assignment4 fundamentals of corporate finance, 4th edition brealey

John wants to receive the first annuity payment at the end of the 15th year from today the same day as his retirement date.. a If you assume that rates do not change over the five-year t

AK/ADMS3530 3.0 Fall 2007 Assignment #1 Solutions

Instructions:

(1) This assignment is to be done individually You must sign and submit the

standard cover page supplied as the last page of this assignment

(2) This assignment is due in class in the week of October 8, 2007

(3) This assignment can be typed or handwritten Work that is too difficult to

read due to messiness and poor handwriting will receive zero credit You must show your work to receive full credit

(4) This assignment carries a total mark of 100 points

(5) For Internet section students, the assignment must be uploaded to the

Centre for Distance Education:

http://www.atkinson.yorku.ca/cde/assignmentupload and identified

precisely in accordance with the course outline by Tuesday, October 9, midnight

(6) Late assignments will not be accepted whether for technical or any other

reason

Notations

We may denote the PV and FV annuity factors respectively by PVIFA(r,n) and FVIFA(r,n), i.e.:

r

r n

r FVIFA r

r n

r PVIFA

n

) , (

; )

1 ( 1 ) ,

≡ +

−

Question 1 (TVM) (15 marks)

This question consists of the following three independent parts

(a) John plans to retire in 15 years, and he wants to have an annuity of $50,000 a year for 20 years after retirement John wants to receive the first annuity payment

at the end of the 15th year from today (the same day as his retirement date) How much must John invest today in order to have his retirement annuity if the annual interest rate is 6%? (5 marks)

(b) How long would it take to accumulate $1 million if you begin putting $100 in the bank every week starting one week from now at an EAR of 8% with weekly compounding? There are 52 weeks in a year (5 marks)

(c) Jen is going to deposit $250 into an account at the beginning of each of the next 5 years starting today The account will then be left to compound for another

10 years (between the end of year 5 and the end of year 15) At the end of year

16, Jen will start receiving a perpetuity from the account If the account pays 7% annually, how much each year will she receive from the perpetuity? (5 marks)

Solution

(a) The present value of the 20-year annuity due over John’s retirement is:

607,905.82

$

] ) 06 0 1 ( 06 0

1 06

0

1 [ ) 06 0 1 ( 000 , 50

$

] ) 1 (

1 1

[ ) 1 (

$

20

=

+

×

−

× +

×

=

+

−

× +

×

r r r r C

PV

This will also be the future value of John’s savings account in 15 years time from today Therefore, the principal amount that John should deposit into his account today is:

253,657.86

$ ) 06 0 1 (

82 905 , 607

$

$

) 06 0 1 (

$ 82 905 , 607

$

) 1 (

$

15

15

= +

=

∴

+

×

=

+

×

=

D

D

r D

(b) First, we need to find the corresponding weekly interest rate:

%

1481 0 1 ) 08 0 1 (

1 ) EAR 1

( r

52 / 1

m / 1 week

=

− +

=

− +

=

The savings account is an annuity for n weeks:

% 1481 0

1

%) 1481 0 1 ( 100

$ 000 , 000 , 1

r

1 ) r 1 ( D

$ FV

n week

n week

− +

×

=

− +

×

=

Solving for n, the number of weeks, by your financial calculator, you

should get:

PV = 0; FV = 1,000,000; PMT = -100; I/Y = 0.1481;

Compute N = 1,865.42 weeks, or roughly 35.87 years

(c) The account is an annuity due of 5 years followed by a single sum for another 10 years The FV of this account at the end of the 15th year (or the beginning of the 16th year) is the PV of the perpetuity

3,026.11

$

) 07 0 1 ( ] ) 07 0 1 ( 07 0

1 07

0

1 [ ) 07 0 1 ( 250

$

) 1 ( ] ) 1 (

1 1

[ ) 1 (

$

10 5

=

+

× +

−

× +

×

=

+

× +

−

× +

×

r r r r D

FV

The annual payment from the perpetuity beginning at the end of Year 16 is:

