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 Financial Mathematics Old FM Forum

#51
05-23-2019, 01:13 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,167

If each payment was 1000 higher than the pattern shown, wouldn’t your approach give exactly the same PV? That can’t be right. Figure out why, and what you need to do to fix your answer.
#52
05-23-2019, 01:42 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

Quote:
 Originally Posted by Gandalf If each payment was 1000 higher than the pattern shown, wouldn’t your approach give exactly the same PV? That can’t be right. Figure out why, and what you need to do to fix your answer.
Thanks. Yes, I forgot to add the 300 extra level payment for 10 terms. 800- 50 x 10 = 300
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#53
05-26-2019, 03:47 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

An investor deposits 1000 at the beginning of each year for five years in a fund earning 5% effective. The interest from this fund can be reinvested at only 4% effective.

Show that the total accumulated value at the end of ten years is

1250(s11|0:04 − s6|0:04 − 1)

My question is I can find AV(lets call it X) of 5 years = 1000 * [(5 + i/j(s6|j- 6)]
n=5, i=0.05, j =0.04

Is AV of 10 years = X (1+i)^5 or X (1+j)^5?

Thanks!
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#54
05-26-2019, 05:08 PM
 Gandalf Site Supporter Site Supporter SOA Join Date: Nov 2001 Location: Middle Earth Posts: 31,167

Quote:
 Originally Posted by jubair07 An investor deposits 1000 at the beginning of each year for five years in a fund earning 5% effective. The interest from this fund can be reinvested at only 4% effective. Show that the total accumulated value at the end of ten years is 1250(s11|0:04 − s6|0:04 − 1) My question is I can find AV(lets call it X) of 5 years = 1000 * [(5 + i/j(s6|j- 6)] n=5, i=0.05, j =0.04 Is AV of 10 years = X (1+i)^5 or X (1+j)^5? Thanks!
Neither, because 5000 of that X is earning interest at 5% (but only for 1 year, then reinvested at 4%; the rest is earning 4%).

I don't even know where any of these formulas come from, but in many FM problems where there are payments in some string of years but not others, a good strategy (that I think would work here):

What is the value if payments (deposits of 1000) are made every year)?
Call it Z. Probably a calculation similar to what you did for X, just a different period.

What is the value if payments (deposits of 1000) are made only in the years that no payments (deposits) were made? Call it Y. There's a real good chance your number will be what you got for X, because you are evaluating Y at time 10, but it is directly at the end of 5 years of payments/deposits.

Then the answer to the question is Z - Y, the value from deposits every year less the value from deposits that were not made.
#55
05-26-2019, 07:28 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

Quote:
 Originally Posted by Gandalf Neither, because 5000 of that X is earning interest at 5% (but only for 1 year, then reinvested at 4%; the rest is earning 4%). I don't even know where any of these formulas come from, but in many FM problems where there are payments in some string of years but not others, a good strategy (that I think would work here): What is the value if payments (deposits of 1000) are made every year)? Call it Z. Probably a calculation similar to what you did for X, just a different period. What is the value if payments (deposits of 1000) are made only in the years that no payments (deposits) were made? Call it Y. There's a real good chance your number will be what you got for X, because you are evaluating Y at time 10, but it is directly at the end of 5 years of payments/deposits. Then the answer to the question is Z - Y, the value from deposits every year less the value from deposits that were not made.
Thanks for the reply. I did your method and really needed to give it a lot of thought since subtracting the AV (t=0 to 10) - AV (t=0 to 5) could mean so many things. Then drew the time diagram and it helped.

The formula is in Finan's book for reinvestement: Accumulated Value at the end of the n periods is the sum of the annuity payments and the accumulated value of the interests for annuity-due.

$AV = n + i \frac{s_{\overline{n+1}|j} -(n+1)}{j}$

AV (t=0 to 10) = s|11 -3
AV (t=0 to 5) = s|6 -2

AV = 1250 [(s|11 - 3) - (s|6 -2)] = 1250[s|11 - s|6 - 1]
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Last edited by jubair07; 05-27-2019 at 01:14 PM..
#56
05-27-2019, 03:38 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

An investor puts $100 into a mutual fund in the first year and$50 in the second year. At the end of the first year the mutual pays a dividend of $10. The investor sells the holdings in the mutual fund at the end of the second year for$180. Find the dollar weighted return.

100(1+i)^2 + 50(1+i) + 10(1+i) = 180

i = 15.51%. Answer on book is 15.65%. I used simple interest method and i = 15.38%.

Where did I go wrong or there another method I didn't apply?
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Last edited by jubair07; 05-27-2019 at 05:50 PM..
#57
05-27-2019, 05:15 PM
 Academic Actuary Member Join Date: Sep 2009 Posts: 8,567

The $10 dividend should have a negative sign. It is a payment from the fund to you. If i is the effective annual rate you would have 100(1+i)^2 - 40(1+i) = 180. #58 05-27-2019, 05:50 PM  jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147 Quote:  Originally Posted by Academic Actuary The$10 dividend should have a negative sign. It is a payment from the fund to you. If i is the effective annual rate you would have 100(1+i)^2 - 40(1+i) = 180.
Thanks. Now I got the correct answer.
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#59
05-27-2019, 08:07 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

An investment manager’s portfolio begins the year with a value of 100,000. Eleven months through the year a withdrawal of 50,000 is made and the value of the portfolio after the withdrawal is 57,000. At the end of the year the value of the portfolio is 60,000. Find the time-weighted yield rate less the dollar-weighted yield rate.

I = 60,000+ 50,000 - 100,000 = 10,000
A=100,000
C=-50,000

$id = \frac{10,000}{100,000- 50,000(1-10/12)}= 0.1091$

$it = \frac{57,000}{100,000-50,000} * \frac{60,000}{57,000} -1 = 0.2$

it-id = 0.0956

I checked it*id = 0.0218 = 0.022. But didn't the question ask less and not times?

Answer on the book is 0.022. Where did I go wrong?
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Last edited by jubair07; 05-27-2019 at 09:19 PM..
#60
05-27-2019, 09:16 PM
 jubair07 Member SOA Join Date: Jan 2016 Studying for Exam FM College: BEng Aerospace Engineering Posts: 147

The problem is attached below.

I calculated:

DW = 27.9%
TW=30%

$NPV = \frac{-1100}{(1+i)^{1/3}} + \frac{900}{(1+i)^{2/3}}= 0$

This gives IRR = -45.22%

Ans: TW > IRR > TD

Where in NPV did I go wrong?
Attached Images

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