I'll assume youve been taught standard actuarial notation (ie among other things):
The scenario you are given is:
Step 0: You have $10k
Step 1: you save $P per year for 15 years (interest 12%) in arrear (15 payments total). At the end of this you are 45 years old.
Step 2: At age 45, you stop paying into your savings plan. The existing value grows at 10%
Step 3: At age 60, you use the accumulated value of your plan to buy a $100k annuity, payable in arrear for 20 years.
The easiest way to work through these problems is to work backwards
Think of the question as:
"How much money must you have on your 45th birthday, in order to pay for $100k per year for 20 years, with the first payment delayed for 16 years".
First note: The interest rate at all times from age 45 is 10%, so use i=0.1 when working this out
Start by working out the lump sum required at age 60 to pay for the annuity. This is just a $100k annuity for 20 years, payable in arrear:
Now, work out what you need at age 45 in order to have a lump sum of S at time 60:
Value of Z in 15 years = :
This is the answer to the question.