25872 Interest Rates and Credit Risk Models

Assignment

Assignment due on October 13, 2019 as per the Subject Outline. Late assignments attract a penalty of 5 marks (out of 50) per day.

Students, who submit a draft of their assignments by September 25, 2019, will receive feedback by October 4, 2019 (and may subsequently amend their draft).

Your assignment must be submitted electronically (Microsoft Excel or Open Office workbook and a text document — Microsoft Word or PDF) via UTS Online before midnight on the due date. If you choose to write/use any additional computer code (in any language) to solve this assignment, this computer code must also be supplied. The text document must be typeset (using for example LATEX or Microsoft Word). Handwritten, scanned assignments will not be accepted. Please make sure your spreadsheets (and any additional computer code, if applicable) are sufficiently documented so that someone reasonably familiar with the underlying theory is able to use them — in addition to correctness, this is a criterion by which the spreadsheets will be marked.

On all tasks, implement your calculations in a Microsoft Excel or Open Office workbook (you may use VBA or another programming language if you wish) and in an accompanying document (PDF, RTF or Microsoft Word) supply a brief explanatory text, providing where appropriate a mathematical justification for your calculations and an interpretation of your results. Please note that not every question necessarily has a single “right” answer. Answers to such questions will be assessed as to whether they are well argued, and the arguments substantiated (for example by mathematical or analytical derivation, or references to the literature).

Any sources used in the preparation of the assignment must be properly cited. Please also refer to the “Statement on plagiarism” in the Subject Outline.

1. Consider the data in the Microsoft Excel workbook YieldCurveData.xlsx. It contains the United States Treasury’s “Daily Yield Curve” data (sourced from www.treasury.gov) for the years 2007 to end of July 2019. These rates are what the US Treasury calls “CMT rates” (see https://www.treasury.gov/resource-center/faqs/Interest-Rates/Pages/faq.aspx).

(a) (2 marks) Convert these rates into continuously compounded yields.

(b) (3 marks) Assuming a one–factor Gauss/Markov HJM model with mean reversion fixed at a = 0.1, use the data for the six-month yields to determine an appropriate choice of the volatility level s. Do this separately for two data sets covering one year each, for 2008 and 2018.

(c) (5 marks) Assuming a one–factor Gauss/Markov HJM model, use the data for the onemonth and thirty-year yields to determine an appropriate choice of the mean reversion parameter a and volatility level s. Do this separately for two data sets covering one year each, for 2008 and 2018. Discuss the results in comparison with your results for (b) above.

(d) (10 marks) Suppose that at the beginning of January 2008 you have sold an at–the– money caplet covering a three-month accrual period beginning in three months’ time. If you hedge this caplet with daily rebalancing using the natural hedge instruments, what would be your profit/loss when this option matures? Assume that caplet price and hedges are calculated using a one–factor Gauss/Markov HJM model with the

1

parameters determined in (c). Assume that any profit/loss is invested/borrowed by buying/selling the zero coupon bond maturing at the same time as the option. Repeat this for a caplet sold at the end of, respectively, March 2008, June 2008, September 2008, March 2018, June 2018, and September 2018. Discuss the possible causes of profit/loss.

(e) (4 marks) Repeat (d) for the 2018 caplets, but using the 2007 model parameters. Discuss the result — in particular, comment on what you would expect to happen (and why), and what actually happens.

(f) (4 marks) Repeat (d) for all caplets, but using as hedge instruments two zero coupon bonds, one of which initially has a time to maturity of six months, and another which initially has a time to maturity of thirty years. Discuss the result — in particular, comment on what you would expect to happen (and why), and what actually happens.

2. Suppose that the dynamics of the default–free interest rate term structure are given by aone–factor Gaussian HJM model with volatility parameter s = 0.025 and mean reversion a = 0.08. The current term structure is flat at 5% continuously compounded for all maturities. Suppose further that the CDS spreads (annual, in arrears, in basis points) for a corporate entity A are deterministic and given by:

Maturity 1 yr 2 yrs 3 yrs 5 yrs 10 yrs

Spread 120 130 140 150 170

Assume that default, recovery in default, and default–free interest rates are mutually independent. Expected recovery is 40%. You may ignore accrued interest in the case of default. Entity A approaches a bank of negligible default risk, with the wish of A to enter into a 10–year interest swap, where A pays floating and receives fixed annually, with simple compounding.

(a) (10 marks) Taking into account counterparty credit risk, what is the level of the fixed coupon on this swap, which results in a zero initial mark–to–market value of the swap? If some of the above assumptions are relaxed, in which case would the bank be exposed to “wrong–way risk” in this transaction?

(b) (12 marks) Suppose the bank has an existing swap with the same counterparty, where the bank pays floating and receives fixed annually, with simple compounding. This swap has exactly three years left to run, and the fixed leg was set at 4.75% (simple compounding).

i. In this situation, what is the answer to (a) if there is no netting agreement inplace? If there is a netting agreement in place?

ii. Suppose further that the market price interest rate risk relevant in this contextis constant sHˆ = 0.15. With and without netting, determine the potential future exposure (PFE) for the bank for a one–year time horizon, with 99% confidence level, just before and just after entering into the 10–year swap with A. What is the impact of the market price of risk on the PFE?

