Annuity investment, find future value

[pbonnie]

Homework Statement

(attached)

Homework Equations

Sn = (a(1-r^n))/(1-r)

The Attempt at a Solution

This was my attempt, but I think I've done this wrong. I'm not really sure how to account for the fact that the payments are made at the end of the year. Unless that means I would act as if it were a 19 year investment?
n = 20 a = 2000 r = 8.5%/4 = 0.085/4 = 0.02125 Sn = (a(1-r^n))/(1-r)
S20 = (2000(1-〖(1+0.02125)〗^20))/(1-(1+0.02125)) S20 = 49 204.22
The final amount would be $49 204.22

 

[haruspex]

Yes, you can treat it as a 19 year investment, just adding an extra 2000 at the end (no interest). But you have another problem. You have effectively quartered the interest rate. There are 80 interest periods, not 20. I suggest you first work out the equivalent interest rate for compunding annually.

 

[Ray Vickson]

Homework Statement

(attached)

Homework Equations

Sn = (a(1-r^n))/(1-r)

The Attempt at a Solution

This was my attempt, but I think I've done this wrong. I'm not really sure how to account for the fact that the payments are made at the end of the year. Unless that means I would act as if it were a 19 year investment?
n = 20 a = 2000 r = 8.5%/4 = 0.085/4 = 0.02125 Sn = (a(1-r^n))/(1-r)
S20 = (2000(1-〖(1+0.02125)〗^20))/(1-(1+0.02125)) S20 = 49 204.22
The final amount would be $49 204.22

To second what haruspex has said: the way that financial institutions typically operate is to divide an annual (nominal) interest rate by the number of compounding periods. That makes the *actual* annual rate different from the nominal one. For example, if we are dealing with a (nominal) rate of 12% per annum, compounded monthly, the actual annual rate would be
$$r = \left( 1 + \frac{0.12}{12} \right)^{12} - 1 = .126825030 \approx 12.68\%.$$

 

[pbonnie]

Oh okay great, thank you both. I originally had 80 periods but I thought I was doing it wrong. So I would do 76 compounding periods, and add 2000 to the final answer?

 

[pbonnie]

So then the final answer to a) would be:
S19 = (2000(1-〖(1+0.02125)〗^76))/(1-(1+0.02125)) S19 = 164 772.79
The final amount would be $164 772.79 + $2000 = $166 772.79

and b) would be:
S20 = (2000(1-〖(1+0.02125)〗^80))/(1-(1+0.02125)) S20 = 174 203.37
?

 

[Ray Vickson]

Oh okay great, thank you both. I originally had 80 periods but I thought I was doing it wrong. So I would do 76 compounding periods, and add 2000 to the final answer?

That would be doing it the hard way: the payments are yearly, while the compounding is quarterly. This mismatch makes some of the formulas harder and more intricate. If I were doing it I would just do 20 yearly periods, but I would use the "true" annual interest rate in the calculation.

 

[pbonnie]

Thank you :)