Please follow the instructions on how to answer the questions and please provide all steps and working. Thank you.
This assignment contains three parts. · The first part asks you to compute 15 basic questions (3% each). · The second part asks you to compute 10 examples (3.3% each). · The third part is a full problem with 8 questions (4% each). The three parts are independent. If no indication is provided, give the results with 2 digits. Remember to read carefully the instructions to avoid mistakes! Instructions for due date: · Exams should either be (clearly) handwritten and scanned or written in word or LaTeX, where pictures can be included for providing mathematical justifications to the answers. · Please provide readable pictures or scans. Care about your writing and the lighting (you should have plenty of time to do so). PART I – Basic Course Questions (45%): For each question, provide your answer in the clearest and shortest possible way. Justify every result. Each question is worth 3%. 1. Circle the true statements in the following list: a. The expectation hypothesis (EH) and the no-arbitrage hypothesis are the same things. b. EH implies that excess returns are always null. c. EH implies that forward rates can be interpreted as forecasts of future spot yields. d. EH does not usually hold in the data. e. A world where bond investors are neutral to risk is incompatible with EH. 2. Assume the spot yield curve is given by . Provide the formula for calculating the excess returns of holding the 10y bond for one year. 3. After running the following regression of future excess returns onto the current slope of the yield curve: I find the following estimates along with their estimated standard deviations. Coefficient Estimate -1.41 0.78 Standard deviation 0.51 0.62 Circle the true statements in the following list: a. I can reject EH. b. Investors are risk-averse. c. The slope of the yield curve significantly predicts future returns at 5% significance level. d. The average risk premium is negative. e. Investors fear increases of the short-term interest rate. 4. We assume that the 1y zero coupon yield moves according to the following dynamics. For each year that go by, it has 50% chance of staying the same, 25% chance of going down by 10bps, 25% chance of going up by 10bps. Explain why the yield curve is flat under the expectation hypothesis. 5. We assume the 1y zero coupon yield has following risk-neutral dynamics. For each year that go by, it has 50% chance of staying the same, 20% chance of going down by 10bps, 30% chance of going up by 10bps. Comparing with the dynamics of the previous question, do investors fear spot rates going up or down? Explain. 6. Assume time is continuous. What does the following quantity represent economically? 7. Assume the instantaneous spot rate dynamics are given by: How is the parameter called? Comment on the dynamics of the short rate if is close to zero. 8. In Vasicek model, the loading associated with the instantaneous short rate is given by: Where is between 0 and 1, and is the maturity of the zero-coupon bond. We are interested in a very long maturity bond (). What is the impact of movements of the spot rate on that bond yield? Explain. 9. Consider a floater with 10y to maturity paying annually. Coupon has just been paid. What is the price of the bond? 10. The computation for a (continuously compounded) forward rate relies on a particular no-arbitrage relationship between two riskless strategies. Provide the equation summarizing this relationship. 11. Assume a 2-year swap paying on an annual basis. Provide the theoretical formula of the swap rate at issuance as a function of . 12. Assume Nelson-Siegel model holds with tuning parameter . What is the level and slope durations of a 10y zero-coupon bond? 13. Explain what bullet hedging is. Is a bullet-hedged portfolio insensitive to movements on yields? Why? 14. Assume that the central bank moves interest rates according to a Taylor rule (without smoothing). After providing the equation summarizing its reaction function, provide qualitative guidance into what could push the central bank to raise interest rates in the context of the model. 15. You own a TIPS maturing in the next month. Explain why there is no uncertainty about the cashflows it provides at maturity. PART II - Pricing examples (33%): For each question, provide your answer in the clearest and shortest possible way. Justify every numerical result by detailing your computations. Each question is worth 3.3%. 1. Convert the following prices into yields in continuous compounding convention (2 digits): Maturity 0.5y 1y 1.5y 2y 2.5y 3y Price 99.75 99.25 98.51 97.63 96.80 96.03 Yield 2. Price a coupon bond paying semi-annually a coupon rate of 3%, with maturity 2.5y. Deduce the value of long position in a swap with maturity 2.5y and swap rate equal to 3% (semi-annually compounded). [Note: a long position in a swap has cashflows = Floating - Fixed] 3. Provide the duration of the coupon bond of the previous question. Deduce the duration of the swap. 4. Compute the following continuously compounded forward yields using yields of question II-1: a. The 2y in 1y b. The 1y in 2y c. The 2.5y in 6m d. The 6m in 2.5y 5. You own a 2.5y zero coupon bond. Assuming the yield curve does not change over time and is given by the table of question II-1, compute the (log) excess returns of selling your bond after 1.5y has passed. 