time value of money,Simple interest and compound interest

This lecture about engineering economics.

1. The Time Value of Money
A fundamental idea in finance
that money that one has now is
worth more than money one will
receive in the future.
Because money can earn interest
or be invested, (earning
power) it is worth more to an
economic actor if it is available
immediately.
Money has a time value because
its purchasing power
changes over time (inflation).
For example, 100 dollars of
today's money invested for one
year and earning 5 percent
interest will be worth 105 dollars
after one year.

2. The Concepts Interest
Interest is the cost of money—a cost to the borrower and an
earning to the lender
Simple Interest Rate:The total interest earned or charged
is linearly proportional to the initial amount of the loan
(principal), the interest rate and the number of interest
periods for which the principal is committed.
I = ( P ) ( i) ( N )
P = principal amount
N = number of interest
i= interest rate

3. The Concepts Interest
Simple Interest
P= $1,000, i= 10%, N= 3 years
F= $1,000 + (0.10)($1,000)3
= $1,300
End of
Year
Beginning
Balance
Interest
earned
Ending
Balance
0 $1,000
1 $1,000$100$1,100
2 $1,100$100$1,200
3 $1,200$100$1,300()
where
= Principal amount
= simple interest rate
= number of interest periods
= total amount accumulated at the end of period
F P iP N
P
i
N
FN


4. The Concepts Interest
Compound Interest
Rate:Wheneverthe
interestchargeforany
interestperiodisbased
on theremaining
principalamountplus
anyaccumulatedinterest
chargesuptothe
beginningofthatperiod.
Compoundinterestis
alwaysassumedinTVM
problems.
P= $1,000, i= 10%, N= 3
years
F= $1,000(1 + 0.10)
3
= $1,331
End
of
Year
Beginning
Balance
Interest
earned
Ending
Balance
0 $1,000
1 $1,000$100$1,100
2 $1,100$110$1,210
3 $1,210$121$1,331

5. The Time Value of Money
If a investor invests a sum of RS. 100 in a fixed deposit
with interest rate 15% compounded annually. The
accumulated amount at the end of the year is
Year end interestCompounded amount
0 100.00
1 15.00 115.00
2 17.25 132.25
3 19.84 152.09
4 22.81 174.90
5 26.24 201.14

6. The Time Value of Money
Alternatively If we want RS. 100 at the end of 5 years what is
the amount that we deposit now at a given interest rate say 15%
Year endPresent worthCompounded amount
0 100
1 86.96 100
2 75.61 100
3 65.75 100
4 57.18 100
5 49.72 100

7. Future Value
Future Valueis the amount of money that an investment with a
fixed, compounded interest rate will grow to by some future date.
The investment can be a single sum deposited at the beginning of
the first period, a series of equally-spaced payments (an annuity),
or both.Since money has time value, we naturally expect the
future value to be greater than the present value.
The difference between the two depends on the number of
compounding periods involved and the going interest rate. (1 )
N
F P i
“The greatest
mathematical
discovery of all
time,”
Albert Einstein

8. What is the future value of $34 in 5 years if the interest rate is
5%? (i=.05)
FV= PV ( 1 + r )
t
FV= 34 ( 1+ .05 ) 5
FV= 34 (1.2762815)
FV= 43.39.
Determine Future Value Compounded Monthly
What is the future value of $34 in 5 years if the interest rate is
5%? (iequals .05 divided by 12, because there are 12 months
per year. So 0.05/12=.004166, so i=.004166)
FV= PV ( 1 + i)
N
FV= 34 ( 1+ .004166 )
60
= 34 (1.283307) = 43.63.
•The interest rate and payment periods must have the same time unit for
the factors to correctly account for the time value of money.
Future Value

9. Present Value
Present Valueis an amount today that is equivalent to a
future payment, or series of payments, that has been
discounted by an appropriate interest rate.
The future amount can be a single sum that will be
received at the end of the last period, as a series of equally-
spaced payments (an annuity), or both.
Since money has time value, the present value of a
promised future amount is worth less the longer you have
to wait to receive it.
The time value of money principle says that future dollars
are not worth as much as dollars today.

10. PV = FV/(1 + r)
t
FV = Future Value
PV = Present Value
r = the interest rate per period
t= the number of compounding periods
What is the present value of $8,000 to be paid at the end
of three years if the correct (risk adjusted interest rate) is
11%?
PV = 8,000/(1.11)
3
= 8,000/1.36
= 5,849
Present Value

11. Equivalence from
Personal Financing Point
of View
Alternate Way of Defining
Equivalence
IfyoudepositPdollarstoday
forNperiodsati,youwill
haveFdollarsattheendof
periodN.
FdollarsattheendofperiodNis
equaltoasinglesumPdollars
now,ifyourearningpoweris
measuredintermsofinterestratei.N
F
P
0N
iPF )1( N
F
P
0(1 )
N
P F i


N0
Equivalence

12. Solving for i
Given: F= $20, P= $10,
N= 5 years
Find: i
Solving for N
Given: P= $6,000, F= $12,000,
andi= 20%
Find: N2 (1 0.20)
2 1.2
log2 log1.2
log2
log1.2
3.80 years
N
N
F P P
N
N
  




Types of Common Cash Flows in
Engineering Economics

13. Rule of 72
Approximating how long it will
take for a sum of money to
double
To get a feel for Compounding,
try the rule of 72. What’s that?
If you divide a particular annual
return into 72, you ’llfind out
how many years it will take to
double your money. Thus, at 10%
a year, an investment will double
in value in a tad over seven years72
interest rate (%)
72
20
3.6 years
N



14. Equal Payment series compounded amount
If we want to find the future worth of n equal payments
which are made at the end of year of the nth period at an
interest rate of i compounded annually.
FV= A
(1 +i)
n
–1
i
A person who is now 35 years old is planning for his
retired life. He plans to invest an equal sum of Rs. 10,000
at the end of every year for next 25 years. The bank gives
20% interest rate compounded annually. Find the maturity
value of his account when he is 60 years old.

