Solve Annuity Word Problem: Debt of $82423.55

[jackscholar]

Homework Statement

A business is paying off a debt by paying an installment of $8000 at the end of each year. Interest is being charged at 5.5%. What is the outstanding debt if the business has just paid an installment and there remain 15 further installments of $8000 and a final repayment of $5000 at the end of the 16th year.

The Attempt at a Solution

Originally I thought this to be a future value problem but I realized it was a present value problem. I used the Present value annuity equation:

P.V.=[(1-(1+i)^-n)/i]*R, where R=8000
Where i is 0.055 and n=15
This gave me a present annuity factor of 10.0375... which was then multiplied by 8000 to give $80300.64. Then I believe because this is annuity due it has to be multiplied by 1+i, which gives $84717.18. Then the last payment of $5000 has to be made. This should have interest calculated on it shouldn't it? If so then it is $5275. This should be subtracted from the value found. This gives $79442.18. The answer is $82423.55 apparently. So where did I go wrong?

 

[Chestermiller]

Homework Statement

A business is paying off a debt by paying an installment of $8000 at the end of each year. Interest is being charged at 5.5%. What is the outstanding debt if the business has just paid an installment and there remain 15 further installments of $8000 and a final repayment of $5000 at the end of the 16th year.

The Attempt at a Solution

Originally I thought this to be a future value problem but I realized it was a present value problem. I used the Present value annuity equation:

P.V.=[(1-(1+i)^-n)/i]*R, where R=8000
Where i is 0.055 and n=15
This gave me a present annuity factor of 10.0375... which was then multiplied by 8000 to give $80300.64. Then I believe because this is annuity due it has to be multiplied by 1+i, which gives $84717.18. Then the last payment of $5000 has to be made. This should have interest calculated on it shouldn't it? If so then it is $5275. This should be subtracted from the value found. This gives $79442.18. The answer is $82423.55 apparently. So where did I go wrong?

Mutliplying by 1+i was incorrect. You already correctly calculated the present value of the 15 payments of $8000. The present value of the $5000 payment at the end of year 16 has to be added to get the total present value of the future payments. What is the present value of the $5000 final payment at the end of year 16?

 

[jackscholar]

Would that be used in the present value annuity equation where n=1 like so..
P.V.=5000*(1-(1.055)^-1)/0.055?

 

[Ray Vickson]

Would that be used in the present value annuity equation where n=1 like so..
P.V.=5000*(1-(1.055)^-1)/0.055?

Forget about using annuity formulas if you do not fully understand them; just proceed from first principles. For an interest rate of 100r %, the PV of $1 received 1 year from now is 1/(1+r) ($). The future value of $1 in one year from now is (1+r) ($). For n periods in the future, the PV is 1/(1+r)^n and the FV is (1+r)^n. For a stream of payments the PVs and FVs are the sum of the separate PV or FV values of the different payments. You could, if you wanted to, express the FV or PV of a steady stream of payments as the sum of a geometric series and use the corresponding summation formulas, but it is often easier to just do the computations directly, without using any formulas; for example, in spreadsheet computations, the direct approach is easiest (and, in some cases, more accurate! because it avoids subtractive roundoff errors).

 

[Chestermiller]

Would that be used in the present value annuity equation where n=1 like so..
P.V.=5000*(1-(1.055)^-1)/0.055?

No. The $5000 is a single payment paid out 16 years from now. So you don't use the annuity equation. Its present value is simply $5000/(i+1)¹⁶. Try that in your solution, and you will see that your results match the "answer".

Chet