Quantitative Methods for Finance
DIRECTIONS
Term Structure of Risk-Free Rates, Defaultable bond pricing, and Hedging. You should turn in an Excel or Matlab file containing all your computations as well as a Word file with your detailed answers and the main empirical results of your analysis. Whenever a discussion is requested, please provide a thoughtful explanation of your findings.
It's December 31, 2015 and you work in the finance department of a large bank headquartered in NY. In order to raise $500 million, the Bank decides to issue bonds with maturity 10 years. The rating of the Bank is "BBB" and the estimated recovery rate on its debt is 50%.
a) Term Structure of Risk-Free Rates First, your boss asks you to tackle the modeling of the Term Structure (TS) of risk-free rates.
To this end, you gather data on Treasury Constant Maturity Rates (CMT) that are populated by the Federal Reserve at St. Louis.
1) From the FED St. Louis (https://research.stlouisfed.org/fred2/release?rid=18), download monthly series for the following maturities: 1-/3-/6-month and 1-/2-/3-/5-/7-/10-/20-year spanning the 2000-2015 period. Notice that the CMT rates are "bond equivalent yields" for securities that pay semiannual interest, see https://www.treasury.gov/resourcecenter/faqs/Interest-Rates/Pages/faq.aspx#2. Thus, you need to convert them to annual percentage yields and then to continuously compounded.
2) Compute unconditional moments (up to the fourth) and summary statistics (min, max, ...).
Also, compute the volatility and correlation of yield changes, and analyze their relation to the level of yields. Is the 2000-2015 evidence consistent with the stylized facts we saw in class? Did the slope/shape of the TS change over the sample period? What do these statistics suggest would be the pros and cons of using a Vasicek versus CIR one-factor model of the TS? Discuss.
3) Estimate the risk-adjusted parameters of the one-factor CIR model by minimizing the sum of squared residuals between theoretical and observed prices for 3-/6-month and 1- /2-/3-/5-/7-/10-/20-year bonds based on the TS that is observed on Dec, 2015. (Note that r, the short rate, will also a parameter to estimate). Discuss your findings. Hint: You may want to impose economic priors as constraints (e.g. σ>0) to ease the solver and start from ‘reasonable' initial guesses.
4) Consider now the following approach: Use the parameters you found in 3) (that use the cross-section as of Dec, 2015) and the 1-month series as a proxy for r to construct the theoretical prices for the CIR model for the full panel of data on 2000-2015. How well do these theoretical prices match the moments of the data? Discuss.
b) Defaultable Bond Pricing
Next, your boss would like to know which coupon should the Bank offer in the bond assuming that:
i) the bond will be issued today (Dec 31, 2015);
ii) at par value - that is, issuing price is $100 per $100 notional; iii) the bond will pay annual coupons. In sheet "BBB" of Excel file "Final_Plazzi.xls" you find the current term structure of yields on BBB-rated corporate bonds.
5) Estimate the term structure of credit spreads assuming that the yields quoted in the market for BBB firms are representative of those which should be applied to the bond issued by the Bank.
To do so, you use a CIR model in which you assume that the instantaneous risk-free rate and the instantaneous credit spread are uncorrelated. You keep the parameters of the CIR model fixed at the estimates you found in point 3) and find the parameters for ds by minimizing the sum of squared residuals. (Notice that the expected recovery rate RR on the quoted yields is 40%).
6) Calculate the term structure of cumulative, unconditional and conditional default probabilities which are relevant for the bond issued by the Bank. (Notice that the expected recovery rate RR on the bonds issued by the Bank is now 50%).
7) You can now go back to your boss and tell him the coupon which should be paid by the Bank on the 10-year bond.
c) Hedging
As you did a great job at pricing, your boss asks for some advice about the hedging of this liability. Assume that the hedging of the credit risk component of the 10-year bond issued by the Bank is not an issue - i.e., assume that the term structure of credit spreads remains fixed at its current level.
8) Use a set of Treasury bonds quoted in the market (see sheet "T-Bonds") to determine a portfolio of Treasury bonds which would allow the Bank to hedge the interest rate risk implicit in the 10-year bond according to the duration matching technique.
9) Simulate an increase of x bps in the YTM, with x = 10bps/50bps/150bps, and discuss how well does the proposed hedging strategy work.
Attachment:- quantitative-methods.xlsx