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1 Concepts from the example problems ( F / P, i, N) Problem 1 ( P / F, i, N) Problem 2 ( F / A, i, N) Problem 3 ( A / F, i, N) Problem 4 ( P / A, i, N) Problem 5 ( A / P, i, N) Problem 6 ( P / G, i, N) Problem 7 ( A / G, i, N) Problem 8

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2 Example 1 How much would you have to deposit today to have $2000 in 4 years if you can get a 12% interest rate compounded annually? GIVEN: F 4 = $2 000 i = 12% FIND P: P = F 4 (P/F,i,n) = 2 000(P|F,12%,4) = 2 000(0.6355) = $ n=4 P? $2 000 DIAGRAM:

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3 Different Ways of Looking at P/F Different Ways of Looking at P/F From previous example, if you can earn 12% compounded annually, you need to deposit $1271 to have $2000 in 4 years. You are indifferent between $1271 today and $2000 in 4 years, assuming you can earn a return on your money of 12%. The present worth of $2000 in 4 years is $1271 (i = 12% cpd ‡ annually). ‡ my abbreviation for compounded If you could get 13% on your money, would you rather have the $1271 today, or $2000 in 4 years ?

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4 Example 2 Tuition costs are expected to inflate at the rate of 8% per year. The first year’s tuition is due one year from now and will be $10,000. To cover tuition cost for 4 years, a fund is to be set up today in an account that will earn interest at the rate of 5% per year, compounded annually. How much must be deposited into the fund today in order to pay the 4 years of tuition expenses?

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5 Present Given Gradient (Geometric) g is the geometric gradient over the time period (time period: Time 0 to Time n, 1st flow at Time 1) P is the present value of the flow at Time 0 (n periods in the past) i is the effective interest rate for each period Note: cash flow starts with A 1 at Time 1, increases by constant g% per period 0n A1A1 P ? 123 P = A 1 (P/A,g,i,n) g = %

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6 Example 2 Tuition costs are expected to inflate at the rate of 8% per year. The first year’s tuition is due one year from now and will be $10,000. To cover tuition cost for 4 years, a fund is to be set up today in an account that will earn interest at the rate of 5% per year, compounded annually. How much must be deposited into the fund today in order to pay the 4 years of tuition expenses?

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7 Example 2 - Concept If your rich Aunt Edna wanted to put a sum of money in the bank today to pay for your next four years of tuition, that sum would be $39,759 assuming 5% return on investment and tuition that begins at $10,000 increasing by 8% per year. This problem assumes tuition is due at the end of the year.

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8 Complex Cash Flows Complex Cash Flows – Break apart (or separate) complex cash flows into component cash flows in order to use the standard formulas. Remember: You can only combine cash flows if they occur at the same point in time. (This is like building with LEGOs!)

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9 Problem 3 A construction firm is considering the purchase of an air compressor. The compressor has the following expected end of year maintenance costs: Year 1 $800 Year 2 $800 Year 3 $900 Year 4 $1000 Year 5 $1100 Year 6 $1200 Year 7 $1300 Year 8 $1400 What is the present equivalent maintenance cost if the interest rate is 12% per year compounded annually?

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10 Problem 3 – Alt Soln 1 GIVEN: MAINT COST 1-8 PER DIAGRAM i = 12%/YR, CPD ANNUALLY FIND P: P = P A + P G + P F = A(P/A,i,n) + G(P/G,i,n) + F(P/F,i,n) = $700(P/A,12%,8) + $100(P/G,12%,8) + $100(P/F,12%,1) = $700(4.9676) + $100( ) + $100(0.8929) = $5014 P A ? n=8 $700 4 NOTE: CAN BREAK INTO 3 CASH FLOWS: ANNUAL, LINEAR GRADIENT, AND FUTURE DIAGRAM: P ? n=8 $700 $100 $700 4 $200 $300 $100 P G ? n=8 $100 $700 $200 $ $100 P F ? n=1

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11 Problem 3 – Alt Soln 2 GIVEN: MAINT COST 1-8 PER DIAGRAM i = 12%/YR, CPD ANNUALLY FIND P: P = P A + P G (P PG ) = A(P/A,i,n) + G(P/G,i,n-1)(P/F,i,1) = $800(P/A,12%,8) + $100(P/G,12%,7)(P/F,12%,1) = $800(4.9676) + $100( )(0.8929) = $5014 DIAGRAM: P ? n=8 $800 $100 $600 4 $200 P A ? n=8 $800 4 P G ? 012 n=7 3 $100 $600 $200 NOTE: P G MUST BE OFFSET ONE YEAR – SO BRING THE OFFSET YEAR BACK TO TIME ZERO 0 P G ? P PG ? 1

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12 Problem 4 A young couple has decided to make advance plans for financing their 3 year old daughter’s college education. Money can be deposited at 8% per year, compounded annually. What annual deposit on each birthday, from the 4 th to the 17 th (inclusive), must be made to provide $7,000 on each birthday from the 18 th to the 21 st (inclusive)? GIVEN: WITHDRAWALS = $7 000 i = 8%/YR, CPD YEARLY FIND A 4-17 : yrs $7 000 A ? DIAGRAM: P 17 = A(F/A,i,n) = A(P/A,i,n) = A(F/A,8%,14) = 7 000(P/A,8%,4) = A( ) = 7 000(3.3121) A = $957 STRATEGY: CAN BREAK INTO 2 CASH FLOWS, SO PICK A CONVENIENT POINT IN TIME AND SET DEPOSITS EQUAL TO WITHDRAWALS…

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