time value of money ahmed hassan

Dr. Kamrul Hasan Dr. Kamrul Hasan

Assistant Professor, AIUBAssistant Professor, AIUB

Time Value of Time Value of MoneyMoney

Boier Desh PublicationsBoier Desh PublicationsFoundation of Financial Management, 1/eFoundation of Financial Management, 1/e

Chapter OutlineChapter OutlineIntroduction

Basic of Time Value of Money

Present value

Future Value

Concept of Interest

Concept of Annuity

Rule of 72

Rule of 69

Perpetuity and Growing perpetuity

EAR and APR

Summary and Conclusion


“ “ Time is more value than money. You can Time is more value than money. You can

get more money but cannot get more get more money but cannot get more

time.” time.”

– – Jim RohnJim Rohn


Time Value of Time Value of Money Money


The basic idea behind the concept of time value of

money is: TK.1 received today is worth more than Tk.1 in the

future OR OR Tk.1 received in the future is worth less than Tk.1

today

Why?Why?

because interest can be earned on the money

The connecting piece or link between present (today)

and future is the interest or discount rate

∆ If you invest your 1 taka for 1 year at a 12% annual

interest rate. (12% interest rate & one-year period)

∆ Then at the end of one year you will accumulate 1.12

taka, you can say that future value of 1 taka is 1.12.

∆ So, present value of the 1.12 taka you expect to

receive in one year is only 1 taka.

∆ A key concept of time value of money is that:

∆ A single sum of money or a series of equal, evenly spaced

payment or receipts promised in the future can be converted to

an equivalent value today.

Time Value of Money Time Value of Money


Time Value of Money Time Value of Money

Time Value of Money (TVM) concept is used to

measure and evaluate many business and financial

transactions, including:

Investment analysis Capital budgeting decisions Stocks, bonds and other securities Long-term Leases Long-term capital assets Accounts receivable Accounts payable Pensions and retirement plans Mergers and acquisitions Assets depreciation Working capital management


Identify the VariablesIdentify the VariablesThere are five variables every time value of money has.

One should identify the variables. Those variable are:

Present Value (PV):

Any value that occurs at the beginning of the problem is a Present Value. It is an amount today that is equivalent to a future payment, or series of payments, that has been discounted by an appropriate interest rate.

Future Value (FV):

Future Value is the amount of money that an investment with 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.


Identify the VariablesIdentify the VariablesInterest Rate (r):

Interest is a charge for borrowing money, usually stated as a percentage of

the amount borrowed over a specific period of time. Interest is two types: (1)

Simple interest & (2) Compound interest.

Annuity Payment (Pmt):

An annuity payment is a series of two or more equal payments that occur at

regular time intervals.

Number of Periods (t):

The number of periods is the total length of time that the investment will

be held. Periods are evenly spaced intervals of time.


Present Value is Related Present Value is Related to…….to…….

PV is positively related to FVPV is inversely related to the interest ratePV is inversely related to the number of periodsPV is positively related to annuity payment


Two Concepts of Interest: Two Concepts of Interest: Simple & Simple & CompoundCompound

Simple interest refers to situation when interest is earned on a principle only.

Compound interest refers the situation the interest is earned on the original principle amount as well as any interest earned that accumulated with the principal.

Compound and simple interest differs on one principle the simple interest won’t earn interest whereas compound interest does earn interest on interest along with principal.

On the other hand, Continuous compound interest counts on interest earned on principal daily and also counts the interest on that interest earned principle compounded daily.

Continuous compounding happens when interest is charged against principle and compound continuously, that is the interest is continuously added to principle to be charged interest again.

Continuous compounding can be used to determine the future value of a current amount when interest is compounded continuously


Concepts of InterestConcepts of InterestInterest rate (12%) Year 1 Year 10 Year 50

Simple 1.12 2.20 7.00

Compound 1.12 3.105 289.00

Continuous compound

1.127 3.319 403.031

Here in the chart you could see the expected return of 1 taka which

earns 12% annually what should be the

expected return on various time line at simple, compound and

continuous compound

interest rate.


Calculating the Future Value Using Simple Calculating the Future Value Using Simple Interest RateInterest Rate

Problem 2.1

If you deposit 10,000 taka in a bank account that earns 12% If you deposit 10,000 taka in a bank account that earns 12%

simplesimple interest for 5 years then how much money would you interest for 5 years then how much money would you

have after that period?have after that period?