6. You own a basic floater paying semi-annually, which has 15months to maturity. Three months ago, the yield curve was given by the table of question II-1, and the current yield curve is flat at 3% continuously compounded. Provide the value of your floater. 7. You own an FRA where observation date is 6 months from now, and payment date is 2 years from now. Its fixed rate is 2%. Using the yields of question II-1, show that the FRA has been issued in the past, and provide the value of the FRA. 8. You own a 2y coupon bond paying semi-annually a coupon rate of 4%. You want to replace all your fixed coupon payments by floating payments. Provide the instrument, the position, and the notional that you need to perform such an exchange. Interpret your aggregated position as a combination of floaters and zero-coupon bonds. 9. Using the inputs of the previous question, provide the duration of the 2y coupon bond and that of the aggregated position with the swap. 10. Assume the BEI curve is flat at 2% continuously compounded. Compute the price of a 2y TIPS bond paying semi-annually [Note: we neglect the inflation lag and the embedded deflation floor]. PART III- Problem (32%): For each question, provide your answer in the clearest and shortest possible way. Justify every numerical result by detailing your computations. Each question is worth 4%. In this problem, we study the implications of a standard Taylor rule on the relative pricing of nominal bonds and TIPS. We assume that the 1y (cont. comp.) yield is given by: Where is the yearly inflation rate, is the yearly output gap, and is a monetary policy shock of (risk-neutral) mean zero. We also have the risk-neutral dynamics of inflation and output that are such that their risk-neutral expectations are given by: 1. Using the fact that forwards are risk-neutral expectations of future spot yields, show that the forward yield as a function of , and is given by: 2. Justify the following two following equalities: 3. Deduce the yield of a zero-coupon bond as a function of its maturity , and of the current values of and . Notes: The following result will come in handy. For any , 4. Deduce the inflation and output-gap duration of a standard zero-coupon bond as a function of its maturity. 5. Using the fact that the yield of a -maturity TIPS is given by: Compute the general formula for TIPS yields as a function of current inflation, output gap, and maturity. 6. Deduce the inflation and output gap duration of TIPS. Compare with that of nominal bonds. Does the result make sense? Notes: We will assume that the basic level duration of a TIPS is equal to its maturity. 7. Assume that the objective (true) forecasts are given by: Compute the risk premium of nominal bonds and TIPS as a function of and , and deduce the inflation risk premium as the difference between nominal and TIPS premia. When is this premium high? 8. Fill in the following table, assuming the current inflation is at and output is at . Maturity 1y 10y Nominal yield TIPS yield Nominal risk premia TIPS risk premia Inflation risk premia BONUS QUESTION: Price the term structure of output-gap linked bonds, which work in the same way as TIPS but adjust principal using output-gap instead of inflation. Use the model presented previously. FIXED INCOME SECURITIES FIXED INCOME SECURITIES -- Chapter 12-- Risk management I: Duration and convexity Guillaume Roussellet – Winter 2021 Motivating example: QE transmission channels Asset valuation channel: it posits that [risk] premium is a function of the stock of long-term bonds held by the private sector… A policy of asset purchases replaces longer-term and/or riskier assets with short-term and safe central bank reserves. The duration risk channel posits that the bond risk premium is increasing in the exposure of [private] bond holders. By reducing private sector holdings of such bonds, central bank purchases should reduce exposure to duration risk and thus lead to a decline in yields. Duration risk denotes the exposure of long-term bonds to unexpected changes in policy interest rates. Such risk induces a premium on bond yields. Long-duration bonds are riskier, because they are more sensitive to interest rate risk… The capital relief channel suggests that the increase in asset prices that [a purchase programme] generates is akin to a capital injection for leverage-constrained institutions, the higher prices of sovereign bonds induced by the asset purchase programme should benefit banks through the ensuing increased valuation of their bonds holdings. Signaling channel: Quantitative easing is akin to forward guidance, namely an announcement that policy interest rates will remain at the lower bound for a longer period. The marginal benefit over forward guidance is to make the announcement more credible. Credibility is higher because large-scale purchases of long-term assets expose the central bank to the risk of losses on its balance sheet, in case short-term rates are abruptly increased. This provides an incentive for keeping