Lecture 4(Ch4)-NCBA&E

8/2/2019 Lecture 4(Ch4)-NCBA&E

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Financial Management-2

Spring Semester 2012NCBA&E

Instructor - Jamal Nasir Khan


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Learning Objectives What is FV & Compounding?

Simple Interest

Compound Interest

What is PV & Discounting?

How to find the Return on Investment?


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What is FV & Compounding?

Which would you choose?

If Govt guarantees the following

Real Estate land which triples in 10 years?

Real Estate land which pays a compoundedrate of 15% pa at end of 10 years?


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What is FV?

Financial Managers must be able to answer

How to determine the value today of cash flows

expected in the future?


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What is FV?

Time Value of Money

Basic Concept

Dollar Today is worth more than a Dollar in future

Why?


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Basic Concept

Dollar Today is worth more than a Dollar in future

Why?

Time & Interest Rate

Time = opportunity Interest Rate = opportunity cost

What is FV?Time Value of Money


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Time?TIME allows you the opportunity to:

a) Postpone Consumptionb) Earn INTEREST.


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Interest?

What is Interest?:


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Interest is:

PRICE OF MONEY OTHERS ARE WILLING TO PAY

Interest?


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2 Types of Interest

Simple Interest

Compound Interest


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2 Types of Interest

Simple Interest

Interest paid on only the original amount,

OR principal amount

Compound Interest

Interest paid on principal AS WELL AS any earned interest.


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What is FV?

What is Future Value?


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What is FV?

What is Future Value?

Cash Value of an investment at some point in the

future.

The amount an investment is worth after 1 or moreperiods.

The amount of money an investment will grow to oversome period of time at some interest rate.


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What is FV?Example of FV?

Timeline

T=o 1 2 3 4 5

(100) 20 20 20 20 120

FV FV FVFVPV


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What is FV?

What is Simple Interest?

Interest Earned only on the original principal amount

invested.


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Simple Interest: Future Value of $100 at 10 Percent

Year Beginning Amount Interest Earned End Amount(FV)

1 $100.00 $10.00 $110.00

2 100.00 10.00 120.00

3 100.00 10.00 130.00

4 100.00 10.00 140.00

5 100.00 10.00 150.00

Total interest 50.00


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Simple Interest Formula

Formula FV = PV ( 1 + r.n)

FV: Simple Interest FV

PV: Deposit today (t=0)

r: Interest Rate per Period

n: Number of Time Periods

Assuming same interest rate over all periods


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What is FV?

What is Compound Interest?


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Process of accumulating INTEREST in an investment

over time to earn more interest.

Also Interest on Interest: Interest earned on the re-investment of previous interest payments.

What is FV?What is Compound Interest?


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Compound Interest: Future Value of $100 at 10Percent Compounded

Year Beginning Amount Interest Earned End Amount (FV)

1 $100.00 $10.00 $110.00

2 110.00 11.00 121.00

3 121.00 12.10 133.104 133.10 13.31 146.41

5 146.41 14.64 161.05

Total interest $61.05


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Deriving the formula: Compounding: Future Value for a Lump Sum

Notice that 1. $110 = $100 (1 + .10)

2. $121 = $110 (1 + .10) = $100 1.10 1.10 =$100 1.102

3. $133.10 = $121 (1 + .10) = $100 1.10 1.101.10

= $100 ________

DERIVING THE FORMULA FOR COMPOUNDING


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COMPOUNDING: Future Value for a Lump Sum Notice that

1. $110 = $100 (1 + .10)

2. $121 = $110 (1 + .10) = $100 1.10 1.10 =$100 1.102

3. $133.10 = $121 (1 + .10) = $100 1.10 1.10

1.10= $100 (1.10)3

In general, the future value, FVn, ofPVinvested today at r% forn periods is

FVn = PV x (1 + r)n

The expression (1 + r)n is thefuture value interest factor.


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What is FV?

The equation of FV Compounded at r

FVn= PV. (1 + r)n


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What is FV?

Applying to a problem set:

Simple Interest Practice


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SIMPLE INTEREST BASIC PROBLEM

Assume you are offered a return of13% Simple Interest on adeposit, but you must keep the money in the investment for aperiod of 3 years.

What is the future value after 3 yrs if you invest Rs. 42,000?


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SIMPLE INTEREST BASIC PROBLEMAssume you are offered a return of13% Simple Interest on a deposit, butyou must keep the money in the investment for a period of 3 years. What

is the future value after 3 yrs if you invest Rs. 42,000?Step 1:Setting up the problem:

PV= 42,000N=3R = 13%

Step 2:Using formula: FV = PV ( 1+ r.n)= 42,000 (1+ .13*3)= 42,000 ( 1.39)= 58,380


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COMPOUND INTEREST BASIC PROBLEM # 1

Suppose you locate a two year investment paying 14% per year.

If you invest $325,

a) how much will you have at the end of the 2 yrs?b) How much is compound interest?


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Compound INTEREST BASIC PROBLEMSuppose you locate a two year investment paying 14% per year. If you invest $325,

a) how much will you have at the end of the 2 yrs?b) How much is compound interest?

Step 1

PV = 325R= 14%N = 2 yrs

Step 2Use formula FV = PV(1+i)^n

= 325 (1+.14)^2=325 (1.14)^2

= 325 (1.29)=422.37Break down: 1st year interest is 325*(1.14)=370.5

2nd year interest is 370.5*(1.14)=422.37


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Youve located an investment which pays 12%. The rate sounds goodto you, so you invest $400.

a) How much will you have in three years?

b) How much in seven years?

c) At the end of seven years, how much interest have you earned?


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a) How much will you have in three years?