The equation for simple interest rate is quite straightforward.

FV using simple interest rate =

Principal + Principal*interest rate*time or P+P*r*tPrincipal + Principal*interest rate*time or P+P*r*t

So, you get 10,000 + 10,000 * 0.12 * 5 = 10,000 + 6,000 =

16,000 tk


Simple formula of Calculating Future ValueSimple formula of Calculating Future Value

Future Value (FV)= Present Value (PV) + [Present Value (PV) * Interest Rate (r)] for 1 year.

However, if the investment is for two years then,

FV= PV + [PV*r] + {[PV + (PV*r)]*r}

Or, FV= PV(1+r+r+1²) , [1²+2.1.r+r² = (1+r)²]

Or, FV = PV (1+r)² for two years.

So the generalized equation for the future value formula will be:

Future Value = Present Value * (1+interest rate)time

Or, FV = PV (1 + r)t

So if we try to solve the problem 2.1 using compound interest rate then the expected return would be:

FV = 10,000 (1+0.12)5 = 17,623.42 taka

You could see the difference of 1623.42 taka due to earning interest on interest. The number looks astronomical once you use a longer time period like the graph has shown.


Calculating Future Value Using Compound Calculating Future Value Using Compound InterestInterest


Monthly CompoundedMonthly Compounded


Continuously CompoundingContinuously Compounding


Continuously CompoundingContinuously Compounding

Formula, Where, Here,

FV = PV(1+r1)(1+r2)(1+r3) FV = Future Value FV = ?

PV = Present Value PV =

100,000

r1 = Year-1 interest rate r1=

0.12

r2= Year-2 interest rate r2=

0.18

r3= Year-3 interest rate r3=

0.08

FV= PV (1+ rFV= PV (1+ r11) (1+ r) (1+ r22) (1+ r) (1+ r33) )

So. FV= 100,000 (1+ 0.12) (1+ 0.18) (1+ 0.08)So. FV= 100,000 (1+ 0.12) (1+ 0.18) (1+ 0.08)

Or, FV= 142,732.8 takaOr, FV= 142,732.8 taka


Calculating Present Value Using Compound Calculating Present Value Using Compound InterestInterest

If, FV = PV (1+r)t then we can rewrite,

PV= FV/(1+r)t

PV= 2000000/(1+0.12)5

PV= 1,134,854 taka

Here, we can see that 1,134,854

taka is equivalent to 20 lac after 5

years if you are earning 12% interest

annually.

The process by which

future cash flows are

adjusted to reflect

these factors is

called “discounting”.

It is used to calculate

PV.Boier Desh PublicationsBoier Desh Publications

Foundation of Financial Management, 1/eFoundation of Financial Management, 1/e

Calculating Present Value Using Compound Calculating Present Value Using Compound InterestInterest


Finding the Interest RateFinding the Interest Rate

Interest rate and the number of periods must always agree

as to the length of a time period.

Example: semi-annually (2), quarterly (4), weekly (52), or

even daily (365) then you need to adjust accordingly.

Interest rate problem could be solved by using basic TVM

formula.

Here, FV = PV (1+r)t

Or, (1+r)t = FV/PV

Or, (1+r)t/t = (FV/PV)(1/t)

So, r = (FV/PV)r = (FV/PV)(1/t) (1/t) -1-1


Finding the Interest RateFinding the Interest Rate

Solution: r = (FV/PV)(1/t) -1Since, you know the interest rate of first bank, all you need to find the rate for the other two banks.

For the 2nd bank that doubles your money in 5 years the interest rate will be:

r = (20000/10000)(1/5)-1 = 1.1486-1 = 14.86%

If you want to triple your money in 8 years the interest rate will be: r = (30000/10000)(1/8)-1 = 1.1472-1 = 14.72%

So, it is better to put money in the second bank, bcz second bank double your money after 5 years & you are earning the highest interest rate of 14.86% among the three banks. Boier Desh PublicationsBoier Desh Publications


Finding the Time PeriodsFinding the Time PeriodsTo find the time periods we use the basic FV formula to derive the equation for t.Here, FV = PV (1+r)t

Or, (1+r)t = FV/PV

ln (1+r)t = ln (FV/PV) t.ln (1+r)t = ln (FV/PV)

So, t = ln (FV/PV)/ln (1+r)

Problem- 2.8Problem- 2.8If you want to Invest 5000 taka today at an annual interest rate of 11%,

how long do you have to wait to receive 20,000 taka?

