Introduction & Time Value of Money

Omesh KiniFOUNDATIONS OF FINANCE - I

Omesh KiniFOUNDATIONS OF FINANCE - I

Agenda Overview of course outline

Organization forms

Goal of financial manager

Time value of money

Overview of course outline

Organization forms

Goal of financial manager

Time value of money

Foundations of Finance - I 2

Forms of Business OrganizationsType Ownership Ability to Raise

CapitalLiability Taxation

SoleProprietorship

100% owned by asingle individualOwner usuallymanages company

Difficult Unlimitedpersonal liability

Single


100% owned by asingle individualOwner usuallymanages company

Partnership Two or moreindividuals

Less difficult thansoleproprietorship

Unlimitedpersonal liability

Single

Corporation Separation of ownersand managers

The least difficultof all forms

Limited toowners initialinvestment

Double

Role of The Financial ManagerFinancialmanager

Firm'soperations

Financialmarkets

(1)(2)

(4a)operations markets

(1) Cash raised from investors(2) Cash invested in firm(3) Cash generated by operations

(3)

(4a) Cash reinvested(4b) Cash returned to investors

(4b)

Role of Financial Manager (Continued) Investment decisions

Financing decisions

Working capital management

Foundations of Finance - I5

Investment decisions

Financing decisions

Working capital management

Goal of corporatefinancial management / financial managerMaximize the wealth of stockholders/ownersMaximize the wealth of stockholders/owners

ororMaximize price per shareMaximize price per share

Why? Stockholders invest in the firm because theywant to make money.

Maximizing price per share not necessarily same asmaximizing earnings per share


Maximize the wealth of stockholders/ownersMaximize the wealth of stockholders/ownersoror

Maximize price per shareMaximize price per shareWhy? Stockholders invest in the firm because theywant to make money.

Maximizing price per share not necessarily same asmaximizing earnings per share

Cash and Value Relationship between cash (cash flows) and value:

The value of an investment is determined by the futurestream of cash flows that the investment generates forthe investor


Relationship between cash (cash flows) and value:

The value of an investment is determined by the futurestream of cash flows that the investment generates forthe investor

Time value of moneyYou are asked to choose from the following: Receive $100 today Receive $100 one year from now

Would you choose 1 or 2?

Basic principle: A dollar today is worth more than adollar tomorrow because the dollar today can beinvested to grow to an amount greater than a dollartomorrow


You are asked to choose from the following: Receive $100 today Receive $100 one year from now

Would you choose 1 or 2?

Basic principle: A dollar today is worth more than adollar tomorrow because the dollar today can beinvested to grow to an amount greater than a dollartomorrow

The Timeline A timeline is a linear representation of the timing ofpotential cash flows.

Drawing a timeline of the cash flows will help youvisualize the financial problem.

A timeline is a linear representation of the timing ofpotential cash flows.

Drawing a timeline of the cash flows will help youvisualize the financial problem.

The Timeline (continued) Assume that you made a loan to a friend. You will berepaid in two payments, one at the end of each yearover the next two years.

Relation between nominal rate, real rate, and expected inflation rate (1)

Foundation of Finance - I 11

Real vs. Nominal Interest Rates (2)Relation between nominal and real rates:

(1 + rnominal) = (1 + rreal) x (1 + i),

or, rearrangingrreal =

As a principle, if cash flows are in nominal (real)terms, then the discount rate should be thenominal (real) rate of return.


Relation between nominal and real rates:

(1 + rnominal) = (1 + rreal) x (1 + i),

or, rearrangingrreal =

As a principle, if cash flows are in nominal (real)terms, then the discount rate should be thenominal (real) rate of return.

11

1 min

i

r alno

Time Value of Money Basics


Time Value Basics


The Formula for Future ValueFuture Value Number of periods

NrN

N FVIFPVrPVFV ,)1(


Present Value Rate of return ordiscount rate orinterest rate or

growth per period

NrN

N FVIFPVrPVFV ,)1(

The Formula for Future Value The formula lets you convert a current cash flow intoits future value.

This process is called compounding.

What about the reverse process? How do we convertfuture cash flows into their present values?

