Chapter 4 Principles of The Time Value of Money (Part 2)
Finance
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Future Value of Multiple Payment Stream You are an auto dealer you have two choice to sell a car take $14,000 cash now , or receive three payments: $2,000 now and $3,000 at the end of the first year $4,000 at the end of second year and $5,000 at the end of third year . If interest rate of money you can earn is 5%, which deal do you prefer?
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Future Value of Multiple Payment Stream You are an auto dealer you have two choice to sell a car take $14,000 cash now , or receive three payments: $2,000 now and $3,000 at the end of the first year $4,000 at the end of second year and $5,000 at the end of third year . If interest rate of money you can earn is 5%, which deal do you prefer?
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Future Value of Multiple Payment Streams With
unequal periodic cash flows, treat each of the cash flows as a lump sum and calculate its future value over the relevant number of periods. Sum up the individual future values to get the future value of the multiple payment streams.
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Topics Covered Annuity Due Perpetuity Future
and Present Value of Annuity Due
Future
and Present Value of Perpetuity
Applications
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Annuity Question
Answer
What is annuity??
A series of equal cash flows at regular periods.
What is ordinary annuity??
Cash flows at end of periods.
What is annuity due??
Cash flows at start of periods.
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Future Value of Annuity Stream Annuity ??? A series of equal cash flows at regular interval across time is an annuity.
Brain Test Equal Car Installments Electricity bill House rent monthly Monthly Gas bill Ordinary Annuity
Yes No Yes No End of the period CF
Payments or receipt at the end of each period
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Future Value of Annuity Stream Toyota motors has following car sale plan. Receive $5000 at the end of every year for five years, average investment rate in the market is 10%. Calculate the future value of the plan?
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Future Value of Annuity Stream On sale of a laptop Dell have following receipt options i.
Receive cash payment of $3400
ii.
Receive $1000 year-end payments for next 5 years,
Market investment rate is 5% per annum, evaluate which option is better?
FV
$3400 (1 .05)
$4339
5
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Future Value of Annuity Application
The formula for calculating the future value of an ordinary annuity stream is as follows:
FV = CF X (1+r)n - 1 r FVIFA
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Future Value of Annuity Application cont. Jill has been faithfully depositing $2,000 at the end of each year since the past 10 years into an account that pays 8% per year. How much money will she have accumulated in the account?
Use Formula
FV=$2000*[((1.08)^10 - 1)/.08] = $28,973.13
Use Table FV=$2000*14.4866 = $28,973.13
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Ordinary Annuity Vs. Annuity Due ! #$%& '()* %+,-$. %/#& $% ,-0+1 (-$%-1 $02 30%/,$0#- 4$5.-0+%1 *&3#& 306)(6-% -7/$( 4-,3)23# #$%& '()*% +&$+ 8-930 ,39&+ $*$5 ), $+ +&- 8-9300309 )' -$#& +3.- 30+-,6$( 3% :0)*0 $% $0 !""#$%& (#)*
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Future Value of Annuity Due
Add one more period to previous formula
FV Annuity Due > FV Ordinary Annuity
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Future Value of Annuity Due cont. Example – Retirement Plan You plan to deposit your saving $3,000 at the start of every year for 20 years and then retire. Given a 8% rate of interest, what will be the FV of your retirement account? Formula
Table
FV= $3000(45.7620) (1.08) =$148,268.76
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PV of Multiple Cash Flows Example Your auto dealer gives you the choice to pay $15,500 cash now, or make three payments: $8,000 now and $4,000 at the end of the following two years. If your cost of money is 8%, which do you prefer? Immediate pay ment 8,000.00
PV 1
(1.08) 3,703.70
PV 2
(1.08) 3,429.36
4 , 000
1
Total PV
4 , 000
2
$15,133.06
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PV of Multiple Cash Flows PVs
can be added together to evaluate multiple cash flows.
PV
C 1
(1 r )
(1 ) .... C 2
1
r
2
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Present Value of Annuity Stream You are purchasing a car. You are scheduled to make 5 annual installments of $5,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)?