Step 1

PV = 400

R= 12%

N = 3 yrsStep 2

Use formula FV = PV(1+i)^n

= 400 (1+.12)^3

= 400 (1.12)^3

= 400 (1.404) = 561.97


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b) How much in seven years?

Step 1

PV = 400

R= 12%N = 7 yrs

Step 2

Use formula FV = PV(1+i)^n

= 400 (1+.12)^7

= 400 (1.12)^7= 400 (2.21) = 884.27

c) At the end of seven years, how much interest have you earned?

We are getting compound interest total of 884.27 400 = 484.27


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You have two options to choose from: You must invest Rs. 130,000minimum.

a) Govt. guarantees you triple your money in 10 years.b)Govt. guarantees you 15% compounded at end of 10 years.

Which do you choose, now that you know FV?


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a) FV = 130,000 * 3 = 390,000 in 10 years.

b) FV = 130,000 (1.15)^10

= 130000 (4.04)

= 525,922.50

Obviously you should choose option B, since 525,922 > 390,000


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Time Value of Money

Learning Objectives

What is FV & Compounding?

Simple Interest

Compound Interest

What is PV & Discounting?

Valuing Cash Flows

How to find the return on investment?


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What is PV?Financial Managers must be able to answer

How to determine the value today (PV) of cash flowsexpected in the future?

Involves bringing FV C/Fs back to Present


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What is PV?Present Value

PRESENT VALUE?

The current value of cash flows discounted at theappropriate discount rate


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Properties?

PV: C/Fs are discounted at DISCOUNT FACTOR (i)

FV: C/Fs are compounded at INTEREST FACTOR (i)

C/Fs cannot be added together

PV allows you to understand & compare C/Fs at To

Relevant C/F for analysis is PV C/F PV: inversely proportional to interest i

FV: directly proportional to i



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What is PV?Adding C/Fs together?

T=o 1 2 3 4 5

(100) 20 20 20 20 120

FV FV FVFVPV


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What is PV?Adding C/Fs together?

T=o 1 2 3 4 5

(100) 20 20 20 20 120

FV FV FVFVPV

35 15


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Names?

Discount Rate

Discount Factor

PV Factor

DCF (Discount Cash Flow)

PVIF (r,t) Present Value Interest Factor @ r & t


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What is PV?Example

What is $10,000 received in 10 yrs worth today?i=6.5%

Just discount the FV at the i


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What is $10,000 received in 10 yrs worth today?i=6.5%

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What is PV?Example

Compound Interest: Future Value of $100 at 10


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Compound Interest: Future Value of $100 at 10Percent Compounded

Year Beginning Amount Interest Earned End Amount (FV)

1 $100.00 PV

2 110.00 11.00 121.00 3 121.00 12.10 133.10

4 133.10 13.31 146.41 FV

5 146.41 14.64 161.05 FV

Total interest $61.05

REVERSE

COMPOUNDING


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COMPOUNDING: Future Value for a Lump Sum

In general, the present value, PV, ofFVn discounted at r% for

n periods is

PV= FVn / (1 + r)n

The expression

1/(1 + r)n

is the Discount Factor (DF) or PVF

Present Value of $1 for Different Periods


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Present Value of $1 for Different Periodsand RatesDISCOUNT FACTORS

Presentvalueof $1 ($)

Time(years)

r= 0%

r= 5%

r= 10%

r= 15%

r= 20%

1 2 3 4 5 6 7 8 9 10

1.00

.90

.80

.70

.60

.50

.40

.30

.20

.10


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What is PV?

The equation of PV Disounted at r or i then is:

PV= FVn / (1 + r)n


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What is FV?Applying to problem sets:


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PV SINGLE PERIOD BASIC PROBLEM #1

Suppose you need $400 to buy textbooks next year.

U can earn 7% on your money.

How much do you have to put up today?

BASIC PROBLEM #


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BASIC PROBLEM #1

Step 1:

Setting up the problem:

FV= $400

N=1

R = 7%

Step 2:

Using formula: PV = FV/ ( 1+ i)^n

= 400/ (1+ .07)^1

= 400 /( 1.07)

= 373.83


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PV BASIC PROBLEM # 2

Suppose you need to have $1000 in 2 yrs.

If you can earn 7%, how much do you have to invest to make sure you getyour $1000.

PV Multiple Periods #2


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PV Multiple Periods #2

Step 1

FV = 1000R= 7%

N = 2 yrs

Step 2

Use formula PV = FV/ ( 1+ i)^n

= 1000/ (1+.07)^2

=1000/ (1.1449)

=873.44


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You would like to buy a new automobile. You have $50,000, but thecar costs $68,500. If you can earn 9%.

a) How much should you invest today to buy the car in two yrs?

b) Do you have enough?

Assume price stays the same?


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a) How much to invest? #3

Step 1

FV = 68500

R= 9%

N = 2

Step 2Use formula PV = FV/ ( 1+ i)^n

= 68500/ (1+.09)^2

= 400 (1.1881)

= 57,655.08

b) Do u have enough? NO 50000-57655.08 = (7655)


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You need $1000 in three yrs. You can earn 15%. Calculate discount

factor to solve.

a) How much do u have to invest today?

b) What's the discount factor?


