I need help figuring out this problem but I need help understanding how to solve

Question

I need help figuring out this problem but I need help understanding how to solve it I don't just need an answer as really the answer is unimportant its just me knowing how to solve a problem like this.

A plan to build a new tower would involve a building tax that would collect $6,745 each year for all people in a certain city. If the cash flow begins 2 years from now, what is the present worth of the building plan over a 10 year planning period at an interest rate of 6% per year?

Thanks for anyone who can help me understand how to solve this problem with detailed steps and explanations. Calculators cannot be used - I need to see how to actually solve the problem.

Explanation / Answer

EDIT: See Method 2 or Method 3 for "non-calculator" methods.

Ok so the cash flows are all the same, $6,745, and they occur for a period of 10 years at an interest rate of 6%, but they don't start after two years have passed so essentially the cash flows are as follows:

Period 0: $0 (end of year 1)
Period 1: $6,745 (end of year 2)
Period 2: $6,745 (end of year 3)
Period 3: $6,745
Period 12: $6,745

Now, this can be solved one of three ways. The easiest of which is using a financial calculator as such:

Calculator Method 1: (if you don't have, or are not familiar with a finacial calculator, see Method 2 or 3)

First, discount the $6,745 annuity payments to the present value at period 2 (where the cash flows begin). Use the following keystrokes:

[N] = 10

[I/Y] = 6

[PMT] = 6745

[CPT][PV] = $49,643.79 (answer may appear negative)

Now, take this number $49,643.79 and discount it to present value at period 0 using the following keystrokes:

[2nd][CLR TVM]

[N] = 1

[I/Y] = 6

[FV] = 49,643.79

[CPT][PV] = $46,833.76

Method 2 - PV of an Ordinary Annuity Table:

The slightly less accurate way of solving this problem is using a PV of an ordinary annuity table. This is a standard table of multipliers that will discount any amount to the corresponding period.

First, find the number for 10 periods at 6% interest and discount the cash flows over 10 years.

$6,745 * 7.36009 = $49,643.81 (notice the slight rounding error)

Now take this number and discount it back one more year (note: it has already been discounted to the beginning of year 2, that why we only need one more year of discounting). Use the figure from the present value of $1 table (different from the annuity table) for 1 period and 6% interest.

$49,643.81 * .94340 = $46,833.97 (again, theres a small rounding error)

Method 3 - Formula Method:

$6,745/(1+.06)^1 + 6745/(1+.06)^2 + 6745/(1+.06)^3 + 6745/(1+.06)^4 + 6745/(1+.06)^5 + 6745/(1+.06)^6 + 6745/(1+.06)^7 + 6745/(1+.06)^8 + 6745/(1+.06)^9 + 6745/(1+.06)^10 = $49,643.79

Now take this number and discount it back by one more period as in the previous methods (but using a formula instead).

$49,643.79/(1+.06)^1 = $46,833.76

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