The present value of 21,000 discounted two periods at 8% -s 21,000 / (1.08)^2 = 18,004.12
The present value of 22,050 us 22.050 / (1.08)^3 = 17,504
etc. for 30 annual incomes. Add them all up and you have your answer. There may be a formula that takes 5% growth into account but I don't know it.
2. Here again you have a different investment every year so you can find the individual future values 30 times.
1. 1,000(1.08) = $1080
2 (1,050 + 1080)(1.08 = 2,300.40
3. (1,102.50 + 2,300.40)(1.08) = 3,675.13
etc.
2. This is another one with odd periods and individual computations. I draw a time line to see what is happening
1 . . 2 . . -1.5 . .\. . 2. . .\ . . .-1 . . .?
I suggest you put in in a spreadsheet
3. This is complicated because the payments are annual but compounding is semiannual. A good approximation can be obtained by assuming you are making semiannual payments of 2,500 or annual compounding.
FV = 100,000
PMT = $5,000
i% = 10%
N = ??
A financial calculator solves this easily, but you can also use a spreadsheet function or a FV of annuity formula. The answer is 11.5 years with annual compounding.
FV = 100,000
PMT = 2,500
i% = 5%
N = ?? = 11.26 years. with semiannual compounding
So the answer is between 11 and 12 years.
If you want a precise answer you can make some 20 or more individual calculations. A spreadsheet can be set up to do the calculations..
1st: Tom is 30 years old today. His salary next year will be $20,000. He forecasts salary growth of 5%/year and plans to retire at 60.
1) If the discount rate is 8%, what is the PV of his future salary receipts?
2) He plans to save 5% of his salary each year and invest it at 8%. Once retired, he plans to spend it evenly over the next 20 yrs. How much will be able to spend?
3) What if the amount spend in retirement grows at 3%/year. What would be the amount withdrawn in the first year?
2nd: You plan to make a series of deposits in an individual retirement account. You will deposit $1,000 today, $2,000 in year 2, and $2,000 in year 5. If you withdraw $1,500 in year 3 and $1,000 in year 7, assuming no withdrawal penalties, how much will you have after 8 years if the interest rate is 7%? What is the present value of these cash flows?
3rd: How many years to save $100 000 by depositing $5 000 at the beginning of each year, earning 10% compounded semi-annually?
usually its solved by: 5 000 = 100 000/1.05t
20 = 1.05t
but does 5 000 and 10% need to be divided by 2 since they are annually based and the question says its compounded semi-annually?