Solution:Solution:Here, t = ln (FV/PV)/ ln (1+r)Here, t = ln (FV/PV)/ ln (1+r)

Or, t = ln (20,000/5,000)/ ln(1+0.11)Or, t = ln (20,000/5,000)/ ln(1+0.11)Or, t = 1,38629/ 0.10436Or, t = 1,38629/ 0.10436

Or, t = 13.283 years.Or, t = 13.283 years.So, you have to wait 13.283 yearsSo, you have to wait 13.283 years.


The Intuitive Basis for Present Value The Intuitive Basis for Present Value

There are three reason why a cash flow in the future is worth

less than a similar cash flow today. Those reasons are:

1. Individual prefer present consumption to future consumption.

2. Monetary Inflation

3. A promised cash flow might not be delivered for a number of

reasons:

The promised might default on payment

The promisee might not be around to receive payment

Some other contingency might intervene to prevent or reduce the

promised payment

Or any uncertainty (risk) associated with cash flow in the future reduces the

value of the cash flow.


Double Your Money!!!Double Your Money!!!

We will use the “Rule-of-72”We will use the “Rule-of-72”


Years to Double = 72 / iYears to Double = 72 / i72/12% = 6 Years (Approx)72/12% = 6 Years (Approx)

Years to Double = 72 / iYears to Double = 72 / i72/12% = 6 Years (Approx)72/12% = 6 Years (Approx)

Double Your Money!!!Double Your Money!!! A general rule estimating how long it will take for an

investment to double, assuming continuously

compounding interest.

Can calculates this by dividing 69 by the “rate of

return” .

“Rule of 69” is not exact

But provides a quick look at the effects of

compounding on an investment

“Rule of 69” is similar to the “Rule of 72”

But “Rule of 72” is more useful for non-continuously

compounding interestBoier Desh PublicationsBoier Desh Publications


PV dealing with multiple, uneven cash flows

The key to finding the PV of multiple, uneven cash flows are to compound each cash flow separately for the appropriate number of time periods and earned interest rate.

Problem- 2.9Problem- 2.9What is the present value of receiving 50,000 taka in one year, 75,000 What is the present value of receiving 50,000 taka in one year, 75,000 taka in two years, and 10,000 taka in three years if the interest rate is taka in two years, and 10,000 taka in three years if the interest rate is

13.5%?13.5%?

Solution:

PV of 50,000 taka is 50,000/ (1+ 0.135) = 44,052.86 taka

PV of 75,000 taka is 75,000/ (1+ 0.135)²= 58,219.64 taka

PV of 100,000 taka is 100,000/ (1+0.135)³= 68,393.11 taka

So, the PV of the series of cash flows is simply their combined PV:

Total PV = 44,052.86 + 58,219.64 + 63,393.11 = 1,70,665.62 Total PV = 44,052.86 + 58,219.64 + 63,393.11 = 1,70,665.62

taka.taka. Boier Desh PublicationsBoier Desh PublicationsFoundation of Financial Management, 1/eFoundation of Financial Management, 1/e

Types of AnnuitiesTypes of Annuities

An annuity is a series of equal payments (inflows or

outflows) for a certain number of time periods.

There are two types of annuities, those are:

Ordinary Annuity: Payments or receipts occur at the end of

each period, it is also called deferred annuity. Example:

Bond that pays interest on the last day of the month.

Annuity DueAnnuity Due: Payments or receipts occur at the beginning

of each period. Example: A mortgage payment or house

rent which is usually due on the first day of the month.


Parts of an AnnuityParts of an Annuity

0 1 2 3

$1000 $1000 $1000

(Ordinary Annuity)EndEnd ofPeriod 1

EndEnd ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period

Apart

EndEnd ofPeriod 3


PV of an Ordinary AnnuityPV of an Ordinary Annuity

To determine today’s value of a series of future

payments, need to use the formula of present value

of an ordinary annuity.

