Problem 1 - (Using volatility to improve long-only equity protfolios) Intrigued with (and perhaps sceptical of) the constant buzz on CNBC about the \fear index" (aka VIX), you decide to explore for yourself the relationship between equities and volatility of equities. To that end, you collect daily data from January 1, 1999 to April 29, 2010 on the S&P 500 and the VIX. The S&P 500, of course, is the most widely used capitalization-weighted index of the largest 500 companies in the US equity market. And the VIX is the Chicago Board Options Exchange's volatility index, which reflects the market's expectation of 30-day equity volatility (hence the moniker\fear index"). The data you collect are in Columns B and C in the Excel spreadsheet final.20170428.Problem 4.xlsx. The same spreadsheet, in Columns D and E, contains the daily, arithmetic \returns" you compute (as St/(St-1) - 1) for the S&P 500 and for the VIX respectively. For the purposes of your analysis (and for this problem), you decide to ignore the fact that the VIX is an index and to treat it as a tradable asset, thus assuming that you can realize the \returns" you have computed. You further decide to ignore interest rates and inflation.
To test the robustness of your conclusions, you define four sub-periods from your data:
Period 1: Internet \bubble" January 1, 1999 - March 31, 2003
Period 2: Bull market before crisis of 2008 April 1, 2003 - October 31, 2007
Period 3: Crisis of 2008 November 1, 2007 - April 29, 2010
Period 4: Entire data period January 1, 1999 - April 29, 2010
Your plan is to carry out your analysis for each period separately and compare your results (across the different periods) to see whether your conclusions change materially with time.
(a) With the data from Period 4 (entire data period), use the tool we used in class (autocorrelogram) or any other visual tool of your choice to explore wheteher the returns of the S&P 500 and the retruns of the VIX appear to be independent in time. Do your results agree with what you would expect?
(b) With the data from Period 4 (entire data period), use the tool we used in class (autocorrelogram) or any other visual tool of your choice to explore wheteher you find evidence of volatility clustering in the returns of the S&P 500 and the retruns of the VIX. Do your results agree with what you would expect?
(c) For each of the four periods, plot the returns of the S&P 500 against the returns of the VIX. On the basis of your plots, do you think the VIX is a good hedge for the S&P 500? If yes, why? If not, why not?
(d) For each of the four periods, construct a mininum volatility VIX hedge for the S&P 500 with the three methods we discussed in class (correlation formula, linear regression, optimizer) and evaluate the efficiency of your hedge (for each period in question). Do your results agree with what you expect? Do the methods that should produce identical results do so? Does the size and the efficiency of your hedge make sense, given the data in each period? Are the results from the different methods in reasonable agreement with each other?
(e) Do your mininum volatility hedges vary significantly with the period? Is the variation in the hedges consistent with the nature of the periods? How effective are the hedges outside the period from which they are derived? Evaluate a representative hedge from Period 1 in Period 2 and a representative hedge from Period 2 in Period 3.
On the basis of your findings so far, you set out to construct a portfolio that is constantly rebalanced (that is, rebalanced every day) so that it maintains fixed allocations 1 w and w to the S&P 500 and to the VIX respectively.
(f) For each of the four periods, find the VIX allocation wMAD that minimizes the M(edian) A(bsolute) D(eviation) (MAD) of your portfolio. Interpreting this minimum MAD VIX allocation as a \hedge," evaluate the efficiency of your hedge (for the period in question). Are your MAD hedges comparable to the minimum volatility hedges you obtained in part (d)? If so, why? If not, why not? (Hint: Be careful in how you asnwer this question!) Compare the variation of your MAD hedges (across periods) with the variation of your minimum volatility hedges: which variation is larger, and is what you observe consistent with what you would expect? Why?
(g) For each of the four periods, find the VIX allocation wsemi-variance that minimizes the semi-variance of your portfolio. Interpreting this minimum semi-variance VIX allocation as a "hedge," evaluate the efficiency of your hedge (for the period in question). Are your minimum semi-variance hedges closer to your minimum volatility hedges (from part (d)) or to your minimum MAD hedges (from part (f))?
(h) For the four periods, use the formula we developed in class to calculate the (per period) rate of growth of the S&P 500, the VIX, and your portfolio with w = 15%. Tabulate the observed (per period) rate of growth of these three investments. Do your analytical results agree with your empirical results?
(i) For the four periods, use an optimizer to obtain the optimal growth allocation wgrowth that maximizes the growth of your portfolio. Does the variation in your estimates of wgrowth make sense according to the nature of the different periods?
(j) Compare your optimal growth allocation estimates from (i) to the \hedges" you have obtained in (d), (f) and (g). Are the sizes of the respective allocations to the VIX comparable?