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a) How much to invest?#4

Step 1

FV = 1000

R= 15%

N = 3Step 2


= 1000/ (1+.15)^3

= 1000/ (1.5209)

= 1/1.5209 = .6575 * 1000 = 657.50


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Your company proposes to buy an asset for $335. This investment is

very safe. You will sell off the asset in three yrs for $400. U know youcan invest the $335 elsewhere at 10 percent with little risk.

a) Which do u think is a better choice?

a) Option 1 & 2 (comparing FVs for valuation) #5


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Step 1

FV = 400

PV= 335N = 3 (i=10% elsewhere)

Step 2

Use formula Fv = PV* ( 1+ i)^n

= 335* (1+.1)^3

= 335* (1.331)= 445.89

Since the 1st option gives FV = 400 after 3 yrs

& 2nd option gives FV= 445.89 > 400, then elsewhere 10% is better.

a) Another way(looking at PV of 400 at 10%) #5


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Step 1

FV = 400

PV= 335N = 3 (i=10% elsewhere)

Step 2


= 400/ (1+.1)^3

= 400/ (1.331)=300.53

Since the 1st option gives PV = 300.53 after 3 yrs

& 2nd option gives PV=335 > 300.53, then elsewhere 10% is better.


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PV BASIC PROBLEM # 6 FINDING i

You are considering a one year investment. If you put up $1250, you

will get back $1350.

a) What rate is this investment paying?

a) R?#6


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Step 1

FV = 1350

PV= 1250N = 1 (i=?)

Step 2

Use formula FV/PV = ( 1+ i)^n

= 1350/1250 -1

= 1.08 - 1= 8%


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An investment offers to double our money in eight yrs. Cost is $100.

a) What rate is this investment paying?

a) R?#7Step 1


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Step 1FV = 200PV= 100

N = 8 (i=?)Step 2Use formula FV/PV = ( 1+ i)^n

(1+i)^8 = 200/100(1+i)^8 = 2(1+i)^8.(1/8) = 2^(1/8)

1+i = 2^0.125i = 1.09 1 = .09

9%


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Rule of 72 for doubling your money returnsThe time it takes to double your money (n) is given by 72/r%

72/r = n or r = 72/n

In this case

72/r% = 8,

R% = 72/8 = 9%

For discount rates b/w 5 & 20%, r can be approximated for doubling.


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An investment promises to double your money every 10 yrs.

a) What rate is this investment paying using rule of 72?


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An investment promises to double your money every 10 yrs.

a) What rate is this investment paying using rule of 72?

R = 72/n

= 72/10

= 7.2%

NOTE: check with calculator & other formula method


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CALCULATING R METHODS

There are 3 ways to calculate r.

a) Use Financial Calculator OR Excel

b) Solve equation for (1+r) taking roots both sidesc) Use FV tables


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You estimate u will need about $80,000 to send your child to college

in 8 yrs. You have $35,000 now. If you earn 20%pa

a) Will you have enough for college?

b) At what rate will you just make it?

a) Will u make it?#9

Step 1


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Step 1

FV = ?

PV= 35000N = 8 i=20%

Step 2

Use formula FV=PV* ( 1+ i)^n

= 35000*(1.20)^8

= 150493.59 YESb) Minimum Rate

FV = 35000 * (1+r)^8

R =(80000/35000)^(1/8) = 2.2857^0.125 = 1.1089 1 = 10.89%


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You would like to retire in 30 years as a millionaire. If you have

$50,000 today to invest.

a) What rate of return do you need to make it?

a) Will u make it?#10

Step 1


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Step 1

FV = 1000000

PV= 50000N = 30 i=?

Step 2

b) Minimum Rate

1000000 = 50000 * (1+r)^30

(1+r)^30 = 1000000/50000

(1+r) = 20^(1/30) = 20^.0333

r = 1.105 = 0.105 = 10.5%

10.5% = i


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Changing the texture of the problem

PV BASICPROBLEM # 11 - VALUATION

Your apartment house has burned down leaving you with a


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Your apartment house has burned down, leaving you with avacant lot worth $50,000 and a check for $300,000 from thefire insurance company. You consider rebuilding, but yourreal estate agent suggests putting up an office buildinginstead. The construction cost would be $300,000 & therewould be the cost of the land, which otherwise may be soldfor $50,000. On the other hand your advisor foresees ashortage of office space and predicts that a year from nowthe new building would fetch $400,000 if you sold it. Thus

you would be investing $350,000 now in the expectation ofrealizing $400,000 a year hence. You should go ahead if thePV of the expected $400,000 payoff is greater than theinvestment of $350,000. US Govt securities are paying 7%with one year maturities.

PV BASIC PROBLEM # 11 - VALUATIONYour apartment house has burned down leaving you with a vacant lot worth $50 000


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Your apartment house has burned down, leaving you with a vacant lot worth $50,000and a check for $300,000 from the fire insurance company. You consider rebuilding,but your real estate agent suggests putting up an office building instead. Theconstruction cost would be $300,000 & there would be the cost of the land, which

otherwise may be sold for $50,000. On the other hand your advisor foresees a shortageof office space and predicts that a year from now the new building would fetch$400,000 if you sold it. Thus you would be investing $350,000 now in the expectationof realizing $400,000 a year hence. You should go ahead if the PV of the expected$400,000 payoff is greater than the investment of $350,000. US Govt securities arepaying 7% with one year maturities.

a) What is the value tdy of $400,000 to be received one yearfrom now based on market returns?

b) Is the PV greater than $350,000?c) What is the rate of return for the office bldg project?d) What decision should be made & why?

a) Will u make it?#10Step 1 PROJECT Govt Security


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FV = 400000 FV = 400000PV=350000 PV = ? (373,832)

N = 1 i=? (14.28%) I = 7% n=1Step 2b) Minimum Rate PV = FV/(1+i)^n

400000 = 350000 * (1+r)^1 = 400000/(1.07)^1(1+r)^1 = 400000/350000 = 373,832(1+r) = 1.1428^1 PV = 373832 > 350000r = 1.1428-1 = 0.1428 = 14.28% 373.832 will need to be

invested to reachFV of 400,000.

