## Interest-related problems

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rolo91
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### Interest-related problems

Here I am again. I was going to post in my former thread, but this problem is about something completely different so I thought it would be better discussed in a different thread.

I'm doing some exercises related about dynamic systems, and its practical uses on financial problems. As I stated in my last thread, this exercises aren't homework, I'm doing them for myself.

The problem I'm facing is mainly about the terminology, I think: I can't understand quite well the diference between APR, nominal rate, nominal annual rate,nominal interest capitalized trimestrally, and so on. The names seem like a contradiction to me (how can something be annual AND nominal? ).

I'll show you one of the sample exercises I have as an example:

You are offered a financial product which gives you a 2.5% nominal interest, capitalized semestrally, during 15 years. How much money you should invest, if you want to have, after those 15 years, 20.000€ ?

(excuse the translation- as I said I'm not a native English speaker)

If I understand the problem, then for a given system A(n) = 1.025^n*A(0), I have to find A(0) so that A(180) equals 20.000. (180 being 15 years * 12 months/year)

If that is true, then A(0) = 20.000/(1.025^180) = 234.8€

However, I suspect that the result is wrong, because of how small the quantity is, and because I'm not using the "capitalized semestrally" thing.

So, can somebody please enlighten me? elasto
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### Re: Interest-related problems

Well, 1.025^180 would imply it was 2.5% interest a month - so that bit's definitely wrong.

Can't help you on the rest though!

Snark
Posts: 425
Joined: Mon Feb 27, 2012 3:22 pm UTC

### Re: Interest-related problems

Take "nominal" to mean "in name only". That helped me out to understand it.

rolo91 wrote:You are offered a financial product which gives you a 2.5% nominal interest, capitalized semestrally, during 15 years. How much money you should invest, if you want to have, after those 15 years, 20.000€ ?

Use this formula:
A = P(1 + (r/n))n*t
A = Accumulated amount or future value
P = Present value
r = nominal rate
n = number of interest compoundings per year
t = number of years
r/n is the actual interest rate per period, (1 + r/n)n - 1 is the actual annual interest rate, and n*t is the number of times that interest is compounded over the life of the loan

For your problem:
20000 = P(1 + .025/4)4*15
Solve for P
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rolo91
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### Re: Interest-related problems

So, then, if I'm told something like "X% nominal annual interest rate, composed trimestrally", that X% is means what my unknow trimestral rate would be, if it were annual instead of trimestral. Is that right?

Also, talking about rates, when it's not specified, I have to assume it's annual?

Snark
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### Re: Interest-related problems

"X% nominal annual interest rate, composed trimestrally" sounds like X is equal to r from our formula where n = 3. Interest is compounded three times a year, each time at a rate equal to X/3. Your true effective annual interest rate is i = (1 + X/3)3 - 1. I can't state it plainer than that. Sorry.

And yes, you usually assume the interest rate is the effective annual rate unless otherwise stated. Usually but not always.
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### Re: Interest-related problems

rolo91 wrote:(how can something be annual AND nominal? ).

Then 'nominal annual interest rate' just means the rate the bank advertises. For example, a bank might say the 'nominal annual interest rate' as 4.5% per annum. But the way they actually calculate interest is different from just adding 4.5% after 1 year. 'Nominal' means 'in name' or 'so-called' etc...

liveboy21
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### Re: Interest-related problems

What you need to know for your problem is the difference between a 'nominal' interest rate and an 'effective interest rate. I will use a simple example using a 12% nominal rate per annum compounded monthly.

The nominal rate is written as 12% nominal rate per annum compounded monthly.
The effective interest rate is 1% per month. We get 1% by taking the nominal rate and dividing by the number of periods (in this case 12% divided by the number of months in a year).

If we start with an investment of A(0), after 1 month it will have increased in value to A(0) * (1 + effective interest rate) = 1.01*A(0)

And if we want to know what the value of the investment after a year is, we use A(0) * (1 + effective interest rate)^(number of periods)
which in this example gives us A(0)*(1.01)^12

I hope that this was clear enough to give you the tools to work our the problem in the opening post and other similar problems.

rolo91
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### Re: Interest-related problems

Those last posts have actually confused me a little...

Snark wrote:"X% nominal annual interest rate, composed trimestrally" sounds like X is equal to r from our formula where n = 3. Interest is compounded three times a year, each time at a rate equal to X/3. Your true effective annual interest rate is i = (1 + X/3)3 - 1. I can't state it plainer than that. Sorry.

And yes, you usually assume the interest rate is the effective annual rate unless otherwise stated. Usually but not always.

I think I get what you are trying to say, but... shouldn't n be equal to 4, then? I mean, if the interest is compounded trimestrally, then it should be compounded 4 times a year (once every 3 months), right?

liveboy21 wrote:What you need to know for your problem is the difference between a 'nominal' interest rate and an 'effective interest rate. I will use a simple example using a 12% nominal rate per annum compounded monthly.