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Present Value of Annuity Stream cont. 1 1 1 r
n
PV
PMT
r
1 1 1 r n
PVIFA
r
Present Value Interest Factor of an Annuity: The present value of $1 a year for each of t years
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Present Value of Annuity Application cont.
1 1 1 r PMT
n
PV
r
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Perpetuities & Annuities A Perpetuity is an equal periodic cash flow stream that will never end. PV of Perpetuity Formula
PV PMT = Cash payment r = interest rate
PMT r
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Perpetuities & Annuities Example - Perpetuity In order to create an endowment fund, which pays $100,000 per year, forever, how much money must be set aside today in the rate of interest is 10%?
PV
100 , 000 .10
$1,000,000
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Perpetuities & Annuities Example - continued If the first perpetuity payment will not be received until three years from today, how much money needs to be set aside today?
PV
1, 000 , 000 (1 .10 )
3
$751,315
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Example: Annuity You are purchasing a car. You are scheduled to make 3 annual installments of $4,000 per year. Given a rate of interest of 10%, what is the price you are paying for the car (i.e. what is the PV)? 1 1 1 r
n
PV
PMT
r
PV
4,000
PV
$9,947.41
1 .10
.10 (1.10 ) 1
3
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Three Loan Payment Methods Loan payments can be structured in one of 3 ways: 1)
Discount loan •
2)
Interest-only loan •
3)
Principal and interest is paid in lump sum at end Periodic interest-only payments, principal due at end.
Amortized loan •
Equal periodic payments of principal and interest
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Three Loan Payment Methods (cont.) Example: Discount versus Interest-only versus Amortized loans Abdullah wants to borrow $40,000 for a period of 5 years. The lenders offers him a choice of three payment structures: 1) Pay all of the interest (10% per year) and principal in one lump sum at the end of 5 years; 2) Pay interest at the rate of 10% per year for 4 years and then a final payment of interest and principal at the end of the 5th year; 3) Pay 5 equal payments at the end of each year inclusive of interest and part of the principal. Under which of the three options will Abdullah pay the least interest and why? Calculate the total amount of the payments and the amount of interest paid under each alternative.
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Three Loan Payment Methods (cont.) Abdullah wants to borrow $40,000 for a period of 5 years. Method 1: Discount Loan. Since all the interest and the principal is paid at the end of 5 years we can use the FV of a lump sum equation to calculate the payment required, i.e. FV = PV x (1 + r)n FV5 = $40,000 x (1+0.10)5 = $40,000 x 1.61051 = $64, 420.40 Interest paid = Total payment - Loan amount Interest paid = $64,420.40 - $40,000 = $24,420.40
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Three Loan Payment Methods (cont.) Abdullah wants to borrow $40,000 for a period of 5 years.
Method 2: Interest-Only Loan. Annual Interest Payment (Years 1-4) = $40,000 x 0.10 = $4,000 Year 5 payment = Annual interest payment + Principal payment = $4,000 + $40,000 = $44,000 Total payment = $16,000 + $44,000 = $60,000 Interest paid = $20,000
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Three Loan Payment Methods (cont.) Abdullah wants to borrow $40,000 for a period of 5 yr. Method 3: Amortized Loan. n = 5; I = 10%; PV=$40,000
1 1 1 r PMT
n
PV
r
PMT = $10,551.86 Total payments = 5*$10,551.8 = $52,759.31 Interest paid = Total Payments - Loan Amount = $52,759.31-$40,000 Interest paid = $12,759.31
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Three Loan Payment Methods (cont.) Loan Type
Total Payment
Interest Paid
Discount Loan
$64,420.40
$24,420.40
Interest-only Loan $60,000.00
$20,000.00
Amortized Loan
$12,759.31
$52,759.31
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Amortization Schedules Example: Loan amortization schedule. Prepare a loan amortization schedule for the amortized loan option given in example above. What is the loan payoff amount at the end of 2 years? PV = $40,000; n=5; i=10%; FV=0; PMT = $10,551.89
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Amortization Schedules (cont.) Year
Beg. Bal
Payment
Interest
Prin. Red
End. Bal
;
<=1===>==
;=1??;>@A
<1===>==
B1??;>@A
CC1<<@>;;
D
CC1<<@>;;
;=1??;>@A
C1C<<>@;
E1D=E>=@
DB1D<;>=C
C
DB1D<;>=C
;=1??;>@A
D1BD<>;=
E1ADE>EA
;@1C;C>D<
<
;@1C;C>D<
;=1??;>@A
;1@C;>CD
@1ED=>?E
A1?AD>BE
?
