Business problems, Please help

Shifty98 · #1 $Old$ September 29th, 2009, 06:30 AM

Hi All, I have a lab due soon, And I was wondering if anyone could give me any help. I don't directly need the answers, of course, I am just wondering how I should go about doing them. The questions are as follows:

Jean-Guy Renoir wanted to leave some money to his grandchildren in his will. He decided that they should each receive the same amount of money whey they each turn 21. When he died, his grandchildren were 19, 16 and 13 respectively. How much will they each receive when they turn 21 if Jean-Guy left a lump sum of 50,000 to be shared among them equally? Assume the interest rate will remain at 7.75% p.a. compounded semi-annually from the time of Jean-Guy’s death until the youngest grandchild turns 21?

3. The Central Bank pays 7.5% compounded semiannually on certain types of deposits. If interest is to be compounded monthly, what nominal rate of interest will maintain the same effective rate of interest?

Wilmer · #2 $Old$ September 29th, 2009, 08:31 AM

Quote:

Originally Posted by Shifty98 $View Post$

Jean-Guy Renoir wanted to leave some money to his grandchildren in his will. He decided that they should each receive the same amount of money whey they each turn 21. When he died, his grandchildren were 19, 16 and 13 respectively. How much will they each receive when they turn 21 if Jean-Guy left a lump sum of 50,000 to be shared among them equally? Assume the interest rate will remain at 7.75% p.a. compounded semi-annually from the time of Jean-Guy’s death until the youngest grandchild turns 21?

Get annual equivalent rate i: i = (1 + .0775/2) - 1 ; ~7.9%

Let equal amounts = x
50000(1 + i)^8 = x(1 + i)^6 + x(1 + i)^3 + x
50000(1 + i)^8 = x[(1 + i)^6 + (1 + i)^3 + 1]

x = 50000(1 + i)^8 / [(1 + i)^6 + (1 + i)^3 + 1]

OK?

3. The Central Bank pays 7.5% compounded semiannually on certain types of deposits. If interest is to be compounded monthly, what nominal rate of interest will maintain the same effective rate of interest?

let i = nominal rate

(1 + .075/2)^2 = (1 + i/12)^12 ; solve for i

OK?

Assuming you can follow all that; come back with questions if not.

Shifty98 · #3 $Old$ September 29th, 2009, 08:37 AM

Thanks very much for your help. Question 3, I understand... But question 2 I have a bit of difficulty understanding, partially because im not exactly sure what the question is asking, but I'm going to take your word for it. the part that I dont understand is:
Get annual equivalent rate i: i = (1 + .0775/2) - 1 ; ~7.9%

If it's not too much to ask for, could you provide a solution, or even better just an explanation? for some reason i am not understanding this question very well

Wilmer · #4 $Old$ September 29th, 2009, 08:57 AM

Quote:

Originally Posted by Shifty98 $View Post$

Thanks very much for your help. Question 3, I understand... But question 2 I have a bit of difficulty understanding, partially because im not exactly sure what the question is asking, but I'm going to take your word for it. the part that I dont understand is:
Get annual equivalent rate i: i = (1 + .0775/2) - 1 ; ~7.9%

If it's not too much to ask for, could you provide a solution, or even better just an explanation? for some reason i am not understanding this question very well

Leaving the rate at 7.75% would be ok, except you'd need 16 periods
(instead of 8), each period using .0775 / 2 as i

What I did is find the rate that is equivalent to that using 8 periods;
I prefer that way (personal preference).

(1 + .0775/2)^2 - 1 = .079001562....or close to 7.9%
Paying that rate annually is same as paying 7.75 semiannually.

If we go with the semiannual rate instead, then my equation would be:
x = 50000(1 + .0775/2)^16 / [(1 + .0775/2)^12 + (1 + .0775/2)^6 + 1]

Either will provide the correct answer, which is x = 23,958.46137...

OK?

Shifty98 · #5 $Old$ September 29th, 2009, 09:28 AM

Excellent! Thanks very much!! I understand now, it seems you left out the ^2 on the first explanation, which is what confused me lol.

NOW, i need to work on question 3, and my poor algebra skills are dooming me, I haven't needed to do any math in the past... oh 9 months, and it's killing me to get back into it lol.

