Question 2 9 marks You have finally decided that this is the year that you purchase that exotic car Aston Martin Vanquish 007’s car, Porsche Carrerra GT, Bentley Continental etc…etc... O
Trang 1Winter 2007 Question 1 (12 marks)
You became incredibly wealthy one day and a huge contributing factor to your success was the education that you’d received at York/Atkinson As a way to thank the school you set up an endowment today in the amount of $5,000,000 But you don’t just hand over $5,000,000 just like that!!! You stipulate that York use the funds to provide scholarships to ADMS3530 students totaling $250,000 annually The scholarships are to commence one year from today and are to continue for as long as ADMS3530 is around (in effect for ever!!!) The market interest rate for the foreseeable future is expected to be 5% per annum compounded annually
(a) Based on the above information will York be able to meet your stipulation by providing ADMS3530 students with annual scholarships totaling $250,000 for ever? Provide supporting analysis (2 marks)
Answer
This is a perpetuity question The formula should be of the form
PV perpetuity = Annual Payment / Annual Int Rate
OR
Endowment = Annual Scholarship / Annual Int Rate
Endowment = $250,000 / 5% = $250,000 / 0.05 = $5,000,000
Therefore your $5,000,000 endowment is sufficient to fund annual scholarships
of $250,000 forever based on a prevailing annual interest rate of 5%
(b) What if your endowment was $4,000,000 instead of $5,000,000 would York still be able to meet your stipulation? If not what would be the maximum annual scholarship payout? Provide supporting analysis (3 marks)
Answer
Based on the answer above York would NOT be able to meet you stipulation as there would be a shortfall in the endowment of $5,000,000 - $4,000,000 =
$1,000,000
With a $4,000,000 endowment the Annual Scholarship = Endowment * Annual Int Rate
Trang 2Annual Scholarship = $4,000,000 * 0.05 = $200,000
Therefore with a $4,000,000 endowment York would be able to fund a maximum annual scholarship of $200,000 with prevailing annual interest rates of 5%
(c) As we all know the cost of education keeps going up and up To accommodate these inflationary pressures you ask that the $250,000 scholarship grow at a constant rate of 1% annually Would York be able to fund this scholarship for ever with your $5,000,000 endowment? If not what would be the maximum annual scholarship? Provide supporting analysis (4 marks)
Answer
PV perpetuity = Annual Payment / (Annual Int Rate – Growth Rate of Payment)
OR
Endowment = Annual Scholarship / (Annual Int Rate – Growth Rate of Scholarship)
Endowment = $250,000 / (0.05 – 0.01) = $6,250,000
Therefore with an endowment of $5,000,000 there would be shortfall of
$6,250,000 – $5,000,000 = $1,250,000
Annual Scholarship = Endowment * (Annual Int Rate – Constant Growth Rate of Scholarship)
Annual Scholarship = $5,000,000 * (0.05 – 0.01) = $200,000
With an endowment of $5,000,000 York would be able to fund an annual scholarship of $200,000 that would grow at a constant annual rate of 1%
(d) What would your endowment have to be to accommodate annual scholarships of $300,000 with a constant growth rate of 1% annually? Provide supporting analysis (3 marks)
Answer
Endowment = Annual Scholarship / (Annual Int Rate – Growth Rate of Scholarship)
Endowment = $300,000 / (0.05 – 0.01) = $7,500,000
Therefore an endowment of $5,000,000 would be required to fund an annual scholarship of $300,000 that would grow at a constant annual rate of 1%
Trang 3Question 2 (9 marks)
