people, photocopy your work and

submit one pdf version on eclass by 11:30 pm, Dec 16.

❼ Please write down the names and student ID’s of the group members.

❼ Please write your answers clearly, since 0 marks will be assigned to ambiguous answers.

❼ Partial credit will be given for showing logically correct reasoning, even if the final answer is

not correct.

❼ If not otherwise specified, discount rates for bonds are quoted with the compounding frequency

corresponding to the frequency of coupon payment  (eg, semiannual APR for semiannual coupon payment).

❼ If not otherwise specified, put 2 decimal places in the final result for dollar amount (eg, $1.99)

and rates (eg, 1.99%); therefore, you need to retain more decimal places in your intermediate calculation steps (6 decimal places recommended).

❼ For questions about Insured Asset Allocation, ignore all the fees.

❼ For questions about VaR, keep z5% = 1.6449,z1% = 2.3263

1. Mutual Funds I [13 points]

Today, a mutual fund currently holds three stocks A, B and C, with the information given below after the market is closed.

The mutual fund currently has 100,000 number of shares outstanding held by its investors.

(a) [1 points] Calculate the NAV for today.

(b) [2 points] Suppose tomorrow stock A pays a $2 dividend per share with no change in price. Suppose nothing changes about the position on stock B, but C changes in price.  After the daily reconciliation (mark to market), the NAV is still equal to the result you get in (a).  What is the price of C?

(c) [2 points] Continue with (a), today after the market is closed, 500 new investor show up and each bought 20 shares of the mutual fund at today’s NAV in (a). The fund immediately used 50% of the money to buy additional stock B (assuming at today’s price) and the rest to buy additional stock C (assuming at today’s price).  Tomorrow at the market close, C increases by $2 while B increases by $1 in price. Calculate the NAV for tomorrow.

(d) [2 points] Continue with (a), today after the market is closed, 50 existing investors wanted a redemption: each of them sold 100 shares of the mutual fund at today’s NAV in (a). Suppose the mutual fund meets the redemption tomorrow by selling part of their position in stock B at $50 per share.  Tomorrow at the market close, share price of B turns out to be still $50.  Assume nothing has changed for A and C. How many number of shares of B needs to be sold? Calculate the NAV for tomorrow.

(e) [2 points] Continue with (a), today after the market is closed, 50 existing investors wanted a redemption: each of them sold 100 shares of the mutual fund at today’s NAV in (a). Suppose the mutual fund meets the redemption tomorrow by selling part of their position in stock B at $41.875 per share. Tomorrow at the market close, share price of B turns out to be still $50. Assume nothing has changed for A and C. How many number of shares of B needs to be sold? Calculate the NAV for tomorrow. Compare the NAV with that in (d): what explains the change or no change in NAV?

(f) [2 points] In addition to the three stocks in the table above, suppose the fund also holds $20,000 in cash. Recalculate the NAV for today.

(g) [2 points] Continue with (f), today after the market is closed, 50 existing investors wanted a redemption:  each of them sold 100 shares of the mutual fund at today’s NAV in (f).  Instead of selling the stock position, the fund decides to use cash to meet the redemption. How much cash is remaining after the fund meets the redemption? What is the NAV for today after the fund meets the redemption?  Compare the NAV with that in (f):  what explains the change or no change in NAV?

2. Mutual Funds II [10 points]

A mutual fund charges a front load of 5%, and an annual operating expense of 2% that include 1.6% management fees and 0.4% 12b-1 fees. The annual operating expense is charged at year end, based on the average of the value of investment at the year beginning and year end. Assume that the fund’s returns (before annual operating expense) for the invested money for the future 5 years are given below.

You have $10,000 and decide to invest in the mutual fund for 5 years.

(a) [5 points] Please complete the table: for each year, calculate the investment value at year be- ginning, investment value at year end before annual operating expense, and investment value at year end after annual operating expense.  [Note:  investment value at year beginning for the 1st year should be the value after front load is paid.]

(b) [2 points] Ignoring the front load and annual operating expense, calculate the future value of $10,000 in 5 years with the returns provided in the above table. What is the cumulative return in percentage over the 5 years? What is the annualized return in percentage for the 5 years, assuming annual compounding?

(c) [2 points] Considering the front load and annual operating expense, by referencing the table you completed in (a), what is the future value of $10,000 in 5 years? What is the cumulative return in percentage over the 5 years? What is the annualized return in percentage for the 5 years, assuming annual compounding?

(d) [1 points] Compare the cumulative return in (b) vs in (c). What explains that difference?

3. Mutual Funds III [16 points]

With $10,000, you are considering investing in one of the two competing mutual funds A and B,

with the information given below.

