Actuarial Outpost FM inflation/geometric annuity problem help
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#1
05-12-2019, 09:32 AM
 xkang1129 SOA Join Date: Mar 2019 College: University of Missouri St Louis Posts: 6
FM inflation/geometric annuity problem help

Can someone help me with this question?

Suppose you are the actuary for an insurance company. Your company, in response to a policyholder claim involving physical injury, is responsible for making annual medical payments. The first payment will occur on January 1, 2008, and the final payment will occur on January 1, 2031. The first payment will be \$100,000; after that, the payments will increase annually for inflation, at a rate of 5% per year. The real interest rate is 3% per year. Find the present value of these future payments as of December 31, 2005.

My solution (yields the wrong answer, not sure where went wrong):

Payments: 100,000 unit, 1, 1.05, 1.05^2, 1.05^3, ... 1.05^23
PV as of January 1, 2007: 100,000 ( v + 1.05V^2 + 1.05^2V^3......1.05^23V^24)
= 100,000 V * S24(at 1.94%, 1.94=1.05/1.03)
= 2,932,053.088

PV as of December 31, 2005:
2,932,053.088 * V = 2,932,053.088 / 1.03 = 2,846,653.483

Can someone help point out where I did wrong please? Many thanks!
#2
05-12-2019, 09:45 AM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,287

Quote:
 Originally Posted by xkang1129 Can someone help me with this question? Suppose you are the actuary for an insurance company. Your company, in response to a policyholder claim involving physical injury, is responsible for making annual medical payments. The first payment will occur on January 1, 2008, and the final payment will occur on January 1, 2031. The first payment will be \$100,000; after that, the payments will increase annually for inflation, at a rate of 5% per year. The real interest rate is 3% per year. Find the present value of these future payments as of December 31, 2005. Correct Answer: 1,491,363 My solution (yields the wrong answer, not sure where went wrong): Payments: 100,000 unit, 1, 1.05, 1.05^2, 1.05^3, ... 1.05^23 PV as of January 1, 2007: 100,000 ( v + 1.05V^2 + 1.05^2V^3......1.05^23V^24) = 100,000 V * S24(at 1.94%, 1.94=1.05/1.03) = 2,932,053.088 PV as of December 31, 2005: 2,932,053.088 * V = 2,932,053.088 / 1.03 = 2,846,653.483 Can someone help point out where I did wrong please? Many thanks!
You are using a nominal rate of 3% instead of a real rate of 3%.
#3
05-12-2019, 10:06 AM
 xkang1129 SOA Join Date: Mar 2019 College: University of Missouri St Louis Posts: 6

Ahh! got it!! Many thanks, Gandalf!
#4
05-12-2019, 11:14 AM
 xkang1129 SOA Join Date: Mar 2019 College: University of Missouri St Louis Posts: 6

Correct solution:
Nominal rate: 1.05 * 1.03 - 1 = 0.0815
Payments: 100,000 unit, 1, 1.05, 1.05^2, 1.05^3, ... 1.05^23
PV as of January 1, 2007: 100,000 ( v + 1.05V^2 + 1.05^2V^3......1.05^23V^24) (at 8.15%)
= 100,000 V(8.15%) * a24(at 3%, 1/(1.05/1.0815))
= 1,612,908.8

PV as of December 31, 2005:
1,612,908.8 * V = 1,612,908.8 / 1.0815 = 1.4913627

 Tags geometric series