Thus project is better choice

i = 14.28% I = 7% < 14.28%

Problem Practise


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Problem Practise

Rarely Prudent, Inc. has an unfunded pension liability of$425 million that must be paid in 23 years. If the relevantdiscount rate is 7.5 percent, what is the present value of thisliability?

Solution


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Solution

Rarely Prudent, Inc. has an unfunded pension liability of

$425 million that must be paid in 23 years. If the relevantdiscount rate is 7.5 percent, what is the present value of thisliability?

Future value = FV = $425 million

t = 23

r = 7.5 percentPresent value = ?

Solution: Set this up as a present value problem.

PV = $425 million x PVIF(7.5,23)

PV = $80,536,778


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PROBLEM #1


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PROBLEM #1

Assume you deposit $1000 today in an account which pays 8%. Howmuch will you have in

4 yrs? (1.08^4)*1000 = 1.36*1000 = 1360.48

14yrs? 2.93*1000 = 2937.19

50yrs? 46.90*1000 = 46901.61

PROBLEM # 2


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PROBLEM # 2

Suppose you just turned 21. A rich uncle had setup a trust fund which would pay you Rs.100,00,000

at age 40. Relevant discount rate is 12%.a) how much is the fund worth today?b) If the fund was paid to you at age 50, how much would it be

worth?

Compound INTEREST BASIC PROBLEM


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Step 1

Fv = 100kR= 12%

N = 40-21 = 19 yrs

Step 2Use formula PV = FV/(1+i)^n

= 1/8.61 = .1161*100000

=11610.67

Compound INTEREST BASIC PROBLEMFor age 50

S


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Step 1

Fv = 100kR= 12%

N = 50-21 = 29 yrs

Step 2Use formula PV = FV/(1+i)^n

= 1/26.74 = .0373*100000

=3738.32

PROBLEM # 3


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PROBLEM # 3

Youve been offered an investment which willdouble your money in 12 yrs.

a) What rate of return are u being offered?b) Check using rule of 72?

c) If the investment had doubled in half the time,what would the rate be?

PROBLEM # 3


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PROBLEM # 3Youve been offered an investment which will double your money in 12 yrs.

a) What rate of return are u being offered?

(1+i)^12=2, 2^1/12, = 1.059-1, = 5.9%

b) Check using rule of 72?

72/12 = 6%

c) If the investment had doubled in half the time, what would the rate be?n=6, 2^1/6, = 1.1224-1 = 12.24%

72/6 = 12%


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PROBLEM # 4

You have three options to choose from:You must invest Rs. 300,000 minimum. Market interest rate is 12%.

a) DSC offers to give you an interest factor of 4.0 over10 yrs. Whats the rate?

b) Govt. savings scheme guarantees you 15.5%compounded over 9 yrs.

c)A third option provides 10% first 5 yrs & then 20%next 5 yrs (compounded)

Which do you choose?


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PROBLEM # 4

You have three options to choose from:You must invest Rs. 300,000 minimum. Market interest rate is 12% on deposits.

a) DSC offers to give you an interest factor of 4.0 over 10 yrs. Whats the rate?i=4^(1/10)-1 = 1.1486-1 = .1486 = 14.86% (FV=1,200,000)

b) Govt. savings scheme guarantees you 15.5% compounded over 9 yrs.1.155^9 = 3.6579*300000 = 1097385.23*1.12= (1,229,071.46)

c) A third option provides 10% first 5 yrs & then 20% next 5 yrs (compounded)

1.10^5 = 1.6105*300k = 483153 (1.20)^5 = 2.488*483153 = 1,202,239.27Which do you choose? B

PROBLEM #5


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GMAC offered some securities for sale to the public. Under the terms, GMAC

promised to pay the owner of one of these securities $10,000 on Dec 1, 2012. Investorspaid GMAC $500 for this security on dec 2, 1982.

a) What rate is GMAC paying?b) Suppose on dec 1,2000, this securitys price was$4490.22. If an investor had purchased it for $500 at the

offering, what rate does she get?c) If the investor had purchased the security at marketon dec 1, 2000, what annual rate of return would shehave earned?

PROBLEM #5

GMAC offered some securities for sale to the public. Under the terms, GMACpromised to pay the owner of one of these securities $10,000 on Dec 1, 2012.I id GMAC f hi i d 8


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p p yInvestors paid GMAC $500 for this security on dec 2, 1982.

a) What rate is GMAC paying?n=30 1+i^30 = 10k/500 = 20^1/30 = 20^.0333 = 1.1050 1 = 10.5%

b) Suppose on dec 1,2000, this securitys price was $4490.22. If an investor hadpurchased it for $500 at the offering, what rate does she get?n=18 1+i^18 = 4490.22/500 =8.98^1/18 = 8.98^.0555 = 1.1295 1 = 12.95%

c) If the investor had purchased the security at market on dec 1, 2000, whatannual rate of return would she have earned?n=12 1+i^12 = 10000/4490.22 =2.227^1/12 = 2.227^.0.0833 = 1.0689 1 = 6.89%

PROBLEM # 6


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You are interested in a Ferrari costing $120,000. You have $30,000 tdy. If amutual fund pays 10.5% & you want to buy the car in 10 yrs on the day youturn 30,

a) how much must you invest today?b) Do you have enough?

PROBLEM # 6


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You are interested in a Ferrari costing $120,000. You have$30,000 tdy. If a mutual fund pays 10.5% & you want to buy thecar in 10 yrs on the day you turn 30,

a) how much must you invest today?PV = 120k / (1.105)^10 = 120K. (1/2.714) = 120K. 0.36844 =

44213.86b) Do you have enough?

no

PROBLEM # 7


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You are scheduled to receive $24,000 in two yrs. When you

receive it, you will invest it for 6 more yrs at 6% per year.

a) How much will you have in eight yrs?

b) What is the worth tdy of this amount if the discount rate is17%?