You have finally decided that this is the year that you purchase that exotic car (Aston Martin Vanquish (007’s car), Porsche Carrerra GT, Bentley Continental etc…etc.) But your spouse (Salma Hayak or Brad Pitt depending on who you are) insists that you must decide on the maximum purchase price you can afford before you ever look at your first car
Assume that the only source of money to finance your purchase is based on your family income The details of which follow…
Your annual income is $50,000 before taxes which is taxed at a flat rate of 30%
Of the after tax income you can commit 25% annually towards the purchase of the car for the next 5 years (starting 1 year from today)
Your spouse’s annual income is $5,000,000!!! (remember it’s Salma or Brad) before taxes which is taxed at a flat rate of 50% Of the after tax income your spouse will only commit 0.5% annually to the purchase of your car for the next 3 years (starting 1 year from today)
The prevailing rate of interest on auto loans in the market place is 7% compounded annually
Assuming you can get an auto loan, can you afford one of the luxury cars which are priced today at over $250,000? Or will you have to buy a Honda Odyssey (which costs approx $50,000) mini van which is that much more practical as you have 3 small children Show your calculations to support your decision
Answer
There are a number of ways to solve this question I’ve included solutions for two different approaches
The question requires that you calculate the PV of cash that is available to you to finance the purchase of your car If the PV is equal to or greater than $250,000 then you can afford the exotic car If it’s greater than $50,000 but less than
$250,000 then you can afford the Honda Odyssey
Available money = available from my income + available from spouse’s income
Money available from my income = $50,000 * (1 - 0.30) * 0.25 = $8,750 annually for 5 yrs starting one year from today
Money available from spouse’s income = $5,000,000 * (1 - 0.50) * 0.005 =
$12,500 annually for 3 yrs starting one year from today
Solved using the PV formula PV = FV / (1+r)^t
From my income, PV of money available for years 1 to 5
= [8,750 / (1 + 0.07)^1] + [8,750 / (1 + 0.07)^2 ]+[ 8,750 / (1 + 0.07)^3 ]+
Trang 4[8,750 / (1 + 0.07)^4] + [8,750 / (1 + 0.07)^5]
=$35,876.73
From spouse’s income, PV of money available for years 1 to 3
= [12,500 / (1 + 0.07)^1] + [12,500 / (1 + 0.07)^2] + [12,500 / (1 + 0.07)^3]
= $32,803.95
Total PV available = $35,876.73 + $32,803.95 = $68,680.68
Therefore we cannot afford a $250,000 exotic vehicle however we can afford the
$50,000 Honda Odyssey
Solved using the annuity formula
PV of your income = C x [1/r - 1 / r(1+r)^t] = 8750*((1/0.07)-1/(0.07*(1.07^5))) = $35,876.73
PV of your spouse’s income = C x [1/r - 1 / r(1+r)^t] = 12,500*((1/0.07)-1/(0.07*(1.07^3))) = $32,803.95 Total PV of income
Question 3 (15 marks)
RRSP season is coming up and Canadians will be faced with many
investment opportunities Assume that you are a Canadian resident with a
marginal tax rate of 40% You have decided to invest in a $1,000 RRSP
Since this is tax deductible and you will receive a $400 tax refund given your
marginal tax rate (40% of $1,000) You have two options available to you
The first is presented here and the second appears in part (c) below In order
to buy the RRSP you may borrow $1,000 and then pay down your $1,000
loan with the $400 immediate tax refund Throughout this question interest
rate is assumed to be 9% compounded monthly
(a) What is the monthly payment required to pay down the loan in one year?
