Basic finance Present Value and Anuity

[tomas_tuchel]

I've done the problems, but they don't make sense to me. The answers don't really make sense. I'll post the question, the given formulas and my attempts:

Given formulas:
(1) future value to present value:
PV= FV / (1+i)ⁿ
(2) annual payment to present value:
PV = A/i * [1 - 1/(1+i)ⁿ]

Questions:
(1) what is present value of someone paying you $10,000 five years from now? Interest or discount is 2%. What if it was 8%?

My attempt: Use given formula 1.
And you get @ 2%:
PV = 10,000/(1.02)⁵
PV = 10,000/(1.0510) = $9,514.75

@ 8%:
PV = 10,000/(1+.08)⁵
PV = 10,000/(1.4693) = $6,805.96

Does that make sense that the 8% has a lower value?
----------------------------
(2) present value of receiving lottery of $10,000 per year for next 20 years. interest or discount is 3%. what about 7%?
I used formula 1 for this one but I think that's not right.
@ 3%:
PV = 10,000/(1+.03)²⁰
PV = 10,000/(1.8061) = $5,536.79

@ 7%:
PV = 10,000/(1+.07)²⁰
PV = 10,000/(3.8696) = $2,584.25

I should have used equation 2, right?
----------------------------------
(3) cousin wants to borrow $12,300. he will pay you back in 5 years but will pay you $13,000. What if you invest the $12,300 in a bank at rate of 2.6% interest. in 5 years which payoff is better? Your cousin or the investment in the bank?

I think this is equation 2 but when I did the math the answer was insane and I don't think it made sense.
PV = A/i * [1 - (1/(1+i)ⁿ]
PV = (12,300)/(.026) * [1-(1)/(1+(.026)⁵]
PV = (473076.923) * [1 - (1)/(1.13693)]
PV = (473076.923) * [1- (.8795616)]
PV = (473076.923) * [.1204384]
PV = $ 56, 976.63

^^^^ isn't that an insane amount!?

 

[Chestermiller]

I've done the problems, but they don't make sense to me. The answers don't really make sense. I'll post the question, the given formulas and my attempts:

Given formulas:
(1) future value to present value:
PV= FV / (1+i)ⁿ
(2) annual payment to present value:
PV = A/i * [1 - 1/(1+i)ⁿ]

Questions:
(1) what is present value of someone paying you $10,000 five years from now? Interest or discount is 2%. What if it was 8%?

My attempt: Use given formula 1.
And you get @ 2%:
PV = 10,000/(1.02)⁵
PV = 10,000/(1.0510) = $9,514.75

@ 8%:
PV = 10,000/(1+.08)⁵
PV = 10,000/(1.4693) = $6,805.96

Does that make sense that the 8% has a lower value?

Yes. If you had invested the money at 8% rather than at 2%, after five years, which investment would give you more money? The idea of PV is to account for the fact that money you receive in the future is worth less to you than money you receive now, because you could have invested the money now and received interest. The higher the interest rate you could have invested it at, the less the present value. If the guy had given you $6805.86 today, you could have invested it at 8% and received $10,000 in 5 years. If you could only have invested at 2%, the guy would have had to give you $9514.75 in order for you to receive $10,000 in 5 years. So the higher the interest rate you could invest at, the less the payment of $10000 in 5 years is worth to you.

----------------------------

(2) present value of receiving lottery of $10,000 per year for next 20 years. interest or discount is 3%. what about 7%?
I used formula 1 for this one but I think that's not right.
@ 3%:
PV = 10,000/(1+.03)²⁰
PV = 10,000/(1.8061) = $5,536.79

@ 7%:
PV = 10,000/(1+.07)²⁰
PV = 10,000/(3.8696) = $2,584.25

I should have used equation 2, right?

Right. Here again, receiving a series of payments in the future is worth less to you than receiving the whole amount today.

----------------------------------
(3) cousin wants to borrow $12,300. he will pay you back in 5 years but will pay you $13,000. What if you invest the $12,300 in a bank at rate of 2.6% interest. in 5 years which payoff is better? Your cousin or the investment in the bank?