PROBLEM # 7

h d l d h


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You are scheduled to receive $24,000 in two yrs. When you

receive it, you will invest it for 6 more yrs at 6% per year.

a) How much will you have in eight yrs?

24000 (1.06)^6 = 24000. 1.418 = 34044 IN 8 YRS

b) What is the worth tdy of this amount if the discount rate is17%?34044/ (1.17)^8 = 1/3.51*34044 = .2847*34044 = 9695.13

PROBLEM # 8


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You have $10,000 to deposit. Alfalah offers 12% per yearcompounded monthly (1% per month), while MCB offers 12%but will only compound annually. How much will yourinvestment be worth at each bank.

a) Alfalah?b) MCB?

PROBLEM # 8


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You have $10,000 to deposit. Alfalah offers 12% per yearcompounded monthly (1% per month), while MCB offers 12%but will only compound annually. How much will yourinvestment be worth at each bank.

a) Alfalah?=10000* (1.01)^12 = 10k. 1.12682 = 11268.25

a) MCB?

= 10000*(1.12)^1 = 11200



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You need $60,000 in eight yrs. If you can earn .75% per

month,

a) how much will u have to deposit tdy?



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You need $60,000 in eight yrs. If you can earn .75% per

month,

a) how much will u have to deposit tdy?

n=96, PV = 60k / (1.0075)^96, 60k/ 2.0489 = 29283.70

PROBLEM # 10

id i i h i i i l


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You are considering a seven year investment. The initial

amount is 80 lacs for 1 kanal plot. The plot is likely to sell for1 crore in 2 yrs, 1.2 crore in 3 yrs, 1.5 crore in 4 yrs and 3 crore in7 yrs. What re the respective rates of returns? Market interestrate on 7 year DSC is 18%. What would you choose?

a) What rate is this investment paying for respective years?

PROBLEM # 10You are considering a seven year investment The initial


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You are considering a seven year investment. The initialamount is 80 lacs for 1 kanal plot. The plot is likely to sell for1 crore in 2 yrs, 1.2 crore in 3 yrs, 1.5 crore in 4 yrs and 3 crore in7 yrs. What re the respective rates of returns? Market interestrate on 7 year DSC is 21%. What would you choose?

a) Rate for 2yrs?

1+i^2 = 100/80 = 1.25^1/2 = 1.1180-1 = 11.80%b) Rate for 3yrs?1+i^3 = 120/80 = 1.5^1/3 = 1.1447-1 = 14.47%

c) Rate for 4yrs?1+i^4 = 150/80 = 1.875^1/4 = 1.1701-1 = 17.01%

d) Rate for 7yrs?1+i^7 = 300/80 = 3.75^1/7 = 1.2078-1 = 20.78%DSC

PROBLEM # 11 - VALUATIONYour apartment house has burned down, leaving you with avacant lot worth $50,000 and a check for $300,000 from the fireinsurance company. You consider rebuilding, but your real estate


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insurance company. You consider rebuilding, but your real estateagent suggests putting up an office building instead. The

construction cost would be $300,000 & there would be the cost ofthe land, which otherwise may be sold for $50,000. On the otherhand your advisor foresees a shortage of office space and predictsthat a year from now the new building would fetch $400,000 if

you sold it. Thus you would be investing $350,000 now in theexpectation of realizing $400,000 a year hence. You should goahead if the PV of the expected $400,000 payoff is greater than

the investment of $350,000. US Govt securities are paying 7% withone year maturities.

a) What is the value tdy of $400,000 to be received one year from nowbased on market returns?b) Is the PV greater than $350,000?c) What is the rate of return for the office bldg project?d) What decision should be made & why?

PROBLEM # 11 - VALUATIONYour apartment house has burned down, leaving you with a vacant lot worth $50,000 and acheck for $300,000 from the fire insurance company. You consider rebuilding, but your realestate agent suggests putting up an office building instead. The construction cost would be


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estate agent suggests putting up an office building instead. The construction cost would be$300,000 & there would be the cost of the land, which otherwise may be sold for $50,000. Onthe other hand your advisor foresees a shortage of office space and predicts that a year fromnow the new building would fetch $400,000 if you sold it. Thus you would be investing$350,000 now in the expectation of realizing $400,000 a year hence. You should go ahead if thePV of the expected $400,000 payoff is greater than the investment of $350,000. US Govtsecurities are paying 7% with one year maturities.

a) What is the value tdy of $400,000 to bereceived one year from now based onmarket returns?373,832b) Is the PV greater than $350,000?yesc) What is the rate of return for the office bldgproject? 14.28%d) What decision should be made & why?

Project pays higher


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PROBLEM # 12

The office building project will have some changesin the cash flows according to new negotiations


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in the cash flows according to new negotiations

with the contractor.1. $123,000 downpayment now2. $ 76,000 retainer fee now

3. $133,000 final payment when building is readyfor occupancy

a)What are the new cash flows?b)What is the rate on the project now?

c)Which option is better now?

PROBLEM # 12The office building project will have some changes in the cash flows according to thecontractor


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contractor.1. $123,000 downpayment now2. $ 76,000 retainer fee now3. $133,000 payment when building is ready for occupancy

a) What are the new cash flows?123k+76k = 199k cost (t0) PV,400k-133k = 267k (t1) FV

a) What is the rate on the project now?1+i^1 = 267/199 = 1.34-1 = 34%

a) Which option is better now? Project still better


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PV & FV for Multiple cash Flows

Annuities

Perpetuities

Calculating Loan Payments & interest rate on loans

Amortization

What is DCF?


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What could be Multiple Cash Flows?

What is DCF?


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Multiple Cash Flows?