83 211

$ 07 0 11 026 , 3 r PV C

$ r

C

$ PV

=

×

=

×

=

∴

=

Question 2 (TVM) (16 marks)

A 20-year annuity was purchased with $180,000 that had accumulated in an RRSP The annuity provides a semiannually compounded rate of return of 5% and makes equal month-end payments

(a) What is the monthly payment? (4 marks)

(b) What will be the principal portion of Payment 134? (4 marks)

(c) What will be the interest portion of Payment 210? (4 marks)

(d) How much interest will be paid in the 6th year? (4 marks)

Solution

(a) We first need to compute the monthly rate equivalent to the semi-annually compounded rate of 5%, that is we solve for i such as:

(1+i)12 = (1+5%/2)2 or i = 0.4124%

The monthly payment is then given by:

$180,000 = PMT x PVIFA(0.4124%,240) PMT = $1,182.83

(b) The principal portion in the 134th payment is given as the difference between the balances after the 133th and the 134th payments:

PV133 = $1,182.83 x PVIFA(0.4124%,107) = $102,162.72

PV134 = $1,182.83 x PVIFA(0.4124%,106) = $101,401.21

The principal portion in payment 134 is then $761.51

(c) We first need to compute the balance after the 209th payment:

PV209 = $1,182.83 x PVIFA(0.4124%,31) = $34,354.26 The interest portion in the 210th payment is simply:

I210 = PV209 x i = $34,354.26 x 0.4124% = $141.68 (d) The principal portion paid in the 6th year is given as previously shown

(in part b) by:

PV60 = $1,182.83 x PVIFA(0.4124%,180) = $150,080.75

PV72 = $1,182.83 x PVIFA(0.4124%,168) = $143,158.37

The principal portion in year 6 is then $6,922.38 The interest paid in year 6

is given by:

12 x 1,182.83 – 6,922.38 = $7,271.52

Question 3 (TVM) (15 marks)

Consider the following stream of cash flows:

where the payments of X start one year from today and last for 10 years; and the

payments of Y start in 11 years from now and last forever The interest rate is

4%

(a) If X = $100 and Y = $200, what is the present value of this stream? (4 marks)

(b) If the present value of the stream is $4,000 and Y = $100, what is the

value of X? (5 marks)

(c) Assume now that the value of the stream at year 10 is $8,000 and that

Y = 3X What is the value of X? (6 marks)

Solution

(a) One simple way to compute the PV is by considering the following equivalent stream: a perpetuity of $200 starting in one year and a 10-year annuity of $100 The PV is then given by:

PV = $200 / 4% - $100 x PVIFA(4%,10) = $5,000 - $811.09 = $4,188.91 Another way to solve this question is to treat the X-annuity as a simple annuity and the Y-payments as a perpetuity delayed for 10 years Here are the details:

PV = $100 x PVIFA(4%,10) + $200 / 4% x (1+4%)-10

= 811.09 + 3,377.82

= $4,188.91

(b) Following part (a), we can consider a perpetuity of $100 and a 10-year annuity of (X – 100):

PV = $4,000 = $100 / 4% + (X - 100) x PVIFA(4%,10)

= $2,500 + (X - 100) x PVIFA(4%,10)

Or $1,500 = (X - 100) x PVIFA(4%,10)

X – 100 = $184.94

X = $284.94

Another way to solve this question (like part a) is to write:

PV = $4,000 = X x PVIFA(4%,10) + $100 / 4% x (1+4%)-10

= 8.1109 x X + $1,688.91

Or X = (4,000- 1,688.91) / 8.1109 = $284.94

(c) Value at year 10 = $8,000 = X x FVIFA(4%,10) + 3X / 4%

$8,000 = X x [ FVIFA(4%,10) + 3 / 4%]

$8,000 = X x [ 12.01 + 75] = 87.01 x X

Or X = 8,000 / 87.01 = $91.95

Question 4 (Mortgage) (20 marks)

Below is a summary of special mortgage deals from TD Canada Trust as of September 13, 2007:

Special Offers – TD Canada Trust Rates

As a recent graduate of Atkinson’s BAS finance program, you are trying to decide

which mortgage option to choose You need a $300,000 mortgage for your new

luxury 500 square foot condo at the corner of King and Bay, downtown Toronto

You have chosen a 25 year amortization period for the loan

(a) If you assume that rates do not change over the five-year term of the

mortgage, what is the monthly payment under the 5.75% variable rate

option? (4 marks)

(b) If you decide to go with the 6.18% fixed mortgage rate, what is the

remaining balance of your $300,000 loan at the end of year 5? (4 marks)

(c) You have been studying the sub prime market meltdown and are

convinced that 5 year variable mortgage interest rates will actually decline

to 5.05% beginning in year 2 of your mortgage loan but will then increase

to 7.00% for year 3, 4 and 5 If you are correct, which option is best? (6 marks)

(d) You have now decided to take the 6.18% fixed rate mortgage but you

decide to go with a 20 year amortization period instead of a 25 year

period What will be your total interest savings over the entire loan period

(assume rates remain at 6.18%) by taking the 20 year amortization

period? (6 marks)

Solution

(a) Mortgage rates in Canada are APRs for 6-month interest rates, so we

first need to convert them to get the monthly rates With a rate of 5.75%

semi-annually compounded, the equivalent monthly compounded rate

must solve:

EAR = (1 + im)12 - 1= (1 + 5.75%/2)2 - 1= 5.8327%

im = 0.4735%

The annuity formula for the condo price is:

300,000 = PMT x PVIFA(0.4735%,300) PMT = $1,875.07

(b) We follow the same steps as in (a) with now a mortgage rate of 6.18%

The monthly rate is:

EAR = (1 + im)12 - 1= (1 + 6.18%/2)2 - 1= 6 2755%

im = 0.5085%

The monthly payment is given by:

300,000 = PMT x PVIFA(0.5085%,300) PMT = $1,951.63

The remaining balance after year 5 is given by:

PV = 1,951.63 x PVIFA(0.5085%,240) = $270,188.89

(c) You will have to compute the balance remaining at the end of year 5 for the variable rate mortgage and then compare it to your answer in part (b) Going through similar calculations as you did in parts (a) and (b), your calculations should be:

Year (N) Beginning

Balance

Monthly Payment

“N”

remaining

Ending Balance

1 (300) 300,000.00 5.75% 4735% $1875.07 288 $294,401.81

2 (288) 294,401.81 5.05% 4165% $1756.88 276 $287,884.53

3 (276) 287,884.53 7.00% 5750% $2083.42 264 $282,582.00

4 (264) 282,582.00 7.00% 5750% $2083.42 252 $276,901.80

5 (252) 276,901.80 7.00% 5750% $2083.42 240 $270,817.02

If you compare this remaining variable rate balance at the end of 5 years ($270,817.02), with the fixed rate balance, calculated in part (b)

$270,188.89, you would be better off going with the 6.18% fixed rate mortgage as the remaining loan balance is marginally smaller

(d) To calculate the interest savings by going with a shorter amortization period (20 years instead of 25), you will first need to calculate the new monthly payment, based on the 20 year period: (Recall you are going with the 6.18% fixed rate mortgage)

300,000 = PMT x PVIFA(0.5085%,300) PMT = $2166.97

Total payments over a 20 year period: $2166.97 x 240 months =

$520,072.80

Total interest paid on a 20 year loan = $520,072.80 - $300,000 =

$220,072.80

Now compare this number with total interest paid using a 25 year loan amortization:

From part (b) PMT = $1951.63 x 300 months (25 yrs x 12) = $585,489 Total interest paid on a 25 year loan = $585,489.00 - $300,000 =

$285,489.00

Æ Total interest savings = $285,489 - $220,072.80 = $65,416.20!

