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**UNIVERSITY OF TORONTO**

##### ACT240H1F

TERM TEST 1 6:00 pm May 22, 2008

Instructor: Keith Sharp PhD FSA CFA

NOTES:

- Calculators allowed
- It’s OK to write on book. If you also use scrap, please submit it with this book.
- This is a closed book exam.
- Multiple choice: only your blobs on the Scantron sheetwill be graded.
- 10 points correct, two if blank, zero points if wrong
- So expectation if you guess is the same as leaving a blank.
- Timing: 50 minutes
- Make sure you’ve indicated your letter answers on the Scantron sheet before time’s up
- Please stay in your seats and don’t talk till all question papers have been collected.
- Photo ID on desk during exam please.
- Name and student ID at top of this question paper please.
- For purposes of identifying your privacy ID, please code question 11 with the privacy ID (A), (B), (C) or (D) indicated in the footer of this and every page.
- Good luck!

ACT 240H1F Sum08 Term Test 1 Privacy ID A v01

**UNIVERSITY OF TORONTO: ACT240H1F SUMMER 2008 TEST 1**

(t)= 0.01 (1+ 0.4t2)

You invest $10,000 at time t=0. Calculate your account balance at time t=3.

(A) Less than $10,660.000

(B) $10,660.000 but less than $10,670.000

(C) $10,670.000 but less than $10,680.000

(D) $10,680.000 but less than $10,690.000

(E) $10,690.000 or more

(D) Solution:

d/dt A(t) = A(t) (t)

d/dt ln (A(t)) = (t)

A(T) = A(0)exp{ }

*= 10,000 * exp{ 0.01 [t + 0.4t3/3] 0 3 }*

= 10,000 * exp{ 0.01 [3 + 0.4*9] }

= 10,682.27

2. (000000 01-007 ;Sharp401610220840v02) (Assignment for May 15, 2008) Interest rates are 7% per annum effective (compound interest). Jenny pays income tax at a rate of 30% on salary and investment income. However, if the investment is inside a Registered Retirement Savings Plan (RRSP) she doesn’t have to pay any tax on the interest. On her 25th. birthday her savings account has $50,000 in it, and her RRSP has $50,000 in it. Calculate the ratio (RRSP balance)/(savings account balance) on her 65th birthday. (Note that in this question we are not taking account of inflation, of the tax deduction on depositing money into an RRSP or of the tax payable on taking money from an RRSP).

(A) Less than 1.500

(B) 1.500 but less than 2.000

(C) 2.000 but less than 2.500

(D) 2.500 but less than 3.000

(E) 3.000 or more

(C) Solution:

Ratio

*= 50,000 * 1.0740 / 50,000 * (1+ (1-0.3) * 0.07)40*

=50,000 * 1.0740 / 50,000 * 1.04940

= 2.2097

3.

*(000000 01-013 ; B40F06TT1, B40F07TT1 Brov3-Ex. 1.1.02av03)*Unit values in a mutual fund have experienced annual growth rates of 10%, 26%, -7%, 4% and 19% in the past five years. The fund manager suggests that the fund can advertise an average growth rate of 10% over the past five years. What is the actual annual compound growth rate over the past five years?

(A) Less than 9.500%

(B) 9.500% but less than 10.000%

(C) 10.000% but less than 10.500%

(D) 10.500% but less than 11.000%

(E) 11.000% or more

(B) Solution:

[1.10*1.26*0.93*1.04*1.19] 0.2 -1 = 0.097906

4. (000000 01-015 ;Sharp401610421552v00 ;240Sum08TT1) You invested in gold and in silver five years ago. Gold was then priced at $500 per ounce. Now gold is priced at $900 per ounce. The annual effective interest rate earned on silver over the same period was half the rate for gold. Your lump of silver is now worth $25,000. What amount X did you pay for your lump of silver?

(A) Less than $18,000.000

(B) $18,000.000 but less than $18,500.000

(C) $18,500.000 but less than $19,000.000

(D) $19,000.000 but less than $19,500.000

(E) $19,500.000 or more

(B) Solution:

(1+igold)5 = 900/500

Hence igold=(9/5)0.2 -1 = 0.1247461

(1+0.5*igold)5 = 25,000/X

(1+0.5*0.1247461)5 = 25,000/X

X= 18,473.74

5. (000000 01-023 ;Sharp401610421229v00 ;240Sum08TT1) (Question of class May 13, 2008) Your Visa card has a balance of (that is, you owe) $1,600 on February 1. On May 1 you make a $300 payment to the bank to reduce your Visa debt. You make no other payments or purchases. What has the debt become by December 1? Assume an interest rate of 18% per annum compounded monthly.

(A) Less than $1500.000

(B) $1500.000 but less than $1540.000

(C) $1540.000 but less than $1580.000

(D) $1580.000 but less than $1620.000

(E) $1620.000 or more

(B) Solution:

*Bal (May 1) = 1,600 * 1.0153 - 300 = 1373.085*

*Bal (Dec 1) = 1373.085*1.0157 = 1523.91*

6. (000000 01-029 ;Brov3-Ex. 1.4.04 ;Sum06TT1;F06TT1, Sum07Fin) Bruce and Robbie each open up new bank accounts at time 0. Bruce deposits 100 into his bank account and Robbie deposits 50 into his. Each account earns an effective annual discount rate of d. The amount of interest earned in Bruce’s account during the 11th year is equal to X. The amount of interest earned in Robbie’s account during the 16 th year is also equal to X. Calculate X.