Car Payments

Home Mortgage Payments

Credit Card payments

Investments/Projects C/Fs

What is DCF?FV & Multiple Cash Flows


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Suppose u deposit $100 tdy in an account paying8%? Then you deposit another $100 at t=1.

How much will u have in 2 yrs?

Time line structure

Calculating one period at a time by adding at each T

Calculating by compounding for n=2 & 1

What is DCF?Example 1


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U can deposit Rs 4000 at end of each year for nextthree years at an interest of 8%. You already have Rs7000 in your account.





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U can deposit Rs 4000 at end of each year for nextthree years at an interest of 8%. You already have Rs7000 in your account.


7000*1.08 + 4000 = 11560

11560*1.08+4000 = 16484.80

16484.8*1.08+4000=21803.58

How much will u have in 4 yrs? 21803.58*1.08=23547.87

What is DCF?2 ways of calculating FVs


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Compound accumulated balance 1 yr at a time

OR

Find FV of each cash flow to the end & add up



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Lets say you can deposit Rs2000 at end of each yearfor next five yrs. Rate is 10% & beginning balance is0. What's the FV?

Calculate using compounding each period + CF (Begamt+add)



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Lets say you can deposit Rs2000 at end of each yearfor next five yrs. Rate is 10% & beginning balance is0. Whats the FV?

Calculate using compounding each period + CF (Begamt+add)

(0*1.1=0)

(0 + 2000)=2000

(2000*1.1)+2000 = 4200

(4200*1.1)+2000 = 6620 (=9282). =12210.20



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Lets say you can deposit Rs2000 at end of each yearfor next five yrs. Rate is 10% & beginning balance is0. Whats the FV?

Calculate using Fv for each C/F

(2000*1.1)^4

(2000*1.1)^3

(2000*1.1)^2

(2000*1.1)^1 (2000) add up = 12210.20



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Changing C/Fs FV.

If u deposit $100 in one year, $200 in 2 yrs, $300 in 3yrs. Interest rate is 7%.




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Changing C/Fs Fv.



100(1.07)^2 = 114.49

200(1.07) = 214.00

300 = 300

FV =628.49

What is DCF?Example


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Changing C/Fs Fv.



628.49(1.07)^2 = 719.56

Try compounding C/F method for 5 yrs


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PV & FV for Multiple cash Flows

Annuities

Perpetuities

Calculating Loan Payments & interest rate on loans

Amortization

What is DCF?


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What is an Annuity?

A level stream of C/Fs for a fixed period of time

What is DCF?


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Annuity Examples?

Making equal payments

All consumer loans

Car Loans

Home Mortgage Loans


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What is DCF?PV of Annuity


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Suppose an asset promised to pay $500 at end ofeach of next 3 yrs. If we want to earn 10% on themoney, how much should we offer for the annuity?

PV = 500/1.1 + 500/1.1^2+500/1.1^3 = 1243.43



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When there are large no. of c/fs, shortcut helps

Annuity PV = C x ( 1-PV factor/ r)

OR

= C x (1-(1/(1+r)^n)/r)


Using Formula


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Suppose an asset promised to pay $500 at end ofeach of next 3 yrs. If we want to earn 10% on themoney, how much should we offer for the annuity?

PV Factor = 1/1.1^3 = 0.75131

APV Factor = (1-PV factor)/r

= (1-0.75131)/0.1 = 2.48685

PV = 500 x 2.48685 = 1243.43



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U can afford to pay $632 per month towards a newcar. The interest is 1% per month for 48 months.

How much can u borrow?



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U can afford to pay $632 per month towards a newcar. The interest is 1% per month for 48 months.

How much can u borrow?

Annuity PV factor = (1-PV factor)/r

= 1 (1/1.01^48) / .01

= (1 0.6203) / .01

= 37.9740

PV = 632 x 37.9740 = $24,000 (is what u can afford to borrow & repay)



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U can afford to pay Rs 35,000 per month towards anew home. The interest is 13% per year for 30 years.

What price home can you buy?



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U can afford to pay Rs 35,000 per month towards anew home. The interest is 13% per year for 30 years.

What price home can you buy?

N=360 i = 13/12 = 1.083% per month

Annuity PV factor = (1-PV factor)/r

= 1 (1/1.01083^360) / .01083

= (1 .0206954) / .01083

= 0.9793 / .01083 = 90.42

PV = 35000 x 90.42 = Rs 31,64,700 (price of house)


Annuity

FINDING THE PAYMENT AMOUNT C


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Use the same formula & find the C, given the PV isalready known

PV = C x (1- PVF) / r

FINDING THE PAYMENT AMOUNT C


Annuity


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You need to borrow $100,000. You can make 5 equalpayments annually. Interest rate is 18%.

What will the payments be?

PV = C x (1- PVF) / r

Find the relevant annuity factor & solve for C


Annuity


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You need to borrow $100,000. You can make 5 equalpayments annually. Interest rate is 18%.

APV = 100,000 = C x (1- 1/1.18^5) / .18

= C x (1 - .4371)/.18

= C x 3.1272

C = 100,000 / 3.1272 = 31,978

What is DCF?


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FV of Annuities Found the same way as the PV for annuities with the

relevant formula.

FV Annuity = C x (FV annuity Factor 1)/r

Or

FV annuity = C x (( 1+r)^n 1))/r

What is DCF?

Example 4

Annuity FVs

You plan to contribute $2000 every year for 30 yrs.


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Account pays 8%. How much will you have?

FV Factor = FV Factor -1 /r

= (1.08^30 1)/.08

= (10.0627 -1)/ .08

= 113.2832

FV annuity = 2000 x 113.2832 = 226,566.4

What is DCF?


