Mathematics of Finance. TAKE-HOME FINAL EXAM.
Due May 16, 2024, 11.59pm May 16, 23:59. If you are graduating, the take-home exam is due May 15, 23:59
Please write a pledge that final exam solutions represent your own work and that you did not copy solutions from the work of other students or other sources, and did not use any other forbidden means and did follow Columbia code of conduct.
Each Problem is worth 10pt
1. Give the definition of the stock beta. Estimate MSFT stock beta relative to Nasdaq 100 using last 250 days historical regression from March 31, 2025 back 250 trading days. You can use QQQ as a proxy for Nasdaq 100. Please use adjusted price for both and subtract risk free rate (you can use SOFR).
2. Using bootstrap method (Hull) calculate zero coupon yield curve from coupon bearing bonds using continuous compounding.
Bond Principal Maturity (years) Coupon Bond Price
100 0.50 0 98.25
100 1.00 0 96.125
100 1.50 3 96.25
100 2.00 4 96.125
3. Give the definition of duration and convexity of a bond and write the relationship between the change in yield and the bond price.
4.Bond is maturing in 10.5 years and has an annual coupon rate 4.0% coupon paid semiannually and price 91-16
a) Calculate its Yield to Maturity. (You can use Excel and try several yields until you get required price.)
b) Suppose you have 8 Million market value of this bond. What is the Modified Duration, Macaulay Duration, DV01 and convexity of this portfolio?
c) Using Duration and Convexity formula approximation calculate bond price if the Yield to Maturity is increased 10 Basis Points.
d) Calculate the exact Bond price for 10 Basis points increased yield using the full discount formula. Compare the two results c) and d)
5. Which of these bonds is cheaper on a relative value basis i.e. which one has a higher yield: Bond A: Maturity 10yr, Coupon 5.7%, Price 83-12, Or Bond B Maturity 9yr, Coupon 6.1% Price 82-10. For coupons rate is annual and coupons are paid semiannually.
6. Consider an investment consisting of 1,800,000 dollars investment in index A and 2,000,000 investment in index B. Assume that daily volatility of each asset is 2% and correlation of returns is 0.4. Calculate 1 day and 10 days Value-At-Risk with 97 percent confidence.
7.Bonds with higher coupons everything else being equal
a) Have higher modified duration than smaller coupon bonds
b) Have lower modified duration than smaller coupon bonds
c) Can have higher or lower modified duration.
Explain your answer