1 Bond Portfolio Management (30 Points) The file bonds.csv contains Treasury bond data collected using different dates over the last few years. Given the above table, address the following questions: 1. Specifically, the data was collected from 5 different periods. For each period, the data cor- responds to the yield curve for 2, 5, 10, and 30 years maturity. Given historical data on the yield curve, can you identify these five dates? (5 Points) Note: To answer this, you need to download data on Treasury yields of different maturities using the FRED database. In particular, you need to download data for the following codes DGS2, DGS5, DGS10, and DGS30. After merging and dropping missing values, the final dataset is daily and should date between Jan 2nd, 2018 and Nov 4, 2021. 2. Use the pricing equation of a fixed-coupon bond to price each bond from the bonds.csv file. I recommend writing a function that takes yield, coupon, face value, and maturity as its main arguments. The resulting prices should correspond to the ones reported above. Hence, you should plot the computed prices against the given ones. To confirm, you should observe a 45-degree line. (5 Points) 3. Prices should reflect investors' perception of future interest rates. Rather than computing the prices using yields as the case in the previous question, in practice, it is the other way around. We try to deduce yields from market prices. Hence, given a pricing function, you need to find the yield that matches the market price. For each bond, find the implied yield and plot it against the corresponding yield from the bonds.csv data. Again, this should result in a 45-degree line. (5 Points) Hint: This relies on a numerical solution. Recall that the solution for function f is the x* that satisfies f(x*) = 0. Since the price of the bond is a function of yield, i.e. f(y) = P, design a function g(y) = f(y) - Po, where Po is fixed using the values from the given prices. As a result, the implied yield is the solution y* that satisfies g(y*) = 0, i.e. f(y*) = P. In R, you may refer to the uniroot function. In Excel, this can be attained using "goalseek. 4. Compute and report the Macaulay duration for each bond in the bonds.csv data file. This should correspond to 20 numbers. As a summary, report the min and max for each duration by maturity. Your summary should correspond to 4 x 2 table, where each row corresponds to a distinct maturity and columns correspond to statistics (i.e., min and max). How do the results compare within columns? What about within rows? Provide some rationale. (5 Points) 5. Using first order Taylor expansion, calculate the change in the Treasury bond prices, if the yield curve in the US shifts up by 25 bps. Focus only on the recent bond data to answer this part, i.e. bonds numbered 17, 18, 19, and 20. To summarize, plot both the original and new prices against maturity. How do you justify this observation? (5 Points) Note: since you have a pricing function for a fixed coupon bond, you should confirm whether the new price is correct. For instance, if the price P is a function of yield y, then we know that price is P = f(y). To check whether your answer is correct, you should compare your Taylor expansion results with the exact price, which would be P1 = f(y + Ay). 6. Assume that the prices in the above table reflect the dollar price of each bond, e.g. the price of bond 9 is $101.57. As a portfolio manager, you need to allocate $100,000 between bonds 17 and 18 from the bonds.csv data file. If you believe that the Federal Reserve will increase interest rates in the near future, you need to limit your portfolio duration to 3 years. As a result, how many units of each bond you need to purchase to satisfy this? How would your answer change if you target a duration of 6 years instead? Explain why these numbers make sense. (5 Points) 7. Bonus Question Consider the details from the previous question. However, in this case, you need to allocate $100,000 among the four Treasury bonds numbered 17, 18, 19, and 20. If you are targeting a portfolio duration of 6 years, how many units of each bond you need to buy? The position in each individual bond should not be zero. (5 Points) Hint: In this case, you need to satisfy two conditions by choosing four unknowns. This results in an under-determined linear system of equations. To solve this, you need to think in terms of a generalized solution. A possible suggestion is to look into a generalized matrix inverse - for instance, see Moore-Penrose pseudoinverse (Wiki page). As a confirmation, check whether the proposed solution satisfies the two requirements. 1 Bond Portfolio Management (30 Points) The file bonds.csv contains Treasury bond data collected using different dates over the last few years. Given the above table, address the following questions: 1. Specifically, the data was collected from 5 different periods. For each period, the data cor- responds to the yield curve for 2, 5, 10, and 30 years maturity. Given historical data on the yield curve, can you identify these five dates? (5 Points) Note: To answer this, you need to download data on Treasury yields of different maturities using the FRED database. In particular, you need to download data for the following codes DGS2, DGS5, DGS10, and DGS30. After merging and dropping missing values, the final dataset is daily and should date between Jan 2nd, 2018 and Nov 4, 2021. 2. Use the pricing equation of a fixed-coupon bond to price each bond from the bonds.csv file. I recommend writing a function that takes yield, coupon, face value, and maturity as its main arguments. The resulting prices should correspond to the ones reported above. Hence, you should plot the computed prices against the given ones. To confirm, you should observe a 45-degree line. (5 Points) 3. Prices should reflect investors' perception of future interest rates. Rather than computing the prices using yields as the case in the previous question, in practice, it is the other way around. We try to deduce yields from market prices. Hence, given a pricing function, you need to find the yield that matches the market price. For each bond, find the implied yield and plot it against the corresponding yield from the bonds.csv data. Again, this should result in a 45-degree line. (5 Points) Hint: This relies on a numerical solution. Recall that the solution for function f is the x* that satisfies f(x*) = 0. Since the price of the bond is a function of yield, i.e. f(y) = P, design a function g(y) = f(y) - Po, where Po is fixed using the values from the given prices. As a result, the implied yield is the solution y* that satisfies g(y*) = 0, i.e. f(y*) = P. In R, you may refer to the uniroot function. In Excel, this can be attained using "goalseek. 4. Compute and report the Macaulay duration for each bond in the bonds.csv data file. This should correspond to 20 numbers. As a summary, report the min and max for each duration by maturity. Your summary should correspond to 4 x 2 table, where each row corresponds to a distinct maturity and columns correspond to statistics (i.e., min and max). How do the results compare within columns? What about within rows? Provide some rationale. (5 Points) 5. Using first order Taylor expansion, calculate the change in the Treasury bond prices, if the yield curve in the US shifts up by 25 bps. Focus only on the recent bond data to answer this part, i.e. bonds numbered 17, 18, 19, and 20. To summarize, plot both the original and new prices against maturity. How do you justify this observation? (5 Points) Note: since you have a pricing function for a fixed coupon bond, you should confirm whether the new price is correct. For instance, if the price P is a function of yield y, then we know that price is P = f(y). To check whether your answer is correct, you should compare your Taylor expansion results with the exact price, which would be P1 = f(y + Ay). 6. Assume that the prices in the above table reflect the dollar price of each bond, e.g. the price of bond 9 is $101.57. As a portfolio manager, you need to allocate $100,000 between bonds 17 and 18 from the bonds.csv data file. If you believe that the Federal Reserve will increase interest rates in the near future, you need to limit your portfolio duration to 3 years. As a result, how many units of each bond you need to purchase to satisfy this? How would your answer change if you target a duration of 6 years instead? Explain why these numbers make sense. (5 Points) 7. Bonus Question Consider the details from the previous question. However, in this case, you need to allocate $100,000 among the four Treasury bonds numbered 17, 18, 19, and 20. If you are targeting a portfolio duration of 6 years, how many units of each bond you need to buy? The position in each individual bond should not be zero. (5 Points) Hint: In this case, you need to satisfy two conditions by choosing four unknowns. This results in an under-determined linear system of equations. To solve this, you need to think in terms of a generalized solution. A possible suggestion is to look into a generalized matrix inverse - for instance, see Moore-Penrose pseudoinverse (Wiki page). As a confirmation, check whether the proposed solution satisfies the two requirements