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Cell Phone Insurance June 18, 2010

Posted by tomflesher in Economics.
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Yesterday, I bought a new phone. It’s a Samsung Gravity 2 and with a two-year contract it cost $79.99 – it came with some accessories that aren’t of interest for now. The salesman tried to sell me insurance at a whopping $4.99 per month over the course of the contract. I told him I’d do $4.99 total, because I’m an economist, but he didn’t bite. (Sigh.)

How bad a deal is that? Well, I wanted to find out. First, I made some assumptions:

  • The appropriate interest rate is 1.25 APY (.1042 MPY), which is roughly what my bank account is paying. I could put some amount of money in the bank right now and earn interest at that rate and it would be enough for me to pay the insurance. This is called the Net Present Value, and over 24 months at 4.99 per month it’s about $118.34.
  • The likelihood of something happening to my phone is entirely random, so I can’t take it into account when determining whether the insurance is a good buy.
  • My phone depreciates at a rate of e^{-.998058*t} , where t is the number of the month (so this month is month 1, next month is month 2, etc.). This puts my discount rate at exactly my APY. It makes for a quick depreciation, with the phone getting within a dollar of its resale value within about 4 months. It caputres the quick drop in depreciation an the slow leveling off quite nicely.
  • The definition of ‘good value’ is that at the time I turn in a damaged phone, its depreciated value is less than the cost of all the premiums I’ve paid. I chose to use the depreciated value rather than the cost of a new phone because it reflects that I’ve gotten some use out of the phone.

The long and the short of it is that if I damage the phone before about the 7th month, it’s a good value. After that, it’s all gravy for T-Mobile.

I ended up telling the salesman that I’m an economist and so paying that much for insurance is against my religion.

For those who are interested in the chart, it’s behind the cut. It lists monthly payment, month ordinal, the effective interest rate, present value of that payment, NPV as sum of the present values, the depreciated value of the phone, the depreciation factor, and the instantaneous depreciation.

Flow t Interest PV NPV phoneval dexp dep
-4.99 0 1.0000 -4.99 -4.99 79.99 1.00 0.00
-4.99 1 0.9990 -4.98 -9.97 79.99000 0.36826 29.45736
-4.99 2 0.9979 -4.98 -14.95 50.53264 0.13562 6.85312
-4.99 3 0.9969 -4.97 -19.93 43.67953 0.04994 2.18148
-4.99 4 0.9958 -4.97 -24.90 41.49804 0.01839 0.76324
-4.99 5 0.9948 -4.96 -29.86 40.73481 0.00677 0.27590
-4.99 6 0.9938 -4.96 -34.82 40.45890 0.00249 0.10092
-4.99 7 0.9927 -4.95 -39.77 40.35799 0.00092 0.03707
-4.99 8 0.9917 -4.95 -44.72 40.32092 0.00034 0.01364
-4.99 9 0.9907 -4.94 -49.67 40.30728 0.00012 0.00502
-4.99 10 0.9896 -4.94 -54.61 40.30226 0.00005 0.00185
-4.99 11 0.9886 -4.93 -59.54 40.30041 0.00002 0.00068
-4.99 12 0.9876 -4.93 -64.47 40.29973 0.00001 0.00025
-4.99 13 0.9866 -4.92 -69.39 40.29947 0.00000 0.00009
-4.99 14 0.9855 -4.92 -74.31 40.29938 0.00000 0.00003
-4.99 15 0.9845 -4.91 -79.22 40.29935 0.00000 0.00001
-4.99 16 0.9835 -4.91 -84.13 40.29934 0.00000 0.00000
-4.99 17 0.9825 -4.90 -89.03 40.29933 0.00000 0.00000
-4.99 18 0.9814 -4.90 -93.93 40.29933 0.00000 0.00000
-4.99 19 0.9804 -4.89 -98.82 40.29933 0.00000 0.00000
-4.99 20 0.9794 -4.89 -103.71 40.29933 0.00000 0.00000
-4.99 21 0.9784 -4.88 -108.59 40.29933 0.00000 0.00000
-4.99 22 0.9773 -4.88 -113.47 40.29933 0.00000 0.00000
-4.99 23 0.9763 -4.87 -118.34 40.29933 0.00000 0.00000
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