Definition: The present value of a future cash flow Nperiods from today is the amount, if invested today,will grow to equal that future value


The formula lets you convert a current cash flow intoits future value.

This process is called compounding.

What about the reverse process? How do we convertfuture cash flows into their present values?

Definition: The present value of a future cash flow Nperiods from today is the amount, if invested today,will grow to equal that future value

The Formula for Present ValueFrom before, we know that

Solving for PV, we get Unless otherwisenoted, r is anannual rate.

NrNN FVIFPVrPVFV ,1


Solving for PV, we get Unless otherwisenoted, r is anannual rate.

NrNNN xPVIFFVr

FVPV,)1(

The Formula for Present Value


Multi-period, Find PVFind the present value of $6,000 that occurs at t = 6.The discount rate is 14 percent.

Use PV = FV6/(1+r)6

FV6=6000, N = 6, r = 14, PV = ?


Find the present value of $6,000 that occurs at t = 6.The discount rate is 14 percent.

Use PV = FV6/(1+r)6

FV6=6000, N = 6, r = 14, PV = ?

Multi-period, find FVSuppose you deposit $150 in an account today and theinterest rate is 6 percent p.a.. How much will you havein the account at the end of 33 years?

Use FV33 = PV x (1+r)33

PV=150, N=33,r=6, FV33=?


Suppose you deposit $150 in an account today and theinterest rate is 6 percent p.a.. How much will you havein the account at the end of 33 years?

Use FV33 = PV x (1+r)33

PV=150, N=33,r=6, FV33=?

Multi-period, find rYou deposited $15,000 in an account 22 years agoand now the account has $50,000 in it. What wasthe annual rate of return that you received on thisinvestment?

Use r = (FVN/PV)1/N 1.

PV = 15000, N = 22, FVN = 50000, r = ?


You deposited $15,000 in an account 22 years agoand now the account has $50,000 in it. What wasthe annual rate of return that you received on thisinvestment?

Use r = (FVN/PV)1/N 1.

PV = 15000, N = 22, FVN = 50000, r = ?

Multi-period, find nYou currently have $38,000 in an account that has beenpaying 5.75 percent p.a.. You remember that you hadopened this account quite some years ago with aninitial deposit of $19,000. You forget when the initialdeposit was made. How many years (in fractions) agodid you make the initial deposit?

FVN = PV(1+r)NOr,38,000 = 19,000(1.0575)NSolve for N


You currently have $38,000 in an account that has beenpaying 5.75 percent p.a.. You remember that you hadopened this account quite some years ago with aninitial deposit of $19,000. You forget when the initialdeposit was made. How many years (in fractions) agodid you make the initial deposit?FVN = PV(1+r)N

Or,38,000 = 19,000(1.0575)NSolve for N

Value Additivity Principle


Value additivity principleYou make two investments. The first investmentwill give you $5,500 after one year and the secondinvestment will give you $12,100 after one year. Ifyour required rate of return is 10 percent, what isthe current value of your investments?


You make two investments. The first investmentwill give you $5,500 after one year and the secondinvestment will give you $12,100 after one year. Ifyour required rate of return is 10 percent, what isthe current value of your investments?

2-period, 2 cash flows, find FVProblem: You deposit $5,000 in a bank accounttoday. You make another deposit of $4,000 intothe account at the end of the first year. If the bankpays interest at 6 percent compounded annually,how much will you have in your account at the endof two years, that is, at t = 2?


Problem: You deposit $5,000 in a bank accounttoday. You make another deposit of $4,000 intothe account at the end of the first year. If the bankpays interest at 6 percent compounded annually,how much will you have in your account at the endof two years, that is, at t = 2?

2-period, 2 cash flows, find PVProblem:You want to withdraw $3,200 from your account atthe end of one year and $7,300 at the end of thesecond year. How much should you deposit inyour account today so that you can make thesewithdrawals? Your account pays 6 percent p.a.


Problem:You want to withdraw $3,200 from your account atthe end of one year and $7,300 at the end of thesecond year. How much should you deposit inyour account today so that you can make thesewithdrawals? Your account pays 6 percent p.a.