Construct one portfolio with your minimum volatility hedge for Period 4 (from (d)) as your allocation to the VIX and another with your optimal growth allocation for Period 4 (from (i)) as your allocation to the VIX. Compare these two portfolios in terms of \risk" and \return," choosing measures for \risk" and \return" respectively that you think are the most suitable for this comparison. Explain which measures you use and why.
Problem 2 - (Backward and forward pricing in an interest rate lattice) Use the interest rate lattice in worksheet Interest Rate Lattice in final.20180427.Problem 2.xls to answer this question. Assume the risk-neutral probabilities of all up moves and all down moves to be 0.5 and all rates to be continuously compounded rates.
(a) Use the interest rate lattice to determine the price and yield of zero-coupon bonds that mature in years 1, 2, 3, 4, and 5. Use the price-yield formula Pi = e-iy_i, where Pi is the price of an i-year zero-coupon bond and yi is its yield.
(b) Use the interest rate lattice to determine the yield curve one year in the future in the upstate, i.e., determine y(1u) = (y1(1u), y2(1u), . . . , y4(1u)).
(c) Use the interest rate lattice to determine the yield curve one year in the future in the down-state, i.e., determine y(1d) = (y1(1d), y2(1d), . . . , y4(1d)).
(d) Use backwards pricing to price a security that has the following payoffs in year 3: it pays $3 if the rate is 10%, it pays $2 if the rate is 7.7%, it pays $1.5 if the rate is 5.94%, and it pays $1 if the rate is 4.57% (corresponding to the four possible levels of interest rates at year 3).
(e) Use forward pricing to price the security in part (d). In particular what is the price today if rates follow the path up-up-up? What is the price today if rates follow the path up-up-down? Give the path price for each of the eight paths. Take the average of the eight path prices (since all paths are equally likely), and give the result. Does your answer match the result from part (d)?
Problem 3 - (Real-world and risk-neutral probabilities) Today, two bonds trade in the market: a one-year zero-coupon bond and a two-year zero-coupon bond, each with a par value of $1. The current one-year rate is 3%. Using continuous compounding, the market price today of a one-year zero-coupon bond is $0.9704 (= e-0.03) and the current market price of a two-year zero-coupon bond is $0.9371. Each year the one-year rate will either go up by 1% or down by 1% from its starting value (i.e., to 4% or 2% from the initial rate of 3%). The real-world probability of the rate going up is pu = 0.45 and the real-world probability of the rate going down is pd = 0.55. The real-world probabilities are constant through time.
(a) What is the price today of bond A, a one-year bond that pays $2 if the one-year rate goes up to 4% and pays $1 if the one-year rate goes down to 2%?
(b) What is the price today of an interest rate derivative that pays $1 if the one-year rate goes up to 4% and pays $0 if the one-year rate goes down to 2%?
(c) What is the price today of an interest rate derivative that pays $0 if the one-year rate goes up to 4% and pays $1 if the one-year rate goes down to 2%?
(d) Give a formula for the price today of a security which pays $a if the one-year rate goes up to 4% and $b if the one-year rate goes down to 2%. Write your formula in the form of P = e-0.03(qua + qdb) and give numerical values for the risk-neutral probabilities qu and qd.
(e) What is the one-year rate one-year forward (the rate that can be locked-in today for borrowing or lending between years 1 and 2)? What is the (real-world) expected one-year rate one year from today (the expectation using the real-world probabilities)? Are forward rates expectations of future rates?
Problem 4 - (Pricing a cap on an interest rate lattice) Use the interest rate lattice in worksheet Data in final.20180427.Problem 4.xlsx to answer this question. The risk-neutral probabilities of all up moves and down moves are 50%. All rates are continuously compounded rates (i.e., a rate of r means that the price of a one-year maturity zero-coupon bond is e-r).
This question involves a 4-year interest rate cap with a strike of eK with K = 6.0%. The cap payments are based on rates set at year 1, 2, 3, 4, with potential cash payments at years 2, 3, 4, 5. If the one-year rate at year i is ri and the strike is eK, a payment of Lmax(er_i - eK, 0) is made at year i+1, where L is the underlying notional amount. (Since the rates in the lattice are quoted on a continuous compounding basis, we assume for consistency that the cap payments are also made this way.) The notional amount is L = 100.
(a) What are the fair prices today of the cap and the four individual caplets?
(b) A corporation borrows P = 500 from bank A. The term of the loan is five years and the interest is the floating rate of interest given in the interest rate lattice. For example, an interest payment of P(e0.06 - 1) is due at year 1 (i.e., one year from today). The last interest payment in year 5 is accompanied by the repayment of the principal P. From bank B the corporation purchases a 4-year interest rate cap with a strike of eK, with K = 6:0%, on an underlying notional of P. The cost of the cap is the amount determined in part (a). Suppose that the path of interest rates is up, up, up, up. Compute the interest and cap payments from the corporation to bank A and from bank B to the corporation at each of the years 0, 1, 2, 3, 4, and 5.
Attachment:- Assignment Files.rar