(A) Less than $57.000

(B) $57.000 but less than $58.000

(C) $58.000 but less than $59.000

(D) $59.000 but less than $60.000

(E) $60.000 or more

(D)Solution:

Same d so same i.

100 i (1+i) 10 = X = 50 i (1+i)15

2 (1+i)10 = (1+i)15

i=0.148698

Hence X = 100 i (1+i) 10

=59.479

7. (000000 01-035 ;Sharp401610421606v00 ;240Sum08TT1) An account pays the following rates:

2009 / 6% per annum effective

2010 / 6% per annum compounded monthly

2011 / 6% rate of discount

2012 / 6% force of interest

You make a series of deposits of $1 million each on January 1 of each of the four years 2009, 2010, 2011 and 2012. Calculate the amount in your account at the end of 2012. Give the answer to the nearest $10,000.

(A) $4,640,000

(B) $4,650,000

(C) $4,660,000

(D) $4,670,000

(E) The correct answer is not given by (A), (B), (C) or (D)

(C) Solution:

Y / Multiplier within year / Jan 1, Y becomes,Dec31, 2012

2009 / 6% per annum effective / 1.06000= 1.0600000 / 1.2712428

2010 / 6% per annum compounded monthly / 1.00512 = 1.0616778 / 1.1992856

2011 / 6% rate of discount / 1/(1-0.06)=1.0638298 / 1.1296134

2012 / 6% force of interest / exp(0.06)= 1.0618365 / 1.0618365

AV /

**Sum at 2012/12/31**/

**Sum of last column**/

**4.6619783 million**

8. (000000 01-041 ;Sharp401610421240v00 ;240Sum08TT1) (Question of class May 15, 2008) The money in your bank account will currently buy you 3,000 bottles of wine. Wine prices are inflating at f=4% per year effective. You pay income tax of T=35% of any interest income that you earn. You require that you will be able to buy g=2% more bottles of wine a year from now; otherwise you’ll have a big wine party now. Calculate the minimum account interest rate iMIN per annum effective that will persuade you to keep your money in the bank.

(A) Less than 8.000%

(B) 8.000% but less than 8.500%

(C) 8.500% but less than 9.000%

(D) 9.000% but less than 9.500%

(E) 9.500% or more

(D) Solution:

1+g< (1+(1-T)*iMIN) / (1+f)

1.02 < (1+(1-0.35)*iMIN) / 1.04

iMIN = 0.093538 = 9.35%

9. (000000 02-076 ;Sharp401610421606v00 ;240Sum08TT1) You borrowed $100,000 on January 1, 2010. You repay the loan by making 60 monthly payments in the form of an annuity due. The intention is that the payments are to be equal. However, the October 1, 2010 payment is half the usual amount because you are short of money and the lender agrees to this partial payment. As a result, the last payment is increased to 180% of its usual amount. Calculate the usual monthly payment K.

(A) Less than $2,100.000

(B) $2,100.000 but less than $2,200.000

(C) $2,200.000 but less than $2,300.000

(D) $2,300.000 but less than $2,400.000

(E) $2,400.000 or more

(B) Solution:

*Last payment is December 1, 2014, which is 4 2/12 years after 50% is not paid. That 50% becomes an 80% debt.*

(1+i(12)/12) 50= 80/50

i(12)/12= 0.009444392

100,000 = K ä60

*100,000 = K * 1.009444392 * a60| (at 0.9444392%)*

100,000 = K * 46.07442

K=$2,170.40

10. (000000 02-082 ;Sharp401610400748v02) (Assignment for May 22, 2008) You aim to work w=35 years from age 25 to 60 and then to draw pension for p=30 years till age 90. Your pay rate stays the same in real terms (after inflation). You wish to have the same living standard after retirement (no need to save) as before retirement (when you live on pay minus the retirement savings rate). For simplicity, model pay and pension as being paid annually at the end of each year. The real rate of interest, after any taxes, is i real = 1% per annum. Calculate as a percentage the proportion of your pay C that you need to save annually before retirement. Hint: easiest to take an equation of value at the time of retirement; accumulated savings=present value of pension. Notice that the pay rate cancels from the calculation.

(A) Less than 35.000%

(B) 35.000% but less than 36.000%

(C) 36.000% but less than 37.000%

(D) 37.000% but less than 38.000%

(E) 38.000% or more

(E) Solution:

*Accumulated savings at retirement = present value of pension at retirement,*

*If you wish to use actuarial notation this is:*

*ZPROP sw (at ireal)= (1-ZPROP) ap (at ireal)*

*which is the equation of value at retirement (having divided by retirement age pay rate). Or just using geometric series*

*ZPROP { (1+ireal) w-1 + (1+ireal) w-2 + ………+ (1+ireal)1 + 1} =*

*(1-ZPROP) { (1+ireal) -1 + (1+ireal) -2 + ………+ (1+ireal)-p+1+(1+ireal)-p }*

*Sum the geometric series or use the formulas for sw and ap*

*ZPROP { (1+ireal) w - 1} / ireal = (1-ZPROP) { 1- (1+ireal)-p }/ireal*

*ZPROP = { 1- (1+ireal)-p }/ { (1+ireal) w - (1+ireal)-p } (≈p/(w+p) for small ireal)*

*ZPROP(w=35, p=30, ireal =0.01) = 38.25%*

11.For purposes of identifying your privacy ID, please code question 11 with the privacy ID (A), (B), (C) or (D) indicated in the footer of this and every page.

ACT 240H1F Sum08 Term Test 1 Privacy ID A v01