View Full Version : Nominal rate of discount question

whoda

December 4th 2010, 05:51 PM

Hi!

I'm trying to figure out how to answer this question, as my attempt is not proving correct.

Find the present value of 5000, to be paid at the end of 25 months, at a rate of discount of 8% convertible quarterly:

(a) assuming compound discount throughout;

(b) assuming simple discount during the final fractional period.

I have been able to solve part (a):

PV = FV*a(t) =5000*(1-0.08/4)^(4*25/12) = 4225.265 (this is the correct answer)

For part (b):

5000*[(1-0.08/4)^(4*2)+(1-0.08(1/12))] = not the right answer

also, I tried weighting each of the terms in brackets by 24/25 and 1/25 respectively = 4281.86, which is not the correct answer. The correct answer is supposed to be 4225.46.

Help?

brandond

December 6th 2010, 01:26 PM

I'm not getting the correct answer but I know in your solution for part b) you can't just divide the nominal discount rate convertible quarterly by 12. You should convert your nominal discount rater convertible quarterly to an annual effective discount rate, and multiply by (1/12). I still don't get their answer however.

I'm getting 4226.30

beast

December 7th 2010, 01:05 PM

5000 [(1-0.08/4)^8 * (1-0.08/12)] = 4225.456345

brandond

December 7th 2010, 01:45 PM

How can you justify dividing d^(4) by 12? d^(4) is 8%. Dividing by 12 is saying your now assuming the annual effective rate of discount is 8%, where it was just previously defined as the nominal rate of discount convertible quarterly.

beast

December 7th 2010, 02:07 PM

when dealing with simple interest/discount rates, we take the ratio of the rate compared to time. 8% compounded quarterly would be 0.08/4 but for only one month we would have to multiply by 1/3 which is the same as 0.08/12.

Simple interest/discount is a bit tricky when computing FV/PV

for interest it is (1+i/m) and discount it is (1-i/m). m depending on the length of time computing.

Simple interest

FV: multiply by (1+i/m), PV: divide by (1+i/m)

Simple discount

FV: divide by (1-i/m), PV: multiply by (1-i/m)

This is pretty much the same concept as non-simple interest rates. Just remember that interest rates are typically (+), multiply when going forward and dividing when going back. For discount rates are typically (-), DIVIDE when going forward and MULTIPLY when going back. Hope this helps.

brandond

December 7th 2010, 02:16 PM

when dealing with simple interest/discount rates, we take the ratio of the rate compared to time. 8% compounded quarterly would be 0.08/4 but for only one month we would have to multiply by 1/3 which is the same as 0.08/12.

Simple interest/discount is a bit tricky when computing FV/PV

for interest it is (1+i/m) and discount it is (1-i/m). m depending on the length of time computing.

Simple interest

FV: multiply by (1+i/m), PV: divide by (1+i/m)

Simple discount

FV: divide by (1-i/m), PV: multiply by (1-i/m)

This is pretty much the same concept as non-simple interest rates. Just remember that interest rates are typically (+), multiply when going forward and dividing when going back. For discount rates are typically (-), DIVIDE when going forward and MULTIPLY when going back. Hope this helps.

Why are you multiplying by 1/3? The fractional period is only one month, from time t=25 to t=24, so why wouldn't you only want one month.

brandond

December 7th 2010, 02:18 PM

when dealing with simple interest/discount rates, we take the ratio of the rate compared to time. 8% compounded quarterly would be 0.08/4 but for only one month we would have to multiply by 1/3 which is the same as 0.08/12.

Simple interest/discount is a bit tricky when computing FV/PV

for interest it is (1+i/m) and discount it is (1-i/m). m depending on the length of time computing.

Simple interest

FV: multiply by (1+i/m), PV: divide by (1+i/m)

Simple discount

FV: divide by (1-i/m), PV: multiply by (1-i/m)

This is pretty much the same concept as non-simple interest rates. Just remember that interest rates are typically (+), multiply when going forward and dividing when going back. For discount rates are typically (-), DIVIDE when going forward and MULTIPLY when going back. Hope this helps.

Isn't the "i" highlighted here your annual effective?

beast

December 7th 2010, 02:26 PM

Yes, "i" would be our annual effective rate and when dealing with simple interest, we disregard the convertible/compounded time.

brandond

December 7th 2010, 02:33 PM

Yes, "i" would be our annual effective rate and when dealing with simple interest, we disregard the convertible/compounded time.

