(UPS Battery) On January 1, 2015, you bought a backup battery for your office uninterruptible power supply (UPS) system. In each of the next three years, the battery can fail and a new battery needs to be bought (if you want to keep using the UPS system). The maximum lifespan of the battery is four years; i.e., the battery must be replaced after four years. Due to chemical properties, the probability of the battery failure is fixed at 20% in each year. Each year, it is estimated that the battery will save $1,200 in backup power cost. The market interest rate is 8%. Suppose battery failures only occur at end of year. (a) Provide a fair valuation for the battery on January 1, 2015. (12 points) (b) It's now January 1, 2016. Suppose the battery did not fail yet, and because of a new ventilation system you have recently installed in the office, the failure probability of the battery has dropped to 15% in each of the two coming years. What are the normal and abnormal economic earnings in 2015? (8 points)