Question

NOTE that significant marks will be lost if your answer does not include the NUMERICAL FORMULA.

Question 1 (25 marks/ Time Value of Money and WACC)

(a) You need to pay off a car loan within the next two years. The payment will be $4,000 every month. Today you have made a single deposit into a return-guaranteed investment account that will allow you to cope with all the monthly payments. This account earns an effective annual interest rate of 12.68250301%. The first payment will be made in one month.

1. (i) Calculate the corresponding monthly rate for the investment account.

2. (ii) “You need to have at least $96,000 at your account today in order to make all the payments on the car loan in the next two years.” True or false? Briefly explain without doing any time value of money related (i.e. PVA or FVA) calculations.

3. (iii) What is the amount of the single deposit made today?

4. (iv) If your mother is going to make the first year’s repayments for you (as a birthday gift) and thus you don’t need to withdraw the $4,000 every month from the investment account, how much more money will you have in your bank account two years from

now?

Answer 1

(i)

Lets annual nominal interest rate is i.

Compute the annual nominal rate using the equation as shown below:

Effective annual rate = {[ 1 + (i / Number of compounding period) ] ^ Number of compounding period } - 1

12.68250301% = {[ 1 + (i / 12)] ^12} - 1

1.1268250301 = [ 1 + (i / 12)] ^12

1.1268250301 = [( 12 +i )/ 12] ^ 12

Taking log both sides

Log( 1.1268250304) = 12 Log [( 12 +i ) - Log 12]

Log( 1.1268250304) / 12 + Log 12 = Log ( 12 + i)

2.494856981 = Log ( 12 + i)

Taking exponential:

e^2.494856981 = 12 + i

i = e^2.494856981 - 12

= 12%

Hence, the nominal annual interest rate is 12%.

Compute the monthly rate using the equation as shown below:

Monthly rate = Annual nominal rate / 12

= 12% / 12

= 1%

Hence, the monthly rate is 1%.

(ii)

If the deposit of $96,000 is providing $4,000 for 24 months, then this means that the investment is not earning any interest. This is incorrect as the investment Is earning 12.68250301% effective annual interest. So, the actual deposit would be less than $96,000. So, this statement is false.

(iii)

Compute the period using the equation as shown below:

Period = Total years * Number of payment per year

= 2 *12

= 24

Hence, the period is 24.

Compute the PVIFA at 1% and 24 periods, using the equation as shown below:

PVIFA = {1 – (1 + 1%)^-Period}/ 1%

                   = {1 – (1 + 1%)^-24}/ 1%

            = 21.24338726

Hence, the PVIFA at 1% and 24 period is 21.24338726

Compute the amount of single deposit using the equation as shown below:

Deposit = Payment per month * PVIFA Rate, period

= $4,000 * 21.24338726

= $84,973.54904

Hence, the deposit is $84,973.54904.

(iv)

Compute the amount in the bank after year 1 using the equation as shown below:

Amount = Deposit * ( 1 + Monthly interest rate)^Period

= $84,973.54904 * ( 1 + 1% )^24

= $107,893.8594

Hence, the amount in the bank after year 1 is $107,893.8594.

Now, $4,000 would be withdrawn from this account every month.

The present value of the withdrawn should be deducted from the amount in the bank.

Compute the PVIFA at 1% and 12 periods, using the equation as shown below:

PVIFA = {1 – (1 + 1%)^-Period}/ 1%

                   = {1 – (1 + 1%)^-12}/ 1%

            = 11.25507747

Hence, the PVIFA at 1% and 12 period is 11.25507747.

Compute the extra amount in the bank account after 2 years using the equation as shown below:

Extra amount = Amount after one year - ( Withdrawn per month * PVIFA Rate, Period)

= $107,893.8594 - ( $4,000 * PVIFA 1%, 12)

= $107,893.8594 - ($4,000 * 11.25507747)

= $62,873.54952

Hence, the extra amount is $62,873.54952.