(2 marks)
Answer
Since $400 will be paid down immediately we need to calculate the PMT
on a $600 loan balance
PMT =? n = 12, i = 9% compounded monthly (i.e monthly interest rate is
0.75%), PV=$600, then:
Trang 5PV of an annuity =
+
−
r r r
C
) 1 (
1 1
+
−
) 0075 1 ( 0075
1 0075
1
∴ C = $600/11.4349 = $52.47
(b) To what will the single $1,000 contribution grow over the next 20 years? (2
marks)
Answer
PV = $1,000, n = 20, i = 9% compounded monthly, then:
FV = PV× 1( +r)t
0075 1
FV = $6,009.15
(c) Let’s say that you opted rather than borrow the $1,000 and immediately
pay down the $400 and then pay out the loan over a one year period (as
in part (a) above) that you agreed last year to make monthly contributions
to a savings plan in order to realize a $600 payout by the end of the 12th month (i.e today) and then borrow the remaining $400 to purchase the
$1,000 RRSP now, how much would you need to save on a monthly basis? You will immediately pay down the $400 by the end of the first month that it is due (3 marks)
Answer
FV = $600, n = 12, i = 9% compounded monthly, then:
×
r
r C
t 1 ) 1 (
×
0075
1 ) 0075 1
C
C = $600/12.5076 = $47.97
(d) Assuming that you would like to continue your RRSP investment in part
(a) or part (c) annually for the next 20 years until your retirement
Trang 6Compare the two RRSP plans by calculating the PV of both options (4 marks)
Answer
The actual present value of both will be the same The difference will lie in the net expense of carrying the annuity each year for 20 years
PMT = $1,000, n = 20, i = 9% compounded monthly, then:
the monthly interest rate (rmon) = i/12 = 0.75%,
and the EAR = (1 r )m 1 (1 0.75%)12 1 9.38069%
+
20 886 , 8 8862 8 000
,
1
] ) 0938069
1 ( 0938069
0
1 0938069
0
1 [ 000
,
1
] ) EAR 1
( EAR
1 EAR
1 [ PMT annuity
of
PV
20 n
=
×
=
−
×
=
+
−
×
=
(e) What is the benefit of one plan over the other assuming that the interest
rate remains at 9% monthly compounded throughout the next 20 years? (4 marks)
Answer
The difference will lie in the actual cash flow savings and payments
generated by the savings plan versus the interest which needs to be paid
on the loans (ignoring the tax expense deduction for now)
From (a) we can calculate that the annual interest payments as equal to ($52.47)(12) - $600 = $29.64 This should be a negative cash flow per year from the perspective of the RRSP investor
From (c) we can calculate the positive cash flow of $24.36 (= $600 –
$575.64) each year This is beneficial to the RRSP investor So the
savings plan in (c) is preferred to the borrowing plan in (a)
Question 4 (13 marks)
Another Canadian investor is planning his retirement Given the following information please help him with his calculations
(a) What will be the amount in an RRSP after 25 years, at which time he will
retire and live off the proceeds, if contributions of $3,000 are made at each year-end for its first seven years and month-end contributions of $500 are
Trang 7made for the subsequent 18 years? Assume that the plan earns 8% compounded quarterly for the first 12 years, and 7% compounded semiannually for the subsequent 13 years (4 marks)
Answer
For the first 7 years, i = 8%4 = 2%, PMT = $3,000,
n = 1(7) = 7, c = 14 = 4, and
EAR=( )1+i c −1 = (1.02)4– 1 = 0.082432160
Amount in the RRSP after 7 years will be
08243216 0
1 08243216
.
.
= $26,968.51
For the next 5 years, PMT = $500,
n = 12(5) = 60, and
mon = + − = (1.08243216)1 / 12– 1 = 0.00662271
Amount in the RRSP after 12 years will be
FV = PV( )1 +i n + ( )
+ −
i
i PMT
n 1 1
= $26,968.51(1 02)20 + $5001.006622710.00662271−1
60
= $76,761.75
For the last 13 years, PMT = $500, n = 12(13) = 156,
i = 72% = 3.5%, and
EAR=(1.035)2 −1=0.071225
rmon =(1+EAR)c −1 = ( )1 / 12
071225
Amount in the RRSP after 25 years will be
FV = PV( )1 +i n + ( )
+ −
i
i PMT
n 1 1
= $76,761.75(1 035)26 + $5001.005750.00575−1
156
= $313,490.28
(b) Your investor would like to set up a Scholarship Fund in his name at
Atkinson College with a $500 annual award to a deserving applicant The first award will be made at the end of the 15th year of his retirement Subsequent awards will be made at the end of each year perpetually
Trang 8Given the RRSP amount in part (a) above, how much can your investor expect to withdraw at the end of each month, starting the first month after his retirement, and still be able to set up the Scholarship Fund? (Assume the interest rate is 10% compounded monthly throughout his retirement.) (6 marks)
Answer
i First we need to calculate how much will be required to set aside at Year
15 for the scholarship:
i = 1012% = 83333%, C = $500,
c = 121 = 12, and
i2 =( )1 +i c − 1 = ( )12
0083333
PV of a perpetuity =
r C
=
0.104709
500
$
= $4,775.14
ii Then we need to calculate the PV of this amount and deduct it from the fund at the start of the retirement $313,490.28
PV of $4,775.14 for 15 years at 10% compounded monthly is $1,072.18 Subtract this from the fund:
$313,490.28 - $1,072.18 = $312,418.10
(Alternately by using the calculator you could find the answer by allowing for a residual FV at the beginning of Year 15 of $4,775.14.)