The annual operating expense of both funds is charged at year end, based on the value of invest- ment at the year end.  Assume that both funds’return (before annual operating expense) for the invested money for each year is 10%.

(a) [1 points] Assume the load in the table is a front load, if you plan to invest for exactly 5 years, calculate the future value of the $10,000 in 5 years, considering the load and annual expense. Which fund would you pick?

(b) [1 points] Assume the load in the table is a front load, if you plan to invest for exactly 10 years, calculate the future value of the $10,000 in 10 years, considering the load and annual expense. Which fund would you pick?

(c) [2 points] Continue with (a) and (b), using Excel, draw the future values of your investment in fund A vs fund B, with x-axis: number of years from now; y-axis: future value.  There should be two curves for A and B respectively, and each curve contains 11 data points, starting at 0 years from now, and ending at 10 years from now. Label the graph properly.

(d) [2 points] Based on the curve in (c), answer the following: i) if you plan to invest for more than 6 years, which fund would you pick? ii) if you plan to invest for less than 4 years, which fund would you pick?

(e) [1 points] Using the formula, FV = P(1 − L)(1 + r)T (1 − F)T . Approximately at which point in time are you indifferent in investing in the two funds?

(f) [4 points] A back load is a load that only applies if you sell your ownership in funds and get your money back in cash.  Assume back load applies after the annual operating cost is charged. Assume that the load in the table is back load not front load; back load is charged but front load is not.  Assume in addition that annual operating expenses do not change in the table.  For fund A, calculate the future value in cash that you can get back in 5 years.  Compare the result with future value for fund A in (a), what do you find?  For fund A, calculate the future value in cash that you can get back in 10 years.  Compare the result with future value for fund A in (b), what do you find?

(g)[2 points] Continue with (f), using Excel, for fund A, draw the curve for future value in cash that you could possibly get back in each year, for the future 10 years.  X-axis:  number of years from now; y-axis: future value in cash after back load. Label the graph properly.

(h) [2 points] Continue with (g), putting the curve in (g) and the curve in (c) for fund A in the same graph.  X-axis:  number of years from now; Y-axis:  future value; title:  future value in cash (back load vs front load). Label the graph properly. What do you find?

(i) [1 points] Based on the result in (h), write down the formula for the future value in cash that you could possibly get back for a given year, with the notations in (e).

4. Insured Asset Allocation I [10 points]

You are the portfolio manager at Golden State Hedgers, and managing a portfolio for your client

with the following information.

You decide to use a buy-and-hold strategy (consisting positions in risk-free zero-coupon bond and stock index) to protect the downside of the portfolio and achieve some upside potential.

(a) [2 points] Calculate the dollar amount invested today in risk-free zero-coupon bonds and in stock index, respectively.

(b) [4 points] The index level in 2 years are given below, fill in the table with the value of your stock position, bond position and total position in 2 years. (you may provide a screenshot of your result in an Excel spreadsheet)

(c) [2 points] Continue with (a), suppose the face value of a risk-free zero-coupon bond is $1000, what is the price of a bond? How many units of bonds do you need to buy today to implement the strategy?

(d) [2 points] You cannot buy index directly.  Continue with (a), you decide to buy shares of a mutual fund TTT that tracks the exact performance of the index. The NAV for the mutual fund is $40 today. How many shares of TTT do you need to buy to implement the strategy in (a)?

5. Insured Asset Allocation II [12 points]

You are the portfolio manager at Golden State Hedgers, and managing a portfolio for your client with the following information.

According to your analysis, the stock index follows a binomial process with u = 1.2,d = 0.9. You decide to implement a three-period (each period is one year) stop-loss strategy to achieve the ob- jective of the client.

(a) [2 points] Build the binomial tree for the stock index

(b) [2 points] Build the binomial tree for the all-stock portfolio. Specify the thresholds at t=0, 1, 2, 3, at which you need to switch the stock position to a bond position.

(c) [2 points] Assume stock index goes u → d → d, do you need to switch your stock position at any point of time? If you do, at which point of time? What is the value of your stop-loss portfolio at target date?

(d) [2 points] Assume stock index goes d → d → u, do you need to switch your stock position at any point of time? If you do, at which point of time? What is the value of your stop-loss portfolio at target date?

(e) [2 points] Based on the result of (c) and (d), is the stop-loss strategy path independent? Explain the reason.

(f) [2 points] Assume stock index goes d, suppose you are worried that the stock price may jump through the threshold (ie, stock price breaks the threshold), and decides to switch at time 1. What is the value of your stop-loss portfolio at target date?

6. Insured Asset Allocation III [15 points]

You are the portfolio manager at Golden State Hedgers, and managing a portfolio for your client with the following information.

You decide to use a CPPI strategy (consisting positions in stock index and your bank account that earns a risk-free rate) to protect the downside of the portfolio and achieve some upside potential.