I think this is equation 2 but when I did the math the answer was insane and I don't think it made sense.
PV = A/i * [1 - (1/(1+i)ⁿ]
PV = (12,300)/(.026) * [1-(1)/(1+(.026)⁵]
PV = (473076.923) * [1 - (1)/(1.13693)]
PV = (473076.923) * [1- (.8795616)]
PV = (473076.923) * [.1204384]
PV = $ 56, 976.63

^^^^ isn't that an insane amount!?

You don't need to use PV to do this calculation. If you put 12300 in the bank today at 2.6% interest per year, how much do you have after 5 years?

 

[Ray Vickson]

I've done the problems, but they don't make sense to me. The answers don't really make sense. I'll post the question, the given formulas and my attempts:

Given formulas:
(1) future value to present value:
PV= FV / (1+i)ⁿ
(2) annual payment to present value:
PV = A/i * [1 - 1/(1+i)ⁿ]

Questions:
(1) what is present value of someone paying you $10,000 five years from now? Interest or discount is 2%. What if it was 8%?

My attempt: Use given formula 1.
And you get @ 2%:
PV = 10,000/(1.02)⁵
PV = 10,000/(1.0510) = $9,514.75

@ 8%:
PV = 10,000/(1+.08)⁵
PV = 10,000/(1.4693) = $6,805.96

Does that make sense that the 8% has a lower value?
----------------------------
(2) present value of receiving lottery of $10,000 per year for next 20 years. interest or discount is 3%. what about 7%?
I used formula 1 for this one but I think that's not right.
@ 3%:
PV = 10,000/(1+.03)²⁰
PV = 10,000/(1.8061) = $5,536.79

@ 7%:
PV = 10,000/(1+.07)²⁰
PV = 10,000/(3.8696) = $2,584.25

I should have used equation 2, right?
----------------------------------
(3) cousin wants to borrow $12,300. he will pay you back in 5 years but will pay you $13,000. What if you invest the $12,300 in a bank at rate of 2.6% interest. in 5 years which payoff is better? Your cousin or the investment in the bank?

I think this is equation 2 but when I did the math the answer was insane and I don't think it made sense.
PV = A/i * [1 - (1/(1+i)ⁿ]
PV = (12,300)/(.026) * [1-(1)/(1+(.026)⁵]
PV = (473076.923) * [1 - (1)/(1.13693)]
PV = (473076.923) * [1- (.8795616)]
PV = (473076.923) * [.1204384]
PV = $ 56, 976.63

^^^^ isn't that an insane amount!?

(1) It makes sense that the PV at 8% is lower than at 2%; if you put $6,805.96 in the bank at 8% interest at time 0, it grows to $10,000 in 5 years. If you put $9,514.75 in the bank at time 0 and earn 2% interest, that grows to $10,000 in 5 years. If the interest rate is smaller you need to put in more initially to end up with the same future value.

(3) Why are you summing various contributions over time? There is only a single payment, at 5 years time. Use the correct formula and you will be fine.

 

[Chestermiller]

(1) It makes sense that the PV at 8% is lower than at 2%; if you put $2,584.25 in the bank at 8%interest at time 0, it grows to $10,000 in 5 years. If you put $5,536.79 in the bank at time 0 and earn 2% interest, that grows to $10,000 in 5 years. If the interest rate is smaller you need to put in more initially to end up with the same future value.

These present values are not correct. The OP's PVs are correct. The Rule of 72 tells that the amount of time it takes for an investment to double is roughly 72 divided by the annual interest rate. At 8%, this would be 9 years and at 2%, it would be 36 years.

 

[Ray Vickson]

These present values are not correct. The OP's PVs are correct. The Rule of 72 tells that the amount of time it takes for an investment to double is roughly 72 divided by the annual interest rate. At 8%, this would be 9 years and at 2%, it would be 36 years.

Right: I copied the OP's figures from the wrong part. I have corrected the error in an "edit".

 

[tomas_tuchel]

Thank you all so much.

So the first two questions are correct as is, and it makes sense? (Thanks for the explanation, Chester).

And for the 3rd question--going by what Chestermiller said--it would be:
.026 * $12,300 = 319.8 * 60 months (5 years) = $19,188

Does that add up? I'll try and see if any of the formulas would give me that answer.