Simple Compound Interest Calculation Question + [bonus question]

[Garland]

Please help!

I’m studying for a test coming up soon, and I have a simple question (or two) to ask. I will put the questions verbatim. It’s possible I’m reading the problem incorrectly.

(1) Suppose an investment earning an annual effective interest rate of 7.6% remains in an account for 2 years and 23 days. The initial balance is $3,000. What is the balance at the end of the deal?

My initial instinct: Calculate the fraction of a year: 23/365 = 0.063013

$3000(1.076)^2.063013

Internet compound interest calculator agrees with me.

Book answer: Calculate the compound interest for 2 years and then treat the remaining time like simple interest:

$3000(1.076)^2 * (1 + (23/365)0.076) = $3000(1.076)^2 * (1.004789)

Obviously I get different answers with the methods above. Which method is correct?

Bonus Question:

(2) An annuity pays $2,000 at time 0 and $700 per year thereafter. What is the accumulated value in 12 years if the interest rate is 9% compounded yearly?

[I have a book answer that differs from mine and challenges my sanity]

Thank you very much in advance for your help.

Garland

[SlowHabit]

Garland,

Are you preparing for an accounting test at Berkeley?

[Garland]

[ QUOTE ]
Garland,

Are you preparing for an accounting test at Berkeley?

[/ QUOTE ]

No, actuary test FM (Financial Mathematics)

Garland

[Plateau]

First Question,
N=2.06
I=7.6
PV=-3,000
FV=3,488.63

Bonus
N=12
I=9
PV=2000
PMT=700
FV=19,723.83

[NickNick]

On a similar note, this is confusing me:

Initial Investment: $1000
Yearly addition: $1000 for 20 years
Interest Rate: 5%

If interest compounded yearly = $37,372.55
If interest compounded monthly = $37,108.17

http://www.moneychimp.com/calculator...calculator.htm

Why is the final amount less if the interest is compounded more regularly?

Thanks a lot! - Sorry if this is a basic oversight!

[Garland]

[ QUOTE ]
On a similar note, this is confusing me:

Initial Investment: $1000
Yearly addition: $1000 for 20 years
Interest Rate: 5%

If interest compounded yearly = $37,372.55
If interest compounded monthly = $37,108.17

http://www.moneychimp.com/calculator...calculator.htm

Why is the final amount less if the interest is compounded more regularly?

Thanks a lot! - Sorry if this is a basic oversight!

[/ QUOTE ]

It took my mind a while to wrap around the concept when studying:
From time 0 to time 19, there are 20 payments of $1000, one for each year t = 0, t = 1, ..., t = 19. In addition, there's one more for the initial investment at time t = 0, which is also $1000. This all takes place in the span of 19 years, and not 20 like I kept thinking. The 20th payment occurs in 19 years. With that out of the way, here goes!

I could use my calculator:

Mode=BGN for Due (1st payment right away as opposed to a full period from now)

N = 20 (20 payment periods)
I/Y = 5 (5% per year)
PV = 1000 (Initial Investment)
PMT = 1000 (Payment per period)

Compute Future value = $37,372.55

That's correct.

Now, when they calculate only changing the number of compounding periods = 12. This is what they did:

Switching to periods of months instead of years.

N = 240 (20 years * 12 months)
I/Y = (0.05/12)*100 (Interest per period)
PV = 1000
Payment = 1000/12 = 83.333333 for each month

Compute Future Value = $37,108.17

So they also assume you are putting your payments in monthly. But also they are using the faulty 5% effective yearly [called i upper 1 or just “i”] and treating it like it’s convertible monthly. This is only equivalent to 4.8889 convertible monthly [called i upper 12].

We need to convert 5% convertible monthly, to its equivalent effective yearly rate.

In order to convert to get the “right” %, you have to do a little math:

[(1 + (0.05/12)^(12) – 1] = 0.05116 or 5.116%

To get per period, we have to divide by 12: 0.05116/12 = 0.004263 or 0.4263%

So now

N = 240 (20 years * 12 months)
I/Y = (0.05116)/12 * 100 = 0.4263 (Interest per period)
PV = 1000
Payment = 1000/12 = 83.333333 for each month

Compute Future Value = $37,639.42

There’s the extra interest you were looking for. And that’s assuming you’re depositing monthly, rather than yearly, so you get yet another number for that. But it’s really, really late, and I have to sleep.

Garland

[Garland]

[ QUOTE ]
First Question,
N=2.06
I=7.6
PV=-3,000
FV=3,488.63

Bonus
N=12
I=9
PV=2000
PMT=700
FV=19,723.83

[/ QUOTE ]

Thank you for your reply. I've been told that the first cannot be done without being told how to handle the fraction year interest: simple or compound. So the question was ambiguous.

As for the bonus question, I agree with your calculations: $19723.83 through a variety of methods. The book kept saying “16,667.15.” And I know they didn’t accumulate $1300 in their work [The split the initial $2000 deposit to $1300 and $700 ($700 to include in the annuity).

There are a lot of bad questions and answers in this book. Time to throw it away.

Garland

[ImBetterAtGolf]

[ QUOTE ]
[ QUOTE ]
First Question,
N=2.06
I=7.6
PV=-3,000
FV=3,488.63

Bonus
N=12
I=9
PV=2000
PMT=700
FV=19,723.83

[/ QUOTE ]

Thank you for your reply. I've been told that the first cannot be done without being told how to handle the fraction year interest: simple or compound. So the question was ambiguous.

As for the bonus question, I agree with your calculations: $19723.83 through a variety of methods. The book kept saying “16,667.15.” And I know they didn’t accumulate $1300 in their work [The split the initial $2000 deposit to $1300 and $700 ($700 to include in the annuity).

There are a lot of bad questions and answers in this book. Time to throw it away.

Garland

[/ QUOTE ]

I think it's more accurate to say that you cannot do this type of calculation without knowing both the rate and the corresponding quoting convention. different types of fixed income instruments are quoted with different conventions and you can't do value calculations without knowing which one is in use. Having said that, i can't say i'm familiar with the convention assumed by the solution in your book