1. BGL Cumulative Inverse Floater Notes. On April 8, 2004, Banque Gen- erale du Luxembourg (BGL) issued the 12-year Cumulative Inverse Floater Notes described in the term sheet posted on Canvas. The Notes paid coupons semi-annually and the coupon rate in each coupon period was linked to the coupon rate in the previous period and to the 6-month USD LIBOR rate at the end of the current period, as described in the term sheet.
Answer the following questions assuming throughout that USD LIBOR is the appropriate discount rate to account for the credit risk of BGL and that the annualized standard deviation of the changes in USD LIBOR rates of all tenors is 1%.
(a) Figures 1 and 2 show the USD LIBOR rate and the USD swap rate fixings on April 8, 2004. Use these rates to estimate the term structure of USD LIBOR rates on April 8, 2004.1 Plot the estimated term structure for tenors up to 30 years and explain the estimation procedure you followed.
(b) Figure 3 shows the Eurodollar futures settlement prices on April 8, 2004. Re-estimate the term structure of USD LIBOR rates using the data in Figures 1 and 3. Plot the estimated term structure for tenors up to 10 years and explain the estimation procedure you followed.
(c) Produce a plot comparing the term structures estimated in part (a) and part (b) for tenors up to 12.5 years (the relevant range for pricing the BGL Notes).
1If using the Solver in Excel to estimate the term structure, see Appendix I.
(d) Calibrate the Ho-Lee model with = 0.5 to the term structure of USD LIBOR rates you estimated in part (b) and construct the tree for the 6-month rate out to 12 years.
(e) Compute the value of the BGL Notes on April 8, 2004 using the tree for the 6-month rate you calibrated in part (d) and Monte Carlo simulation with at least 1,000 simulated paths.
For simplicity, ignore the fact that the BGL Notes were callable and assume that the coupon rate is linked to the 6-month USD LIBOR rate on the coupon date (rather than to the 6-month USD LIBOR rate 11 business days before the coupon date).
Describe the procedure you used for the simulation and report the number of paths you simulated, the estimated value of the BGL Notes and the 95% confidence interval for the value.
(f) Compute the duration of the BGL Notes on April 8, 2004 and describe the procedure you used for this computation.
2. Back to Assignment II. In Assignment II, you priced the P&G swap using the Ho-Lee and Black-Derman-Toy models. Since it was convenient to set the time step to six months in those models in order to simplify the calculations, the price you computed was based on only two possible values for the random spread on May 4, 1994 and required approximating the price of the on-the-run 30 year Treasury Bond using linear interpolation. In addition, the 5-year TCM rate was estimated rather crudely by assuming a constant spread between the semi-annually compounded 5-year risk-free spot rate and the 5-year TCM rate.
Here you will re-price the P&G swap without relying on those simplifications.
(a) Use the continuous-time Ho-Lee model and Monte Carlo simulation to value the derivative embedded in the P&G swap (as described in Assign- ment 2) on November 4, 1993. Assume that the annualized standard de- viation of the changes in the short-term risk-free rate is as in part (8) of Assignment II. In addition, assume that the 5-year TCM rate equals the 5-year risk-free par yield minus 3 basis points and do not use interpolation to determine the price of the on-the-run 30-year Treasury Bond.2
2See Appendix II for the definition of the par yield and the rationale for simulating the TCM rate using the par yield.
Overnight 1 week 2 week 1 month 2 month 3 month 4 month 5 month 6 month 7 month 8 month 9 month 10 month 11 month 12 month
1.05750 1.07375 1.08000 1.10000 1.12000 1.14000 1.17000 1.19875 1.22500 1.26313 1.30563 1.35000 1.40000 1.45000 1.50000
Figure 1: USD LIBOR rates on April 8, 2004. Source: Fed
1 year 2 year 3 year 4 year 5 year 7 year
10 year 30 year
1.51000 2.20000 2.78000 3.23000 3.60000 4.11000 4.60000 5.35000
Figure 2: USD swap rates on April 8, 2004. Source: Fed
Apr 19, 2004 May 17, 2004 Jun 14, 2004 Jul 19, 2004 Aug 16, 2004 Sep 13, 2004 Dec 13, 2004 Mar 14, 2005 Jun 13, 2005 Sep 19, 2005 Dec 19, 2005 Mar 13, 2006 Jun 19, 2006 Sep 18, 2006 Dec 18, 2006 Mar 19, 2007 Jun 18, 2007 Sep 17, 2007 Dec 17, 2007 Mar 17, 2008 Jun 16, 2008 Sep 15, 2008
98.8625 98.8350 98.7750 98.7000 98.6000 98.5300 98.1950 97.8150 97.4050 97.0450 96.7250 96.4550 96.2000 95.9700 95.7550 95.5700 95.3900 95.2250 95.0600 94.9250 94.7900 94.6650
Dec 15, 2008 Mar 16, 2009 Jun 15, 2009 Sep 14, 2009 Dec 14, 2009 Mar 15, 2010 Jun 14, 2010 Sep 13, 2010 Dec 13, 2010 Mar 14, 2011 Jun 13, 2011 Sep 19, 2011 Dec 19, 2011 Mar 19, 2012 Jun 18, 2012 Sep 17, 2012 Dec 17, 2012 Mar 18, 2013 Jun 17, 2013 Sep 16, 2013 Dec 16, 2013 Mar 17, 2014
94.5500 94.4550 94.3550 94.2750 94.1800 94.1050 94.0200 93.9550 93.8850 93.8250 93.7600 93.7100 93.6550 93.6150 93.5650 93.5250 93.4850 93.4600 93.4150 93.3900 93.3450 93.3250
Figure 3: Eurodollar futures settlement prices on April 8, 2004. Source: CME Group
Appendix I: Excel Solver Settings
The default options for the Solver in Excel are set for speed rather than accuracy. As a result, with the default options, the Solver performs poorly in problems that involve multiple variables.