The nominal rate is written as 12% nominal rate per annum compounded monthly.
The effective interest rate is 1% per month. We get 1% by taking the nominal rate and dividing by the number of periods (in this case 12% divided by the number of months in a year).

If we start with an investment of A(0), after 1 month it will have increased in value to A(0) * (1 + effective interest rate) = 1.01*A(0)

And if we want to know what the value of the investment after a year is, we use A(0) * (1 + effective interest rate)^(number of periods)
which in this example gives us A(0)*(1.01)^12

I hope that this was clear enough to give you the tools to work our the problem in the opening post and other similar problems.

But I can't see how that could be correct.

In your example, if we start with A(0), and after the first month we have 1.01*A(0), then after 12 months we will have (1.01^12)*A(0), as you said.

But 1.01^12= 1.1268, which is more than what it's announced, and it doesn't make sense for a bank to advertise an interest which is actually inferior than what they're going to give.

So, shouldn't the monthly rate be a smaller number X such that X^12 = 1.12, that is, that after 12 months gives an interest equal to the announced nominal rate?

Maybe I'm overthinking this, but it seems weird to me for somebody to advertise a product reducing what they offer gmalivuk
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### Re: Interest-related problems

rolo91 wrote:I think I get what you are trying to say, but... shouldn't n be equal to 4, then? I mean, if the interest is compounded trimestrally, then it should be compounded 4 times a year (once every 3 months), right?
I think there's a mistranslation or miscommunication here about what a "trimester" actually is.

The main point is that, yes, n is the number of times in a year interest is compounded. If it happens 4 times a year, n=4.

But I can't see how that could be correct.

In your example, if we start with A(0), and after the first month we have 1.01*A(0), then after 12 months we will have (1.01^12)*A(0), as you said.

But 1.01^12= 1.1268, which is more than what it's announced, and it doesn't make sense for a bank to advertise an interest which is actually inferior than what they're going to give.
And actually, it does make good business sense, when you consider that a lot more consumers are exposed to the interest rates of *loans*, rather than investments. If your mortgage or car payments are advertised as 12% (nominal) APR, that looks better than 12.68%.

When you invest money at a specified rate, it's often advertised at a particular APY, which is the effective annualized rate, and thus higher than the nominal rate if it's actually compounded more frequently than once per year. (Though if you buy a product that you then can't access for a specified amount of time, such as a CD, you'll typically just get the whole lump sum at the end of that period, which will work out to be equivalent to the APY applied once for every year of the period, regardless of how often the bank actually compounds anything.)
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rolo91
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Joined: Mon Aug 06, 2012 10:21 am UTC

### Re: Interest-related problems

Oh, ok, if you think about loans then it makes sense.

If you don't mind, I'll post here a last exercise, already solved, just to be sure I've understood everything:

Spoiler:
A 38 year old man receives 2.000.000€ after winning the lottery. He decides to save it until he retires,so he invests all that money in a bank account, at 4% nominal interest rate, capitalized every month. After reaching the age of 65, he will start taking away 30.000€ every month. How much time is he going to be able to do that? What is the present value of the first 30.000€ he will retire, taking a 5% rate of annual inflation?

This is how I solved it:

4% nominal interest rate, in a monthly basis, means 4/12 = 0.34% monthly rate. The annual rate is then (1.0034)12-1 = 0.0416, that is, 4.16%

Until the moment of retirement, that is, during 65-38 = 27 years, this is a homogeneous sistem which A(n) = 1.0416n*2000000, where n = number of years.
Then, rounding it a little, A(27) = 6.011.000

From that moment on, the system becomes A(x+1) = 1.0416*A(x)-30.000, where x is now the number of months (counting from the moment of retirement)-

According to my formulas, that means

A(x) = (a0 - P)*1.0034x + P

Where:
a0 = the starting money (at the moment of retirement)
P = the system's point of equilibrium (I don't know how is that named in English)
x = number of months, as previously stated.

Said point of equilibrium would be -30.000/(1-1.0034) = 8.823.529

Then, A(x) will equal 0 when the money runs out. So:

(6.011.000 - 8.823.529)*1.0034x + 8.823.529 = 0

1.0034x = -8.823.529/-2.812.529 = 3.14

taking logarithms:

x = log 3.14/log 1.0034 = 337.1

So the last month will be the 337th. That is, this man can keep taking money from his bank account for 28 years, when he'll be 93 (if he is still alive )

the last question is about the present value of the first 30.000 € he's retiring.

Given a 5% inflation (which I assume is not a nominal value, but a real, annual one) then we take an unknown quantity X that, after being affected by a 5% interest rate during 27 years, becomes 30.000 €:

X * 1.0527 = 30.000 ==> X = 30.000/1.0527= 8.035.4 €

Is everything right?

Thanks to everyone for your answers!

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