Pensions: If John saves $600 at the start of each year for 5 years at 4% per annum...

[Julton]

Hello, I am having some difficulty understanding the basics of financial maths. There are two questions which are confusing me, the first I can work out, the second I get wrong using the same method:

Q1 This works out: If John saves $600 at the start of each year for 5 years at 4% per annum. Calculate the future value of the savings plan and the present value of these payments.

For the future value (FV) I calculated: 600x1.04 + 600x1.04² +...+600x1.04⁵ so John has $3379.79 at the end of five years.

To calculate the present value (PV), I used the following formula (which is how I understand it to work):

PVx(1.04)⁵=FV -> PV=600/(1.04)⁴ +...+600 = $2777.94

Q2 This gives me the wrong answer: What amount of money is needed today to provide a pension of $25000 a year for 20 years assuming an AER of 4%.


My logic:

PVx(1.04)²⁰ = FV

FV= 25000x(1.04)²⁰ +25000x(1.04)¹⁹ + ... +25000x(1.04)

PV=25000 + 25000x(1.04)^-1 + ... + 25000x(1.04)^-19

This equation for PV gives me the wrong answer. The correct answer ($339758.16) uses a PV of PV= 25000x(1.04)-20 +25000x(1.04)-19+ ... +25000x(1.04)^-1 Which I understand would give an original FV of FV= 25000x(1.04)¹⁹ +25000x(1.04)¹⁸ + ... +25000

I don't understand if you were saving 25000 a year for 20 years why the interest isn't calculated as normal Compound Interest. Could anyone please explain where I am misunderstanding?

 

[HallsofIvy]

You say "I don't understand if you were saving 25000 a year for 20 years why the interest isn't calculated as normal Compound Interest."

In this problem, you are not "saving 25000 a year for 20 years"! $25000 pension is being payed out of the fund every year. The amount in the fund is decreasing, not increasing.

 

[Julton]

Thank you for your replies.

Denis, thank you for the formula. I think my biggest problem at the moment is understanding what is being requested.

HallsofIvy, working on from your response I worked out the Present Value to be - the amount I would need now at 1.04% interest to be able to withdraw 25000 next year, plus the amount I would need to have now to withdraw 25000 in two years and so on for the 20 years. That gave me the formula,

PV= 25000/1.04¹ + 25000/1.04² + ... + 25000/1.04²⁰ . This makes sense to me and it gives me the correct answer. However, I then tried another question and I got the wrong answer.

Q: A person wishes to have a fund that could, from the date of retirement, give them a payment of $20000 at the start of each year for 25 years. Calculate the value, on the date of retirement, of the fund required based on a 3% annual growth rate.

I approached it, that the present value required would be the sum needed now at 1.03% to pay 20000 next year, plus the sum needed now at 1.03% to pay 20000 in 2 years and so on.

PV = 20000/1.03 + 20000/1.03² + ... + 20000/1.03²⁵

The correct answer is calculate using a Present Value of 20000 + 20000/1.03 + ... + 20000/1.03²⁴

I figure this is based on the person drawing the first 20000 the year they retire? I would be grateful for any further insight and/or corrections.

 

[Ishuda]

Hello, I am having some difficulty understanding the basics of financial maths. There are two questions which are confusing me, the first I can work out, the second I get wrong using the same method:

Q1 This works out: If John saves $600 at the start of each year for 5 years at 4% per annum. Calculate the future value of the savings plan and the present value of these payments.

For the future value (FV) I calculated: 600x1.04 + 600x1.04² +...+600x1.04⁵ so John has $3379.79 at the end of five years.

To calculate the present value (PV), I used the following formula (which is how I understand it to work):

PVx(1.04)⁵=FV -> PV=600/(1.04)⁴ +...+600 = $2777.94

Q2 This gives me the wrong answer: What amount of money is needed today to provide a pension of $25000 a year for 20 years assuming an AER of 4%.


My logic:

PVx(1.04)²⁰ = FV

FV= 25000x(1.04)²⁰ +25000x(1.04)¹⁹ + ... +25000x(1.04)

PV=25000 + 25000x(1.04)^-1 + ... + 25000x(1.04)^-19

This equation for PV gives me the wrong answer. The correct answer ($339758.16) uses a PV of PV= 25000x(1.04)-20 +25000x(1.04)-19+ ... +25000x(1.04)^-1 Which I understand would give an original FV of FV= 25000x(1.04)¹⁹ +25000x(1.04)¹⁸ + ... +25000

I don't understand if you were saving 25000 a year for 20 years why the interest isn't calculated as normal Compound Interest. Could anyone please explain where I am misunderstanding?

Saying things in a somewhat different way, there are two things you need to remember:
(1) For a single payment at interest rate i (per term) held for n terms,
FV = PV (1+i/100)ⁿ
or, equivalenty,
PV = FV / (1+i/100)ⁿ
(2) S(x,n) = (1 + x + x2 + x3 + ... + xⁿ) = (1-xⁿ⁺¹)/(1 - x) = (xⁿ⁺¹ -1)/(x-1)

As an example, look at Q1, you had 5 single payments of 600 lasting 1, 2, 3, 4, and 5 years respectively from which, using (1), you have a total FV composed of the sum 5 future values of the 5 payments
FV = 600*1.04 + 600*1.04² +...+600*1.04⁵ = 600 * 1.04 * (1 + 1.04 +...+ 1.04⁴) or, with an interest rate of 4% giving an x of x = 1.04, we have
FV = 600 * 1.04 * S(1.04,4)
Note that this is a make first payment immediately and 4 additional payments annually. Using the present value of that 'single payment' FV is (with the same interest rate)
PV = FV / 1.04⁵ = 600 * S(1.04,4) / 1.04⁴
Of, course you can also write that as you have it and change the value of x to 1/1.04 to get the same answer.


Now for Q2: What amount of money is needed today to provide a pension of $25000 a year for 20 years assuming an AER of 4%. Note that the names have changed but the formulas are really the same. For convenience call the present amount of money needed P, let x=1.04 and p=25000: If we have a starting payment immediately we have
(1) money left immediately = P - p
and since that earns money until you get your next payment, we have
(2) money left at end of term 1 = (P - p) x - p = P x¹ - p (1+x¹) = P x - p S(x,1)
(3) money left at end of term 2 = (P - p) x - p = P x² - p (1+x+x²) = P x² - p S(x,2)
...
(n) money left at end of term n = P xⁿ - p (1+x+x2+...+xⁿ) = P xⁿ - p S(x,n)
Since you would have zero money left at the end of the 19th term, we have
P = p S(x,n) / xⁿ

You really should memorize these equation as it takes time to work out the answers for a test. Also the equations can be turned around and used as a payment formula for a huose/car/... payment but, as mentioned by Denis, pay attention to the immediate and regular types. for example, had you delayed the first payment and taken it at the end of the first term above you would have
(1) money left immediately = P
and since that earns money until you get your next payment, we have
(2) money left at end of term 1 = P x - p
(3) money left at end of term 2 = P x² - p (1+x) = P x² - p S(x,1)
...