iii Calculate the monthly payments on the PV of $312,418.10 at 10%
compounded monthly:
+
−
r r r
C
) 1 (
1 1
+
−
) 008333
1 ( 008333
1 008333
1 C
C = $3,357.96
Trang 9(c) What complications might occur in achieving the plan in part (b)? (3
marks)
Answer
The main purpose of the question is to have you discuss the fact that upon retirement no one can really plan when they will die
There are two concerns to be addressed:
1 What will happen if the investor dies before the beginning of the
15th year? A simple solution would include a will that ensures that the annuity is taken over by a 3rd party, either as an inheritance or as an asset in the estate
2 The real issue is what if the investor survives beyond the 15th
year; there is no more money except for the small amount that has been set aside as a scholarship The investor either becomes a ward of the family or the state
Question 5 (14 marks)
Throughout this question consider the following bond: face value of $1,000, coupon rate is 8%, semi-annual coupon payments, 4 years of maturity, and a purchase price of $1,055.69
(a) Calculate the current yield and yield to maturity on the bond as of the date
of purchase (3 marks)
Answer
%
58 7 69 055 , 1
80
$ 69
055 , 1
000 , 1 08
=
price bond
coupon annual
= yield Current
If you use your financial calculator, you will find the yield to maturity (YTM)
to be 6.4% Alternatively, if you use the approximate formula, the YTM is:
Trang 104287 6 2
) 69 055 , 1 000 , 1
) 69 055 , 1 000 , 1 ($
80
$
2
) price current value
face (
maturity
) price current value
face ( coupon annual
YTM
= +
− +
=
+
− +
=
(b) Calculate the current yield and bond price on each anniversary date of the
bond purchase until maturity Suppose on each of these dates the yield to maturity on the bond is 7%, 6.6%, 6.2%, and 6.36%, respectively (6
marks)
Answer
To illustrate, let’s compute the current yield and bond price on the first anniversary The calculations for the other anniversaries follow suit
64 026 , 1 035 1
000 , 1 ] ) 035 0 1 ( 035 0
1 035
0
1 [ 40
$
) 1 (
value face ] ) 1 (
1 r
1 [ coupon price
Bond
6 6
t t
= +
+
−
×
=
+
+ +
−
×
=
%
79 7 64 026 , 1
80
$ 64
026 , 1
000 , 1 08
=
price bond
coupon annual
= yield Current
For the second and third anniversaries and the maturity date, the bond price and current yield (in parentheses) is: $1,025.83 (7.8%), $1,017.2 (7.86%), and $1,000 (8%)
(c) Assume instead of holding the bond until maturity, you sell the bond on
the second anniversary of its purchase (right after you receive the last coupon) Based on your results in part (b) above, what is your total rate of return over this 2-year holding period? What is your annual rate of return over the same period? Assume you can reinvest the previous coupons at
an APR of 10% quarterly compounded (5 marks)
Answer
First, we need to compute the semi-annual interest rate on reinvestment
%
0625 5 1 ) 4
APR 1
(
1 ) rate quarterly 1
( rate annual Semi
2
2
=
− +
=
− +
=
−