Accordingly, if you use the Solver for Question 1, you should set the Options in the Solver Parameters dialog box as specified below:
• In the “All Methods” tab of the “Options” dialog box, set “Constraint Preci- sion” to 1E-6, select “Use Automatic Scaling”, deselect “Show Iteration Results” and select “Ignore Integer Constraints”. Leave all the boxes in the “Solving Limits” panel blank.
• In the “GRG Nonlinear” tab of the “Options” dialog box, set “Convergence” to 1E-8, select “Central Derivatives”, select “Use Multistart” (with a “Population Size” of 10 or larger) and select “Require Bounds on Variables”.
• Back to the main Solver Parameters dialog box, make sure that “GRG Nonlin- ear” is selected and enter an upper and a lower bound in the Constraints box for each parameter. Finally, make sure that the appropriate radio button (i.e., “Max”, “Min” or “Value Of”) is selected.
You can get an idea of what are reasonable upper and lower bounds for the parameters in the Svensson parameterization of the term structure by looking at the historical estimates in the FEDS 2006-28 paper.4 If the Solver finds a solution for one of the parameters that is at the lower or upper bound you specified for that parameter, you need to relax that bound and re-run the Solver.
4Keep in mind that the interest rates in the FEDS 2006-28 paper are expressed in percent: if you instead express the interest rates in decimal form, the parameters 0, 1, 2 and 3 are simply divided by 100, while the parameters ⌧1 and ⌧2 are una↵ected.
Appendix II: Simulation of TCM Rates
Recall that TCM rates reflects the YTMs of the on-the-run Treasury securities and that, as a result, there are two factors that impact the spread between the semi- annually compounded risk-free spot rate and the TCM rate of the same tenor:
1. The TCM rate is a YTM instead of a spot rate;
2. The TCM rate reflects the pricing of the on-the-run T-Note instead of a generic T-Note and the former includes a liquidity premium.
The second factor unambiguously results in a positive spread, while the first factor can result in a positive or negative spread, depending on the shape of the risk-free term structure.
The par yield for a given maturty is defined as the coupon rate such that a bond with the specified coupon rate and maturity would trade at par. It is fairly easy to show that:
1. The par yield for a risk-free bond with maturity Tn and coupon dates T1, . . . , Tn equals the Treasury swap rate as given in equation (7) in Section VIII, i.e.,
1Z(0,Tn) rIRS(0, Tn) = m Pni=1 Z(0, Ti),
where Z is the risk-free discount factor, m is the number of coupon payments per year and Ti = i/m.
2. The YTM of a bond trading at par is equal to the coupon rate.
When the US Treasury issues a new T-Note or T-Bond, it sets the coupon rate to the highest level (in increments of 1⁄8 of a percentage point) that does not result in an issuance price that exceeds par. As a result, absent a liquidity premium for on-the-run securities (the second factor mentioned above), coupon rates and YTMs of on-the-run securities should be very close to the risk-free par yield.
The above prediction is well-supported empirically. Using daily data for the period 1990-2017, the average spread between the 5-year risk-free par yield computed from the FEDS 2006-28 term structure and the 5-year TCM rate is 3 basis points, the standard deviation of the spread is 4 basis points and the R2 for the linear regression of the 5-year TCM rate on the 5-year par yield is 99.96%.
Appendix III: Simulation of Mortgage Rates
A good way to predict the 30-year mortgage rate based on the Treasury term structure is to compute the 10-year Treasury par yield, defined using equation (7) in Section VIII, i.e.,
1Z(0,Tn) rIRS(0, Tn) = m Pni=1 Z(0, Ti),
whereTi =0.5⇤i,n=20,m=2andZ(0,T)denotestheTreasurydiscountfactor. Figure 4 shows a plot of the weekly time series of the 30-year mortgage rate (monthly compounded) and of the 10-year par yield from August 1971 through March 2017, while Figure 5 shows the spread between the two rates. The average spread is
161 basis points, but there is considerable variation through time.
Figure 6 shows a scatter plot of the data, together with the estimated quadratic
polynomial regression model
y = 0.022047 + 0.746495x + 2.06492x2, (1)
where y is the mortgage rate and x the Treasury par yield.
The model fits the data well: the R2 for the regression is 98.1%. Figure 7 shows
a plot of the time series of the actual and predicted 30-year mortgage rate.
Treasury swap rate
Figure 4: 30-year mortgage rate and 10-year Treasury swap rate (percent), 1971-2017
6 5 4 3 2 1 0
1980 1990 2000 2010
Figure 5: Spread between the 30-year mortgage rate and the 10-year Treasury swap rate (percent), 1971-2017
0 5 10 15
Figure 6: Fit of the quadratic polynomial regression model, 1971-2017
Figure 7: Actual and predicted 30-year mortgage rate (percent), 1971-2017
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