Question 5 (Bonds) (16 marks)

Bonds K and L both have a face value of $1,000, pay semi-annual coupons and have 15 years remaining until maturity Their coupon rates are 6% and 8% respectively If the prevailing market rate decreases from 7.5% to 6.5% compounded semiannually, calculate the price change of each bond:

(a) In dollars (4 marks)

(b) As a percentage of the initial price (4 marks)

(c) Are high-coupon or low-coupon bonds more sensitive to a given interest rate change? Justify your response using the results from part (b) (4 marks)

(d) What would be the coupon rate that would lead to the largest price change as a percentage of the initial price? What would be this percentage change? (4 marks)

Solution

(a) The bond K and L prices when i = 7.5% are respectively:

PVK = $30 x PVIFA(3.75%,30) + $1,000 x (1+3.75%)-30 = $866.28

PVL = $40 x PVIFA(3.75%,30) + $1,000 x (1+3.75%)-30 = $1,044.57 The bond K and L prices when i = 6.5% are respectively:

PVK = $30 x PVIFA(3.25%,30) + $1,000 x (1+3.25%)-30 = $952.55

PVL = $40 x PVIFA(3.25%,30) + $1,000 x (1+3.25%)-30 = $1,142.36 Bond K price rises by $86.26 while Bond L price rises by $97.79

(b) Expressed as percentages of the initial prices, the price changes are:

$866.28

$86.26

×100% = 9.9580%

Bond L: $97.79 ×100% = 9.3618%

(c) The results in part (b) demonstrate that the bond having the lower coupon rate (K) undergoes a proportionately larger price change (for a given change in the prevailing market rate) In general, the price of a bond having a low coupon rate is more sensitive to a change in the market rate than the price of a bond having a higher coupon rate

(d) Based on part (c), the lower the coupon, the higher the price change as a percentage So the largest price change would be produced for a zero- coupon bond (coupon rate = 0%), this percentage change is 15.5956%:

For i = 7.5%: PVzc = $1,000 x (1+3.75%)-30 = $331.40

For i = 6.5%: PVzc = $1,000 x (1+3.25%)-30 = $383.09

Percentage change = (383.09 – 331.40)/331.40 x 100% = 15.5956%

The figure below shows the percentage change in the price for different coupon rates between 0% and 20% We can see that the largest percentage change occurs for the bond with 0% coupon rate, which is the zero-coupon bond

6%

8%

10%

12%

14%

16%

18%

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

Coupon Rate

Question 6 (Bonds) (18 marks)

Assume that you are an investor and are in the market for corporate debt You run across XYZ Corporation’s bonds which have a 20 year term, a 7.5% coupon (interest paid semi-annually) and were issued five years ago (Sep 1st 2002) at par of $1,000 Over the last five years, interest rates have fluctuated substantially

and the following table shows the required returns for XYZ bonds for the last 4

years:

Sep 1st., 2003 8.50%

Sep 1st., 2004 6.75%

Sep 1st., 2005 5.95%

Sep 1st., 2006 6.35%

(a) Calculate the rate of return for each of the last four years since the

bonds were issued Assume that the coupons are not reinvested (6 marks)

(b) What relationship do your results in (a) illustrate? (2 marks)

(c) Calculate your current yield and your yield to maturity if you purchase

the bond on September 1st, 2007 at a price of $1,070? (4 marks)

(d) If you purchase the bond on September 1st, 2007 at a price of $1,070,

what are the total rate of return and the effective annual rate of return if

you decide to sell the bond after 3 years at a price of $1,030? Assume

the coupons are reinvested at 5% semi-annually compounded (6 marks)

Solution

(a)

% 8454 1 1000

1000 55 906 75

55 906 0425

1 / 1000 )

38

%, 25 4 ( 50

37 1

38 1

−

=

− +

=

= +

×

=

R

PVIFA P

55 906

55 906 48 1077 75

48 1077 03375

1 / 1000 )

36

%, 375 3 ( 50

37 2

36 2

=

− +

=

= +

×

=

R

PVIFA P

48 1077

48 1077 36

1164 75

36 1164 02975

1 / 1000 )

34

%, 975 2 ( 50

37 3

34 3

=

− +

=

= +

×

=

R

PVIFA P

% 1587 2 36 1164 49

1114 75

49 1114 03175

1 / 1000 )

32

%, 175 3 ( 50

4

=

− +

=

= +

×

=

R

PVIFA P