Sorry sorry sorry. I see what your doing. However, you should still get the same answer if I convert d^(4) to d correct? And I don't, which is why I was confused.

d^(4) = 8% then d = 7.763184% and I should get the correct answer doing

5000(1-.02)^8 * (1- .07763184(1/12)) which I don't.

beast

December 7th 2010, 02:56 PM

In simple interest/discount rates, try not to use any exponents. If d=8% compounded quarterly, simply disregard the "compounded quarterly" and use d=8%. If you want one month then divide the rate by 12: 0.08/12

brandond

December 7th 2010, 02:58 PM

In simple interest/discount rates, try not to use any exponents. If d=8% compounded quarterly, simply disregard the "compounded quarterly" and use d=8%. If you want one month then divide the rate by 12: 0.08/12

Thats my problem, why would we do that. We don't do that with compound interest rates, the convertible periods matter. In this case they are saying that d^(4) = 8%, and in the event we talk about simple interest, then just pretend d^(1) = 8% as well. Seems inconsistent to me.

beast

December 7th 2010, 03:06 PM

Lets say that i=8%, then in simple interest rate for one month it'll be 0.08/12

Now for i=8% compounded quarterly it would be j=2% for 3 months. So we're dealing with 3-month periods now instead of a 1-year period. In order to find one month simple interest from the 3-month period of 2%, we would divide 3. Thus, 0.02/3

This is ultimately the same as 0.08/12, sorry for not explaining thoroughly.

brandond

December 7th 2010, 03:16 PM

Lets say that i=8%, then in simple interest rate for one month it'll be 0.08/12

Now for i=8% compounded quarterly it would be j=2% for 3 months. So we're dealing with 3-month periods now instead of a 1-year period. In order to find one month simple interest from the 3-month period of 2%, we would divide 3. Thus, 0.02/3

This is ultimately the same as 0.08/12, sorry for not explaining thoroughly.

I get this. They are called equivalent rates for a reason though. Whether or not I convert d^(4) to d or d^(2) or whatever, I should get the same answer either way. If the problem states that d^(4)=8%, then d^(4)/4 = 2%, which is the effective per 3 month period, so you divide by 3 again to get one month, ok.

In this case however I should be able to convert d^(4) to just d correct? This would be the effective rate of discount for one year? Now we are just discounting using this simple discount for 1 month, so, why is

(1-d^(4)/4)^4 = 1-d not hold here? If d^(4) is 8%, d should be .07763184, and discounting only one month, it would be (1-.07763184(1/12)), this however doesn't give the same answer.

Unless equivalent rates don't hold under simple interest.

beast

December 7th 2010, 03:46 PM

I get this. They are called equivalent rates for a reason though. Whether or not I convert d^(4) to d or d^(2) or whatever, I should get the same answer either way. If the problem states that d^(4)=8%, then d^(4)/4 = 2%, which is the effective per 3 month period, so you divide by 3 again to get one month, ok.

In this case however I should be able to convert d^(4) to just d correct? This would be the effective rate of discount for one year? Now we are just discounting using this simple discount for 1 month, so, why is

(1-d^(4)/4)^4 = 1-d not hold here? If d^(4) is 8%, d should be .07763184, and discounting only one month, it would be (1-.07763184(1/12)), this however doesn't give the same answer.

Unless equivalent rates don't hold under simple interest.

Well when you think about it, every rate they give you can be a rate of a higher compound. If effective rate of interest is 8%, it is the same as two-year interest of 16% compounded yearly. Which is still 8% a year. Ultimately, you want to find the interest of the smallest period that they assign. If its 10% compounded semiannually, then only look at the 6-month period of 5%. Thus, an 8% compounded quarterly, is the same as 2% for a 3-month period.

brandond

December 7th 2010, 04:13 PM

Well when you think about it, every rate they give you can be a rate of a higher compound. If effective rate of interest is 8%, it is the same as two-year interest of 16% compounded yearly. Which is still 8% a year. Ultimately, you want to find the interest of the smallest period that they assign. If its 10% compounded semiannually, then only look at the 6-month period of 5%. Thus, an 8% compounded quarterly, is the same as 2% for a 3-month period.

Yes, in this case though if you convert d^(4) to d, you don't get the same answer which is my issue.

beast

December 8th 2010, 12:38 PM

Yes, in this case though if you convert d^(4) to d, you don't get the same answer which is my issue.

You should not convert d^4 to d. That is the difference between effective rates and simple rates. As long as you have the rate down to the smallest time period, you can use it to calculate problems using simple rates.

EG. 1) 8% compounded monthly ==> 0.08/12 per month simple rate

2) 20% convertible semiannually ==> 0.1 per 6-month period

For FV in 6 months using rate 1, it will be X(1+0.08/12 * 6)

You should not convert any rate to a 1-year span, just have it at the smallest time period. I don't know how else to explain this. I hope this helps.

brandond

December 8th 2010, 09:08 PM

I get it, thanks.

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