Time value of money

time value of money
,
concept of time value of money
,
significance of time value of money
,
present value vs future value
,
solve for the present value
,
simple vs compound interest rate
,
nominal vs effective annual interest rates
,
future value of a lump sum
,
solve for the future value
,
present value of a lump sum
,
types of annuity
,
future value of an annuity

1. Time Value of Money
“Thevalueofmoneyvariesintermsof
time.”
-Prof.Dr.Md.JahirulHoque

2. Why you need to study?
Accounting: To understand time-value-of-money (T-V-M) calculations
in order to account for certain transactions such as loan amortization,
lease payments, and bond interest rates.
Information systems: To design systems that optimize the firm’s cash
flows.
Management: To make plan of cash collections and disbursements in a
way that will enable the firm to get the greatest value from its money.
Marketing: To ensure funding for new programs and products must be
justified financially using time-value-of-money techniques.
Operations: To identify the optimum ways of investments in new
equipment, in inventory, and in production quantities will be affected by
TVM techniques.

3. Basic Chapter Contents…..
Concept of Time Value of Money,
Significance of Time value of money.
Present Value, Future Value.
Concept & Types of Annuity,
Present Value of an Annuity.
Future Value of an Annuity, Perpetual annuity.
Loan Amortization & Sinking Fund.
Problems and Solutions

4. 9/29/20194
Concept of Time Value of Money
Theideathattoday’saspecificsumofmoneyis
worthmorethanthesameamountinthefuture
becausetimeallowsustheopportunitytopostpone
consumptionandearnreturn.

5. Significance of Time value of money
This chapter introduces the topic of financial
mathematics also known as the time value of
money.
To avoid Inflationary effect in assets
To secure the assets’ return both in short and long-run
by creating proper working capital Management and
capital budgeting decisions
To calculate the cost of capital when a firm is going to
raise capital.
To determine pricing a bond issuance
To find out whether lease financing is applicable or
not.

6. 9/29/20196
Simple vs Compound Interest Rate
SimpleInterestRate:Interestisapplicableonlyonthe
principalamount.
CompoundInterestRate:Interestisapplicableonboth
theprincipalamountandcumulativeinterestearned.

7. 9/29/20197
Nominal vs Effective Annual Interest Rates
i.NominalInterestRate:Thecontractualinterestrate
forayearwhichisnotadjustedforfrequencyof
compounding.
ii.EffectiveAnnualInterestRate:Therateofinterestfor
ayearwhichisadjustedforfrequencyof
compounding.

8. 9/29/20198
Present Value vs Future Value
PresentValue:Itisthecurrentvalueofafuture
amountofmoney,oraseriesofpayments,evaluated
atagiveninterestrate.
FutureValue:Itisthevalueatsomefuturetimeofa
presentamountofmoney,oraseriesofpayments,
evaluatedatagiveninterestrate.

9. Future Value of a Lump Sum
The future value in 2 years of $1,000 earning
5% annually is an example of computing the
future value of a lump sum. We can compute
this in any one of three ways:
Using a calculator programmed for financial math
Solve the mathematical equation
Using financial math tables

10. Solve for the Future Value
The general equation for future value is:
FV
n= PV x (1+i)
n
Computing the future value in the
example:
FV
2= $1,000 x (1+5%)
2
= $1,102.50

11. Present Value of a Lump Sum
How much do you need to invest today so
you can make a single payment of $30,000
in 18 years if the interest rate is 8%? This is
an example of the present value of a lump
sum.
Again we can solve it using a programmed
calculator, solving the math
Md. Azizur Rahman

12. The general equation for present value is:
Computing the present value in the example:
Solve for the Present Value
n
n
i1
FV
PV

  
47.507,7$
8%1
$30,000
PV
18




14. Annuities
•Two or more periodic payments
•All payments are equal in size.
•Periods between each payment are
equal in length.
Md. Azizur Rahman

15. 9/29/201915
Types of Annuity
i.OrdinaryAnnuity:Paymentsorreceiptsoccuratthe
endofeachperiod.
ii.AnnuityDue:Paymentsorreceiptsoccuratthe
beginningofeachperiod.
iii.PerpetualAnnuity:Itisexpectedtobecontinued
forever.

16. Future Value of an Annuity
Supposeyouplantodeposit$1,000
annuallyintoanaccountattheendofeach
ofthenext5years.Iftheaccountpays
12%annually,whatisthevalueofthe
accountattheendof5years?Thisisa
futurevalueofanannuityexample.
Wecansolvethisproblemusinga
programmedcalculator,solvingthemath,
orusingTable5.3.

17. Solve for the Future Value of an Annuity
The general equation for a FV of an annuity is:
The FV of the annuity in the example is:





 

i
1i1
x PMTFVA
n
n  
85.352,6$
12%
112%1
x 000,1$FVA
5
5





 


18. Present Value of an Annuity
Youplantowithdraw$1,000annuallyfrom
anaccountattheendofeachofthenext5
years.Iftheaccountpays12%annually,
whatmustyoudepositintheaccounttoday?
Thisisanexampleofapresentvalueofan
annuity.
Wecansolvethisproblemusinga
programmedcalculator,solvingthemath,or
usingTable5.4.