15. A company has to replace the present facility after 15
years of Rs. 500,000. it plans to deposit an equal amount
at the end of every year for 15 years at an interest rate of
18% compounded annually. Find the equivalent amount
that must be deposited at the end of every year for the
next 15 years.
If we want to find the equivalent amount A that should be
deposited at the end of year of the nth period to realize a
future sum F at an interest rate of i compounded annually.
A = FA
(1 +i)
n
–1
i
Equal Payment series Sinking Fund

16. Equal Payment Series Present Worth Amount
If we want to find the present worth of an equal payment
made at the end of year of the nth period at an interest rate
of i compounded annually.

 P = A
(1 +i)
n
–1
i (1 + i)
n
A company wants to set up a reserve to an annual
equivalent amount of Rs. 10,000 for next 20 years. The
reserve is assumed to grow at the rate of 15% annually.
Find the single payment that must be paid now as the
reserve amount.

17. If we want to find the annual equivalent amount A which
is to be recovered at the end of year of the nth period for a
loan P which is sanctioned now at an interest rate of i
compounded annually.
A = P
(1 +i)
n
–1
i(1 + i)
n
Equal Payment SeriesCapital Recovery Amount
A Bank give a loan to a company worth Rs. 10,00,000 at
an interest rate of 18% compounded annually. This
amount should be repaid in 15 yearly equal installments.
Find the installment amount that company has to pay to
the bank.

18. Uniform Gradient Series Annual Equivalent Amount
The objective of this mode of investment is to find the
annual equivalent amount of a series with an amount of
A1 at the end of 1
st
year and with an equal increment (G)
at the end of each of the following years with an interest
rate i compounded annually.
A = A1 + G
(1 + i)
n
-in -1
i(1 +i)
n
–i
A person is planning for his retired life. He has 10 more years
of service. He would like to deposit Rs. 4,000 at the end of 1
st
year and thereafter he wishes to deposit the amount with an
annual increase of Rs. 500 for next 9 years. Find the total
amount at the end of 10
th
year with an interest rate of 15%.

19. Effective interest Rate
Let i be the nominal interest rate compounded annually.
But in practice compounding may occur less than a year
like monthly, quarterly or semi-annually. The formula to
compute effective interest rate is
Effective interest rate, R = (1 + i/C)
C
-1
A person invests Rs. 5,000 in a bank at a nominal interest
rate of 12% for 10 years. The compounding is quarterly.
Find the maturity amount of the deposit after 10 years.
Effective interest rate, R = (1 + i/C)
C
-1
F = P(1 + R)
n

20. 1.YourrichGrandfatherhasofferedyouachoiceofoneofthe
threefollowingalternatives:$10,000.00now;$2,000ayear
foreightyears;or$24,000attheendofeightyears,assuming
youcouldearn11percentannually,whichalternativeshould
youchoose
2.In2010,theNationalHighwayTrafficsafetyAdministration
raisedtheaveragefuelefficiencystandardto35.5milesper
gallonforcarsandlighttrucksbytheyear2016.Theruleswill
costconsumersanaverageof$926extrapervehicleinthe
2016modelyear.AssumeAliwillpurchaseanewcarin2016
andplanstokeepitfor5years.Howmuchwillthemonthly
savingsinthecostofgasolinehavetobetorecoverAli’sextra
cost?Useaninterestrateof0.75%permonth
Economic Equivalence

21. Mr.Usmanwishestosavemoneytoprovideforhisretirement.
Beginningoneyearfromnow,hewillbegindepositingthesame
fixedamounteachyearforthenext30yearsintoaretirement
savingsaccount.Startingoneyearaftermakinghisfinaldeposit,
hewillwithdraw$100,000annuallyforeachofthefollowing25
years(i.e.hewillmake25withdrawalsinall).Assumethatthe
retirementfundearns12%annuallyoverboththeperiodthathe
isdepositingmoneyandtheperiodhemakeswithdrawals.In
orderforUsmantohavesufficientfundsinhisaccounttofund
hisretirement,howmuchshouldhedepositannually.
Economic Equivalence

22. RonJamison,a20-year-oldcollegestudent,consumes
aboutacartonofcigarettesaweek.Hewondershowmuch
moneyhecouldaccumulatebyage65ifhequitsmoking
nowandputhiscigarettemoneyintoasavingsaccount.
Cigarettescost$35percarton.Ronexpectsthatasavings
accountwouldearn5%interest,compounded
semiannually.ComputeRon'sfutureworthatage65.
Economic Equivalence