Question

(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)

Answer 1

a.The UPS battery can be fairly valued as

the total Present Value of the probable annuity of savings in back-up power costs over the next 3 years-- at 8% discount rate

ie.((1200*80%)+(0*20%))*2.57710=                 (P/A,i=8%,n=3 )

2474.02

b.Normal economic earnings

will be the present value as above

ie. 2472.02

Abnormal economic earnings

will be the difference between the PV of the income predicted before & after the introduction of the new system

PV of Income after the introduction of the new system:

ie.(((1200*80%)+(0*20%))/1.08^1)+(((1200*85%)+(0*15%))/1.08^2)+(((1200*85%)+(0*15%))/1.08^3)

2573.08

So,

Abnormal economic earnings=

2573.08-2472.02=

101.06

For the year 2015, both normal & abnormal earnings are the same

ie.((1200*80%)+(0*20%))*0.92593=                 (P/F,i=8%,n=1 )

888.89