(a) [1 points] What is the effective daily interest rate?

(b) [3 points] Assuming the money in your bank account earns effective daily risk-free rate, follow the CPPI strategy with the multiplier m = 2, and fill in the following table for day 1, 2, and 3. The numbers for day 0 have been filled.

(c) [3 points] Random numbers of standard normal distribution have been provided for ϵ in “CPPI Sample.  xlsx”, for 1 path, and 500 days in the future.  Use the following discretized geometric Brownian motion to simulate 1 path of index level for the future 2 years (500 trading days), with parameters S0 = 1, 000,µ = −50%,σ = 80%, ∆t = .

S + ∆S = Se(µ− σ 2 )∆t+σϵ ^∆t

Plot the path of index level for the 501 data points; x-axis: number of days from now; y-axis: index level

(d) [4 points] Based on (c), following the CPPI strategy with the multiplier m = 2, build a table with these columns to store your positions for each trading day, for the 501 days. The numbers for day 0 have been filled. Build 2 charts: 1) plot the portfolio value and the floor over the 501 days;

2) plot the value of the stock index position and bank account position that we need over the 501 days. Label the charts properly. [Note: submit the chart only; do not submit the table.]

(e) [4 points] Based on (d), now let’s analyze the role of the multiplier m. Let the parameter σ = 0

just so we can see it clearly. Draw 4 charts using the 501 data points, with y-axis: portfolio value, and x-axis: number of days from now

i) with parameters µ = −50% and m = 8.

ii) with parameters µ = −50% and m = 1.

iii) with parameters µ = +50% and m = 8.

iv) with parameters µ = +50% and m = 1.

Based on the 4 charts, what is the role of m?

7. Insured Asset Allocation IV [16 points]

You are the portfolio manager at Golden State Hedgers, and managing a portfolio for your client with the following information.

You find a traded call and a traded put on the market with strike price $1,200, and decide to use traded option strategy to achieve the objective of the client.  Notice the traded option strategies

discussed in our class is a special case; it requires the strike price X to be equal to the floor F

which does not apply in this case. Let’s generalize the strategy. We can quantify the objective of the client as the following

max{NST,F}

ie, the portfolio value at target date is at least F, where N > 0 could be interpreted as number of units of stock index

(a) [2 points] Prove: max{NST,F} = N max {ST − F,0} + F

That is, this traded call strategy requires N units of call options on the index, with strike price X = F, and risk-free zero-coupon bonds with total face value of F .

(b) [2 points] Prove: max{NST,F} = N max { F − ST , 0 } + NST

That is, this traded put strategy requires N units of put options on the index, with strike price X = F, and N units of stock index.

(c) [3 points] Based on (a), how many units of call option do you need to buy today?  Fill in the following table for the position values on the target date.

(d) [3 points] Based on (b), how many units of put option do you need to buy today?  Fill the following table for the position values on the target date.

(e) [2 points] Continue with (c), assume the face value of a risk-free zero-coupon bond is $1,000. What is the price of a bond? How many units of bonds do you need to buy today?

(f) [2 points] What is the price of the call and the put under the Black-Scholes Model today, re- spectively?

(g) [2 points] Based on (f), what is the total cost of the traded call strategy and the traded put strategy respectively today? Are they equal?

8. Insured Asset Allocation V [16 points]

You are the portfolio manager at Golden State Hedgers, and managing a portfolio for your client with the following information.

According to your analysis, the stock index follows a binomial process with u  =  1.2,d  = 0.8. Since options with suitable strike prices are not available on the market, you decide to implement a three-period (each period is one year) synthetic option strategy to achieve the objective of the client.

(a) [2 points] Build the binomial tree for the stock index

(b) [2 points] Continue with (a), after the communication with the client, you both agree to use an all-stock portfolio by investing 90% of V0  today as your reference, to achieve the upside potential and downside protection. Build the binomial tree for the all-stock portfolio.

(c) [1 points] Continue with (b), write down the value of your synthetic option strategy at t=3 (target date) for each of the 4 states.

(d) [6 points] Continue with (c), build the binomial tree for the portfolio value of the synthetic option strategy, the number of units ∆ of stock index and the amount of money B in bond.

(e) [1 points] Continue with (d), is the strategy achievable with V0 = 200, 000? Depending whether it is achievable, calculate the amount of money remaining after implementing the strategy, or the additional amount of money that you need to get from the client.

(f)  [2 points] Continue with (d), at time 1, for u-node and d-note respectively, verify the self- financing property.

(g) [2 points] Continue with (f), at time 1, (i) for u-node, calculate the value of bond position we need to sell to rebalance the portfolio; (ii) for d-node, calculate the value of stock position we need to sell to rebalance the portfolio.