PV of ordinary annuity calculates the present value of

coupon payments that will receive in the future.

Simple logic of PV of an ordinary annuity is receiving

a series of payment of 1,000 taka at a different time

period is less today than receiving a lump sum cash

flow of 5,000 tk today.


Assume that interest rate is 14% and the PV of an annuity that pays 1,000 tk per year for five year.

Let’s find the value of this ordinary annuity using formula:

PVordinary annuity = Payment*[1-(1+r/n)-t*n ÷ r/n]

PVordinary annuity = 1000[1-(1+0.14)-5 ÷ 0.14]

PVordinary annuity = 3433.08 taka

PV of an Ordinary AnnuityPV of an Ordinary Annuity

Here,

PV ordinary annuity of Year – 1: 1000(1+0.14)-1 = 877.19PV ordinary annuity of Year – 2: 1000(1+0.14)-2 = 769.46PV ordinary annuity of Year – 3: 1000(1+0.14)-3 = 674.97PV ordinary annuity of Year – 4: 1000(1+0.14)-4 = 592.08PV ordinary annuity of Year – 5: 1000(1+0.14)-5 = 519.36

3433.08 Boier Desh PublicationsBoier Desh Publications


Parts of an AnnuityParts of an Annuity

0 1 2 3

$1000 $1000 $1000

(Annuity Due)BeginningBeginning of

Period 1

BeginningBeginning ofPeriod 2

Today EqualEqual Cash Flows Each 1 Period Apart

BeginningBeginning ofPeriod 3


PV of an Annuity DuePV of an Annuity Due

Assume that interest rate is 14% and the PV of an annuity that pays 1,000 tk per year for five year.

Formula.

PV annuity due = Payment*[1-(1+r/n)-t*n * (1+r/n) ÷r/n]

PV annuity due = 1000[1-(1+0.14)-5 * (1+0.14)÷ 0.14] = 3913.71 Taka

Here, PV annuity due of Year – 0: 1000(1+0.14)-o = 1,000

PV annuity due of Year – 1: 1000(1+0.14)-1 = 877.19PV annuity due of Year – 2: 1000(1+0.14)-2 = 769.46PV annuity due of Year – 3: 1000(1+0.14)-3 = 674.97PV annuity due of Year – 4: 1000(1+0.14)-4 = 592.08

3913.71


FV of an Ordinary AnnuityFV of an Ordinary Annuity FV of an ordinary annuity measures how much you would have in the future given a specified rate of return.

Assume that interest rate is 14% and the FV of an annuity that pays 1,000 tk per year for five year.

Formula, FV ordinary annuity = Pmt*{(1+r/n)t*n-1} ÷r/n]

FV ordinary annuity = 1000*{(1+0.14)4-1} ÷0.14] = 6610.10 taka.

Here,

FV ordinary annuity of Year – 0: 1000(1+0.14)0 = 1,000.00FV ordinary annuity of Year – 2: 1000(1+0.14)1= 1140.00FV ordinary annuity of Year – 3: 1000(1+0.14)2 = 1299.60FV ordinary annuity of Year – 4: 1000(1+0.14)3 = 1481.54FV ordinary annuity of Year – 5: 1000(1+0.14)4 = 1688.96

6610.10 Boier Desh PublicationsBoier Desh Publications


FV of an Annuity DueFV of an Annuity Due

Assume that interest rate is 14% and the FV of an annuity that pays 1,000 tk per year for five year.

Formula.

FV annuity due = Pmt*{(1+r/n)t*n-1}*(1+r/n)} ÷ r/nFV annuity due = 1000*{(1+0.14)5-1}*(1+0.14)} ÷ 0.14 = 7,535 taka

Here, FV annuity due of Year – 1: 1000(1+0.14)1 = 1,140

FV annuity due of Year – 2: 1000(1+0.14)2 = 1299 FV annuity due of Year – 3: 1000(1+0.14)3 = 1481.54 FV annuity due of Year – 4: 1000(1+0.14)4 = 1688.96 FV annuity due of Year – 5: 1000(1+0.14)5 = 1925.41

7535 taka


PV of a Growing AnnuityPV of a Growing Annuity

The PV of growing annuity formula calculates the present

day value of a series of future periodic payments that grow at

a proportionate rate.

It may sometime referred to as an increasing annuity.