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Annuities Due

Annuity Due

An annuity for which the C/Fs occur at the beginning

of the period.Example: apartment lease prepay rent

Ordinary Annuity

An annuity for which the C/Fs occur at the END of the period.

What is DCF?


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Annuities Due

Annuity Due Calculation

Since the C/Fs are one period earlier, we just adjust

by (1+r)multiply by (1+r) the ordinary annuitySteps:

1) Calculate ordinary annuity PV or FV

2) Multiply by (1+r)

What is DCF?

A i i D


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Annuities DueExample

Annuity Due Calculation

Since the C/Fs are one period earlier, we just adjust

by (1+r)Steps:

1) Calculate ordinary annuity PV or FV

2) Multiply by (1+r)


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What is DCF?Example 4 shorter way

An annuity has 5 payments of $400 each. Interest is10%


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10%.

Calculate ordinary annuity for n=5 (time line)

PV factor = (1/(1.1^5))

= .6209

PV annuity = 1-.6209/ .1= 0.379/.1 = 3.7907

PV = 400x3.7907 = 1516.314

Multiply by (1+r) = 1516.314 x 1.1 = 1667.94

NOTE: since we discount by 5 periods (1 too many), we adjust by multiplying

by (1+r)

What is DCF?Perpetuities?


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Its a Special Case


Its a Special Case


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Perpetuity

An annuity in which the cash flows continue forever.

Other names:

CONSOLS (European & Canadian)


Calculation


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Its the easiest case

Perpetuity PV = C / r OR C x (1/r)

Other names:

CONSOLS (European & Canadian)

What is DCF?Perpetuities

Example 1


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An investment offers a C/F of $500 every year forever.Interest rate is 8%.

Whats the value of this investment?


Example 1

$


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An investment offers a C/F of $500 every year forever.Interest rate is 8%.

Whats the value of this investment?

Perpetuity PV = C / r

= 500/.08

= 6250


Example 2

S JNJ ll f d k $100


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Suppose JNJ wants to sell preferred stock at $100 pershare. A similar stock OGDC already has a price of $40 pershare & offers a dividend of $1 every quarter.

What dividend must JNJ offer to give the same return?


Example 2

Suppose JNJ wants to sell preferred stock at $100 per


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Suppose JNJ wants to sell preferred stock at $100 pershare. A similar stock OGDC already has a price of $40 pershare & offers a dividend of $1 every quarter.

What dividend must JNJ offer to give the same return?

Step 1: Calculate the r for $40 stock

PV = C / r 40 = 1 / r r = 1/40 = .025

Step 2: Find C giving the same r

100 = C / .025 C = 100 * .025 C = $2.5

NOTE: the r is quarterly rate, not yearly so adjust in Qs

What is DCF?Different Types of Rates

Stated Interest Rate


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EAR (Effective Annual Rate)

APR (annual percentage rate)




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Stated Interest RateInterest rate expressed in terms of interest payment per period.

Also the quoted interest rate

EAR (Effective Annual Rate)

Interest rate based on per year compounding

APR (annual percentage rate) legally allowed!

Interest rate charged per period x no. of periods per year

What is DCF?Examples

Stated Interest Rate (ONE U NEED TO CONVERT)


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Stated Interest Rate (ONE U NEED TO CONVERT) 10% per year (paid semi-annually 5%)- 10 vs 10.25%

Calculate to see the equivalent rate

EAR (Effective Annual Rate) THE USEFUL ONE

The 10.25% is the actual effective rate

Interest rate based on per year compounding

APR (annual percentage rate) THE TRICKY ONE


What is DCF?Examples



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REALITY: ITS JUST A STATED OR QUOTED RATE

NOTE: It is not necessarily the EAR!!

So, if a bank says it charges 12% APR on car loans,

With monthly payments, it actually means..

1% per month x 12 mths (according to law).

But IN FACT its EAR is 12.6825%!


Example

Bank A: 15% compounded daily


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Bank A: 15%, compounded daily

Bank B: 15.5%, compounded quarterly

Bank C: 16%, compounded annually

Which is best if u want to open a savings account?


Example

Bank A: 15% compounded daily


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Bank A: 15%, compounded daily

EAR = 16.18% n=365

Bank B: 15.5%, compounded quarterly

EAR = 16.42% n=4

Bank C: 16%, compounded annually

EAR = 16% n=1

Which is best if u want to open a savings account?

Bank B

What is DCF?Calculating EAR


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EAR Formula

EAR = (1+ Quoted rate / m )^m -1

where m = no. of times interest is compounded per year

What is DCF?3 Steps to EAR


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Step 1: Divide the quoted rate in decimal by the #of times it is compounded (m)

Step 2: Add 1 to the result & raise by m

Step 3: Subtract 1 to get the rate


Example


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You are offered 12% compounded monthly.

Whats the EAR?


Example


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You are offered 12% compounded monthly.

Whats the EAR?

M = 12 so

EAR = (1+ .12/12 )^12 1 = 1.01^12 1 = 1.126825 1

= 12.6825%


Example 2


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A Bank is offering 12% compounded quarterly. If you put$100 in an account,

how much will you have at the end of the year?

Whats the EAR?.


Example 2

A Bank is offering 12% compounded quarterly. If you put


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$100 in an account,

how much will you have at the end of the year?

Whats the EAR?.

FV = 100*1.03^4

= 112.55

EAR is (1+ .12/4 )^4 -1 = .1255= 12.55%

Lets solve for 2 yrs.


Example 2

A Bank is offering 12% compounded quarterly. If you put$100 i


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$100 in an account,

how much will you have at the end of the 2 years?

Whats the EAR?.2 yrs

FV = 100*1.03^8

= 126.67

EAR is (1+ .12/4 )^4 -1 = .1255= 12.55% same

For 2 yrs it would be 8 quarters.