2

Assignment

Assignment due on October 13, 2019 as per the Subject Outline. Late assignments attract a penalty of 5 marks (out of 50) per day.

Students, who submit a draft of their assignments by September 25, 2019, will receive feedback by October 4, 2019 (and may subsequently amend their draft).

Your assignment must be submitted electronically (Microsoft Excel or Open Office workbook and a text document — Microsoft Word or PDF) via UTS Online before midnight on the due date. If you choose to write/use any additional computer code (in any language) to solve this assignment, this computer code must also be supplied. The text document must be typeset (using for example LATEX or Microsoft Word). Handwritten, scanned assignments will not be accepted. Please make sure your spreadsheets (and any additional computer code, if applicable) are sufficiently documented so that someone reasonably familiar with the underlying theory is able to use them — in addition to correctness, this is a criterion by which the spreadsheets will be marked.

On all tasks, implement your calculations in a Microsoft Excel or Open Office workbook (you may use VBA or another programming language if you wish) and in an accompanying document (PDF, RTF or Microsoft Word) supply a brief explanatory text, providing where appropriate a mathematical justification for your calculations and an interpretation of your results. Please note that not every question necessarily has a single “right” answer. Answers to such questions will be assessed as to whether they are well argued, and the arguments substantiated (for example by mathematical or analytical derivation, or references to the literature).

Any sources used in the preparation of the assignment must be properly cited. Please also refer to the “Statement on plagiarism” in the Subject Outline.

1. Consider the data in the Microsoft Excel workbook YieldCurveData.xlsx. It contains the United States Treasury’s “Daily Yield Curve” data (sourced from www.treasury.gov) for the years 2007 to end of July 2019. These rates are what the US Treasury calls “CMT rates” (see https://www.treasury.gov/resource-center/faqs/Interest-Rates/Pages/faq.aspx).

(a) (2 marks) Convert these rates into continuously compounded yields.

(b) (3 marks) Assuming a one–factor Gauss/Markov HJM model with mean reversion fixed at a = 0.1, use the data for the six-month yields to determine an appropriate choice of the volatility level s. Do this separately for two data sets covering one year each, for 2008 and 2018.

(c) (5 marks) Assuming a one–factor Gauss/Markov HJM model, use the data for the onemonth and thirty-year yields to determine an appropriate choice of the mean reversion parameter a and volatility level s. Do this separately for two data sets covering one year each, for 2008 and 2018. Discuss the results in comparison with your results for (b) above.

(d) (10 marks) Suppose that at the beginning of January 2008 you have sold an at–the– money caplet covering a three-month accrual period beginning in three months’ time. If you hedge this caplet with daily rebalancing using the natural hedge instruments, what would be your profit/loss when this option matures? Assume that caplet price and hedges are calculated using a one–factor Gauss/Markov HJM model with the

1

parameters determined in (c). Assume that any profit/loss is invested/borrowed by buying/selling the zero coupon bond maturing at the same time as the option. Repeat this for a caplet sold at the end of, respectively, March 2008, June 2008, September 2008, March 2018, June 2018, and September 2018. Discuss the possible causes of profit/loss.

(e) (4 marks) Repeat (d) for the 2018 caplets, but using the 2007 model parameters. Discuss the result — in particular, comment on what you would expect to happen (and why), and what actually happens.

(f) (4 marks) Repeat (d) for all caplets, but using as hedge instruments two zero coupon bonds, one of which initially has a time to maturity of six months, and another which initially has a time to maturity of thirty years. Discuss the result — in particular, comment on what you would expect to happen (and why), and what actually happens.

2. Suppose that the dynamics of the default–free interest rate term structure are given by aone–factor Gaussian HJM model with volatility parameter s = 0.025 and mean reversion a = 0.08. The current term structure is flat at 5% continuously compounded for all maturities. Suppose further that the CDS spreads (annual, in arrears, in basis points) for a corporate entity A are deterministic and given by:

Maturity 1 yr 2 yrs 3 yrs 5 yrs 10 yrs

Spread 120 130 140 150 170

Assume that default, recovery in default, and default–free interest rates are mutually independent. Expected recovery is 40%. You may ignore accrued interest in the case of default. Entity A approaches a bank of negligible default risk, with the wish of A to enter into a 10–year interest swap, where A pays floating and receives fixed annually, with simple compounding.

(a) (10 marks) Taking into account counterparty credit risk, what is the level of the fixed coupon on this swap, which results in a zero initial mark–to–market value of the swap? If some of the above assumptions are relaxed, in which case would the bank be exposed to “wrong–way risk” in this transaction?

(b) (12 marks) Suppose the bank has an existing swap with the same counterparty, where the bank pays floating and receives fixed annually, with simple compounding. This swap has exactly three years left to run, and the fixed leg was set at 4.75% (simple compounding).

i. In this situation, what is the answer to (a) if there is no netting agreement inplace? If there is a netting agreement in place?

ii. Suppose further that the market price interest rate risk relevant in this contextis constant sHˆ = 0.15. With and without netting, determine the potential future exposure (PFE) for the bank for a one–year time horizon, with 99% confidence level, just before and just after entering into the 10–year swap with A. What is the impact of the market price of risk on the PFE?

2

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