A1?AD>BE
;=1??;>@A
A?A>DE
A1?AD>BE
=
The loan payoff amount at the end of 2 years is $26,241.03
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Topics Covered
Problems &
Cases
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Problem Q-Sam Hinds, a local dentist, is going to remodel the dental reception area and two new workstations. He has contacted IKEA, and the new equipment and cabinetry will cost $18,000. IKEA will finance the equipment purchase at 7.5% over a sixyear period of time. What will Hinds have to pay in annual payments for this equipment? 1 1 1 r
n
PV
PMT
r
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Problem Q-The
Stack has just written and recorded the single greatest rock song ever made. The boys in the band believe that the royalties from this song will pay the band a handsome $200,000 every year forever. The record studio is also convinced that the song will be a smash hit and that the royalty estimate is accurate. The record studio wants to pay the band upfront and not make any more payments for the song. What should the record company offer the band if they use 5% discount rate, 7.5% discount rate, or 10% discount rate?
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Problem County Ranch Insurance Company wants to offer a guaranteed annuity in units of $500, payable at the end of each year for twenty-five years. The company has a strong investment record and can consistently earn 7% on its investments after taxes. If the company wants to make 1% on this contract, what price should it set on it? Assume that it is an ordinary annuity and that the price is the same as present value. (Use 6% as the discount rate)
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Case 1
Q-What
is the difference between a series of payments and an annuity? What are the two specific characteristics of a series of payments that make them an annuity? Answer-
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Case 2 Q What effect on the future value of an -
annuity does increasing the interest rate have? Does a change from 4% to 6% have the same dollar impact as a change from 6% to 8%?
Answer
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Case 3 Q What
effect on the present value of an annuity does increasing the interest rate have? Does a decrease from 7% to 5% have the same dollar impact as a decrease from 5% to 3%? -
Answer
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Case 4 Q-Is the present value always less than the future value? ?? Answer
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Topics Covered
Problems &
Cases
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MCQ Problem If a perpetuity is worth $1,000 and rate is 15.5%, what is the cash flow?
a. $155 b. $157 c. $150 d. $160
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MCQ Problem The value of a payment, if the payment were made at some point in the future is called the
a. Time value of money b. Principal c. Present value d. Future value
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MCQ Problem An investor deposits £600 in a bank and plans to leave it there for four years. The value of the account after four years if it earns 10 percent interest compounded annually will be £
a. b. c. d. e.
£ 856.78 £ 878.46 £ 915.34 £ 934.23 £ 978.99
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MCQ Problem Which equation represents the general relationship between future values and present values
a. b. d. e.
FV = PV(1+r)^n PV= FV(1+r) ^n PV=FV(1xr) ^n FV=PV(1xr) ^n
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MCQ Problem A (an) ............ is a finite series of equal cash flows made at regular intervals. a. b. c. d.
IRA Annuity Perpetuity Annual regularity
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MCQ Problem An annuity with the first cash flow occurring immediately is called a(n)
a. b. c. d.
First annuity Cash annuity Simple annuity Annuity due
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MCQ Problem If one speaks of an annuity without any qualification, a(n) ……….. is being discussed
a. b. c. d.
First annuity Cash annuity Simple annuity Annuity due
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MCQ Problem Calculate the present value of an annuity of $100 for two periods with a 12% rate of interest
a. b. c.
$ 88.59 $100.45 $123.90
d.
$ 169.05
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MCQ Problem Calculate the present value factor for a five-period annuity with an interest rate of 12% per period
a. 1.2008 b. 2.6554 c. 3.6048 d. 4.7665