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What is DCF?Converting APR

A credit card quotes an interest rate of 18% APR. Monthly

payments are required.


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What is the actual interest rate you pay on this credit card?

EAR = (1+ APR / m )^m -1

= (1+ .18/12)^12 1 = 1.015^12 1 = 1.1956 1 = 19.56%

NOW U KNOW HOW THEY CONFUSE CONSUMERS!!

LOANS

Types of Loans

3 Basic Types


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Pure Discount Loans

Interest Only Loans

Amortized Loans

LOANS

Types of Loans

Pure Discount Loan


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Simplest Form

Borrower receives money today & repays a single lump sum

Example:

A one year, 10% discount loan means borrower pays $1.1dollar for every dollar borrowed

NOTE: Simple compounded FV T-bills are PDLs (short term)

LOANS

Types of Loans

Interest Only Loans


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Borrower pays interest each period & repays the entire principal atsome point in the future.

Example:

A 3 year, 10% interest only loan of $1000, means borrowerpays $100 for 1st & 2nd year & $1100 the 3rd year.

NOTE: Bonds are interest only

LOANS

Types of Loans

Amortized Loans


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Borrower pays part of principal & interest in payments over time.

Process of paying a loan by making regular principal reductions iscalled amortizing.

Example:

A $5000, 5 year, 9% loan is taken out. The borrower paysinterest each year & reduces the loan by $1000 each year.The loan balance declines by $1000 each year, it is paid up

in 5 yrs.

LOANS

Types of Loans

Amortized Loans


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The Principal drops every year & therefore,

The interest payments per year also DROP resulting

in lower yearly payments.

Example:

1st year interest = 5000*.09 = 450

2nd year interest = 4000*.09 = 360

3rd year interest = 3000*.09 = 270

LOANS

Types of Loans

Amortized LoansCalculate the following


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Calculate the following

Beg Bal Total Pymt Interest Paid Principal Paid Ending Bal

1st

year2nd year

3rd year

4th year

5th

year

LOANS

Types of Loans

Amortized Loans different pymt amtsCalculate the following


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Calculate the following

Beg Bal Total Pymt Interest Paid Principal Paid Ending Bal

1st

year 5000 1450 450 1000 40002nd year 4000 1360 360 1000 3000

3rd year 3000 1270 270 1000 2000

4th year 2000 1180 180 1000 1000

5th

year 1000 1090 90 1000 0

LOANS

Types of LoansAmortized Loans


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The Principal drops every year & therefore,

The interest payments per year also DROP resultingin lower yearly payments.

Example:

1st year interest = 5000*.09 = 450

2nd year interest = 4000*.09 = 360

3rd year interest = 3000*.09 = 270

LOANS

Amortized Loans

CONSTANT PAYMENTS


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First determine the CASH Payment on the loan.

Step 1: It is an annuity, so use formula to get C

5000 = C x (1-(1/1.09^5)/.09 = C= 1285.46 or use PMT

Step 2:

Calculate interest on the balance (e.g 5000*.09=450) based on C

Step 3:

Find principal amt. based on interest paid (eg 1285.46-450=835.46)

Complete amortization

Loan Amt $ 5,000Interest Rate 9%

Loan Term 5Loan Payment $1,285.46 Payment is calculated using PMT fn

Year Begin Total Interest Principal Ending


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Year Begin Total Interest Principal EndingBalance Payment Paid Paid Balance

1 $5,000.00 $1,285.46 $450.00 $835.46 $4,164.542 $4,164.54 $1,285.46 $374.81 $910.65 $3,253.893 $3,253.89 $1,285.46 $292.85 $992.61 $2,261.284 $2,261.28 $1,285.46 $203.52 $1,081.94 $1,179.335 $1,179.33 $1,285.46 $106.14 $1,179.32 $0.01

TOTALS $6,427.3 $1,427.3 $5,000.0

Minus =

What are major types of Financial Assets?Intro

Capital Market Securities


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?

What are major types of Financial Assets?Intro

Capital Market Securities


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Fixed Income (Bonds)

Treasuries

Agencies

Municipals

Corporates

Equities

Preferred Stock

Common Stock

What is DCF?PV & Multiple Cash Flows

S d $1000 i 1 & $2000 i 2


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Suppose u need $1000 in 1 year & $2000 more in 2yrs. If you can earn 9%.

How much do u need to invest tdy?

Time line structure

Discounting one period at a time & adding at each T backwards

Calculating by discounting for n=1 & 2

What is DCF?PV Example 4

How much do u have to put up?


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2000/(1.09)^2 = 1683.36

1000/(1.09) = 917.43

PV =2600.79


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An investment pays 1000 at end of every year for next

five years. Discount rate is 6%.


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How much needs to be invested today?

1000/1.06^5 = 747.26

1000/1.06^4 = 792.09

1000/1.06^3 = 839.62

1000/1.06^2 = 890.00

1000/1.06^1 = 943.40

PV = 4212.37

What is DCF?NOTE ON CASH FLOW TIMINGS

Timing is critically important


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Assumption is always:

C/F occurs at end of period

Unless specified otherwise.

What is DCF?PV Uneven C/Fs Example 6

An investment pays 200 in one year, 400 next year,600 the next year & 800 the next year. You can earn12% on very similar investments


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12% on very similar investments.

What should you pay?

What is DCF?PV Uneven C/Fs Example 6

An investment pays 200 in one year, 400 next year,600 the next year & 800 thye next year. You can earn

12% on very similar investments.


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What should you pay?

800/1.12^4 = 508.41

600/1.12^3 = 427.07

400/1.12^2 = 318.88

200/1.12^1 = 178.57

PV = 1432.93

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Lecture 4(Ch4)-NCBA&E

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