Shifty98 · #6 $Old$ September 29th, 2009, 09:38 AM

wait, for question 3, is the answer the same, 7.48% or so?? because if it's not, i'm going to need help with my algebra here lol. sorry.

Also, Isn't it 8, 5 and 2, not 8 6 and 3?

Wilmer · #7 $Old$ September 29th, 2009, 12:32 PM

Quote:

Originally Posted by Shifty98 $View Post$

wait, for question 3, is the answer the same, 7.48% or so?? because if it's not, i'm going to need help with my algebra here lol. sorry.

First, sorry for leaving out the ^2. No wonder you were confused...

7.48 is not correct. We have:

(1 + i/12)^12 = (1 + .075/2)^2

Let the right side = k (easier typing!)
(1 + i/12)^12 = k
1 + i/12 = k^(1/12)
i/12 = k^(1/12) - 1
i = 12 * [k^(1/12) - 1]
i = .073854....~7.39%

OK? Remember that if a^b = c, then a = c^(1/b).

Wilmer · #8 $Old$ September 29th, 2009, 12:49 PM

Quote:

Originally Posted by Shifty98 $View Post$

Also, Isn't it 8, 5 and 2, not 8 6 and 3?

No.
When the 19 year old reaches 21, there will be 6 years left before 8th year.
When the 16 year old reaches 21, there will be 3 years left before 8th year.

After 2 years from the initial $50,000 deposit, the 19 year old gets his
share of $23,958.46.
So its futute value is for 6 years.

After 5 years from the initial $50,000 deposit, the 16 year old gets his
share of $23,958.46.
So its futute value is for 3 years.

After 8 years from the initial $50,000 deposit, the 13 year old gets his
share of $23,958.46 leaving account at zero.
So its futute value is for 0 years.

You can look at this as 4 different accounts.
1: 50,000 new account opened at year 0, untouched for 8 years
2: 23,956.40 new account opened at year 2; so 6 years to go
3: 23,956.40 new account opened at year 5; so 3 years to go
4: 23,956.40 new account opened at year 8; so 0 years to go

Account#1's future value = sum of future values of the other 3.

Shifty98 · #9 $Old$ September 29th, 2009, 06:18 PM

Right, sorry, I meant 7.38, not 7.48. Well the lab is handed in, Thanks for all of your help, I really appreciate it. Thanks very much.

jonah · #10 $Old$ October 2nd, 2009, 01:15 PM

Quote:

Originally Posted by Wilmer $View Post$

Quote:

Originally Posted by Shifty98 $View Post$

Also, Isn't it 8, 5 and 2, not 8 6 and 3?

No.
When the 19 year old reaches 21, there will be 6 years left before 8th year.
When the 16 year old reaches 21, there will be 3 years left before 8th year.

After 2 years from the initial $50,000 deposit, the 19 year old gets his
share of $23,958.46.
So its futute value is for 6 years.

After 5 years from the initial $50,000 deposit, the 16 year old gets his
share of $23,958.46.
So its futute value is for 3 years.

After 8 years from the initial $50,000 deposit, the 13 year old gets his
share of $23,958.46 leaving account at zero.
So its futute value is for 0 years.

You can look at this as 4 different accounts.
1: 50,000 new account opened at year 0, untouched for 8 years
2: 23,956.40 new account opened at year 2; so 6 years to go
3: 23,956.40 new account opened at year 5; so 3 years to go
4: 23,956.40 new account opened at year 8; so 0 years to go

Account#1's future value = sum of future values of the other 3.

Beautiful Sir Wilmer, beautiful!
I wish I'd thought of that first.
However, Shifty98 is not necessarily incorrect on his/her conjecture about that "8, 5 and 2" stuff instead of your personal preference of "8 6 and 3" if the comparison date is the present instead of at the end of 8 years. Thus,

$50,000 = x\left( {1 + {\textstyle{{0.0775} \over 2}}} \right)^{ - \left( {2 \times 2} \right)} + x\left( {1 + {\textstyle{{0.0775} \over 2}}} \right)^{ - \left( {5 \times 2} \right)} + x\left( {1 + {\textstyle{{0.0775} \over 2}}} \right)^{ - \left( {8 \times 2} \right)}$