Example of an growing annuity would be an individual who

receive Tk. 100 the first year and successive payments

increase by 10% per year for a total three years.

[ Tk.100>Tk.110>Tk.121]

Formula of PV of a growing annuity = P÷(r-g)*[1-{(1+g) ÷

(1+r)}t ]

Here, P = first payment, r = interest payment, g = growth

rate, and t = number of year.Boier Desh PublicationsBoier Desh Publications


PV of a Growing AnnuityPV of a Growing AnnuityProblem -2.16

Suppose you have just won the first prize in a lottery. The lottery offers you two possibilities for receiving your prize. The first

possibility is to receive a payment of 10,000 taka at the end of the year, and then, for the next 15 years this payment will be repeated, but it will grow at a rate of 5%. The interest rate is 12% during the entire period. The second possibility is to receive 100,000 tk right

now. Which one out of the two possibilities would you take?

Here,P = 10,000; r = 0.12; g = 0.05; t = 16PV of a growing annuity = P÷(r-g)*[1-{(1+g) ÷ (1+r)}t ] = 10,000÷(0.12-0.05)*[1-{(1+0.05) ÷ (1+0.12)}16 ] = Tk. 91,989.41Tk. 91,989.41 < Tk. 100,000 therefore, you would prefer to be paid out right now.


FV of Growing AnnuityFV of Growing Annuity

The formula of future value of growing annuity is used to calculate the future amount of a series of cash flows, or payments, that grow at a proportionate rate. Fv of a growing annuity = P*[{(1+r)t – (1+g)t} ÷ (r-g)]

Problem – 2.17Problem – 2.17If an employee saves 20000 tk per year which increases by 5% every If an employee saves 20000 tk per year which increases by 5% every year and earns 12% annual interest year and earns 12% annual interest then after 10 years , how much after 10 years , how much

would she have? would she have?

FV of this growing annuity =

20000*[{(1+0.12)10 – (1+0.05)10} ÷ (0.12-0.05)]

= 431,295.35 taka.


PerpetuitiesPerpetuities

Perpetuity is a type of annuity that receives an infinite amount of periodic

payment.

Perpetuity is a constant cash flow at a regular intervals forever.

Formula of PV of Perpetuity = c/r

Here, c = coupon payment & r = interest rate.

Assume that you have a 12% coupon console bond. The value

of this bond, if the interest rate is 9% is as follows:

Value of Console Bond = 120/0.09 = 1,333.33 taka.

The value of a console will be equal to its face value only if the coupon rate is equal to the interest rate.


Growing PerpetuityGrowing Perpetuity A growing perpetuity is the same as a regular perpetuity (C/r), but just like

the cash flow is growing ( or declining) each year.

PV of growing perpetuity = c/(r-g)

Where, C= Initial cash flow or coupon payment, r= Interest rate, g= Growth

rate. Problem 2.18When would you be willing to pay for financial instrument, which promises you to pay cash payment of 25,000 tk at the end of the each year, which

will increase every year by 5%, forever. The annual interest rate is fixed at 12%

What would you be willing to pay if the financial instrument paid out its first 25,000 tk right now, and everything else being same?


Solution PV of Growing Annuity = 25,000 / (0.12 – 0.05) = 357,142.85 taka.PV = [(25000 * 1.05) / (0.12 – 0.05)] + 25,000 = 400,000 taka.

EAR & APREAR & APR

Effective Annual Interest RateEffective Annual Interest Rate

The actual rate of interest earned (paid) after adjusting the nominal rate

for factors such as the number of compounding periods per year.

EAR (%) = (1+rnomibal /n)n – 1

Here, n is number of compounding in a year.

EAR for continuous compounding = er - 1

Annual Percentage RateAnnual Percentage Rate

APR is defined as the period rate × the number of periods per year.

Periodic Rate = rnominal / n

rnominal = Periodic Rate × n = APR.


See Problem – 2.19, Page - 45

Summary and ConclusionsSummary and Conclusions

The financial manager uses the time value of money approach to value cash flows that occur at different points in time

A dollar invested today at compound interest will grow a larger value in future. That future value, discounted at compound interest, is equated to a present value today

Cash values may be single amounts, or a series of equal amounts (annuity)


time value of money ahmed hassan

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