19. Solve for the Present Value of an Annuity
The general equation for PV of an annuity is:
The PV of the annuity in the example is:














i
i1
1
-1
x PMTPVA
n
n  
78.604,3$
12%
12%1
1
-1
x 000,1$PVA
5
5 















21. Perpetuity—An Infinite Annuity
Aperpetuityisessentiallyaninfiniteannuity.
Anexampleisaninvestmentwhichcostsyou
$1,000todayandpromisestoreturntoyou$100
attheendofeachforever!
Whatisyourrateofreturnortheinterestrate?%10
$1,000
$100

PV
PMT
i 

22. The Present Value of a Perpetuity
Anotherinvestmentpays$90attheendof
eachyearforever.If10%istherelevant
interestrate,whatisthevalueofthis
investmenttoyoutoday?Weneedtosolve
forthepresentvalueoftheperpetuity.900$
10%
$90
i
PMT
PV 

23. Compounding Periods Other Than Annual
Future value of a lump sum.
–i
nom= nominal annual interest rate
–m = number of compounding periods per year
–n = number of yearsn x m
nom
n
m
i
1 x PVFV 







24. Compounding Periods Other Than Annual
A $1,000 investment earns 6% annually
compounded monthly for 2 years.2 x 12
2
12
6%
1 x 000,1$FV 





   16.127,1$0.5% 1 x 000,1$FV
24
2 

25. Compounding Periods Other Than Annual
PV of a lump sum uses a similar adjustment to the basic
equation for non-annual compounding.
–i
nom= nominal annual interest rate
–m = number of compounding periods per year
–n = number of yearsn x m
nom
n
m
i
1
FV
PV









26. Effective Annual Rate
Aneffectiveannualrateisanannual
compoundingrate.Whencompoundingperiods
arenotannual,theratecanstillbeexpressedas
aneffectiveannualrateusingthefollowing:
–i
nom= nominal annual rate
–m = number of compounding periods in 1 year1
m
i
1 Rate Annual Effective
m
nom








27. Effective Annual Rate
A bank offers a certificate of deposit rate of 6%
annually compounded monthly. What is the
equivalent effective annual rate?  6.17% 1 - 0.5%11
12
6%
1
12
12








28. SINKING FUND
Vs
AMORTIZATION
SINKING FUND: With the sinking fund we begin with a fund of
zero taka and make periodic deposits into the fund which, along
with the interest earned on these deposits, accumulate to the
total amount of a savings goal.
Where, S= Amount (Future value of annuity) Sinking fund after
n payment.







1)1(
x SR
n
i
i

29. SINKING FUND
Vs
AMORTIZATION
AMORTIZATION: With the amortization of a debt, we begin with a
debt balance of ‘X’ taka and make periodic payments toward the
debt and the interest on the unpaid balance, eventually reducing
the debt balance to zero taka.
Where, A= Amount of Debt (Present value of annuity)













n-
i1
1
-1
A x R
i

30. 1.Calculate the payment per period.
2.Determine the interestin Period t.
(Loan Balance at t-1) x (i% / m)
3.Computeprincipal payment in Period t.
(Payment -Interestfrom Step 2)
4.Determine ending balance in Period t.
(Balance-principal payment from Step 3)
5.Start again at Step 2 and repeat.
Steps to Amortizing a Loan

31. Julie Miller is borrowing $10,000 at a compound
annual interest rate of 12%. Amortize the loan
if annual payments are made for 5 years.
Step 1:Payment
PV
0 = R (PVIFA
i%,n)
$10,000 = R (PVIFA
12%,5)
$10,000 = R (3.605)
R= $10,000/ 3.605 = $2,774
Amortizing a Loan Example

32. Amortizing a Loan ExampleEnd of
Year
PaymentInterestPrincipalEnding
Balance
0 --- --- --- $10,000
1 $2,774$1,200$1,574 8,426
2 2,774 1,011 1,763 6,663
3 2,774 800 1,974 4,689
4 2,774 563 2,211 2,478
5 2,775 297 2,478 0
$13,871$3,871$10,000
[Last Payment Slightly Higher Due to Rounding]

33. Using the Amortization
Functions of the Calculator
Press:
2
nd
Amort
2ENTER
2 ENTER
Results:
BAL = 6,662.91* ↓
PRN =-1,763.99* ↓
INT = -1,011.11* ↓
Year 2 information only
*Note: Compare to 3-82

34. Using the Amortization
Functions of the Calculator
Press:
2
nd
Amort
1ENTER
5 ENTER
Results:
BAL = 0.00 ↓
PRN =-10,000.00 ↓
INT = -3,870.49 ↓
Entire 5 Years of loan information
(see the total line of 3-82)

35. Usefulness of Amortization
2.Calculate Debt Outstanding --The
quantity of outstanding debt may be used
in financing the day-to-day activities of the
firm.
1.Determine Interest Expense --Interest
expenses may reduce taxable income of
the firm.

36. Reference:
Gitman, L.J. (2007) Principles of Managerial Finance (Twelfth Edition).
Boston, MA: Pearson Education, Inc.
Besley, S., & Brigham, E. F. (2008). Essentials of managerial finance.
Thomson South-Western.
Brigham, E. F., & Houston, J. F. (2012). Fundamentals of financial
management. Cengage Learning.
Hoque, Md. Jahirul (2007), Fundamentals of Managerial Finance,
Open University.