Calculating the Net Present Value Calculating the NPV of future cash flows allows us toevaluate an investment decision.

Net Present Value compares the present value of cashinflows (benefits) to the present value of cash outflows(costs).

Calculating the NPV of future cash flows allows us toevaluate an investment decision.

Net Present Value compares the present value of cashinflows (benefits) to the present value of cash outflows(costs).

Net Present Value (NPV) of anInvestment Opportunity

Common streams of cash flows (1)


Common streams of cash flows


Wealthy alumnus example


Wealthy alumnus example (Continued)


Wealthy alumnus example For each of the four cases in the wealth alumnusexample, what is the ending balance in theendowment account each year for the next 20 yearsafter making the payment for the year?

For each of the four cases in the wealth alumnusexample, what is the ending balance in theendowment account each year for the next 20 yearsafter making the payment for the year?


Annuity, find PVProblem: You are considering buying a rentalproperty. The yearly rent from this property is$18,000. You expect that the property will yield (i.e.,generate) this rent for the next twenty years afterwhich you will be able to sell it for $250,000. If yourrequired rate of return is 12 percent p.a., what is themaximum amount that you would pay for thisproperty?


Problem: You are considering buying a rentalproperty. The yearly rent from this property is$18,000. You expect that the property will yield (i.e.,generate) this rent for the next twenty years afterwhich you will be able to sell it for $250,000. If yourrequired rate of return is 12 percent p.a., what is themaximum amount that you would pay for thisproperty?

Annuity, find FVProblem: You open an account today with $20,000 and atthe end of each of the next 15 years, you deposit $2,500 init. At the end of 15 years, what will be the balance in theaccount if the interest rate is 7 percent p.a.?


Problem: You open an account today with $20,000 and atthe end of each of the next 15 years, you deposit $2,500 init. At the end of 15 years, what will be the balance in theaccount if the interest rate is 7 percent p.a.?

Annuity, find I/YYou lend your friend $100,000. He will pay you $12,000per year for ten years and a balloon payment at t = 10 of$50,000. What is the interest rate that you arecharging your friend?


You lend your friend $100,000. He will pay you $12,000per year for ten years and a balloon payment at t = 10 of$50,000. What is the interest rate that you arecharging your friend?

Annuity, find PMTNext year, you will start to make 35 deposits of$3,000 per year in your Individual RetirementAccount (so you will contribute from t=1 to t=35).With the money accumulated at t=35, you willthen buy a retirement annuity of 20 years with equalyearly payments from a life insurance company(payments from t=36 to t=55).If the annual rate of return over the entire period is8%, what will be the annual payment of the annuity?


Next year, you will start to make 35 deposits of$3,000 per year in your Individual RetirementAccount (so you will contribute from t=1 to t=35).With the money accumulated at t=35, you willthen buy a retirement annuity of 20 years with equalyearly payments from a life insurance company(payments from t=36 to t=55).If the annual rate of return over the entire period is8%, what will be the annual payment of the annuity?

Annuity Due 1 Up till now, we have dealt with ordinary annuities. For an ordinary annuity, payment occurs at the endof each period.

For an annuity due, payment occurs at the beginningof each period.

The difference becomes clear when we look at time lines.


Up till now, we have dealt with ordinary annuities. For an ordinary annuity, payment occurs at the endof each period.

For an annuity due, payment occurs at the beginningof each period.

The difference becomes clear when we look at time lines.

Consider an annuity that pays $300 per yearfor three years.If ordinary annuity, time line is:

If annuity due, time line is:T = 0

$300

T = 1

$300 $300

T = 2 T = 3


If ordinary annuity, time line is:

If annuity due, time line is:T = 0 T = 1 T = 2 T = 3

T = 0 T = 1

$300

T = 2 T = 3

$300 $300

Is there a relationship between ordinaryannuity and annuity due?Yes !

PV of annuity due= (PV of ordinary annuity) x (1 + r)

FV of annuity due= (FV of ordinary annuity) x (1 + r)

Note: The terms ordinary annuity and regular annuity mean the samething.


Yes !PV of annuity due

= (PV of ordinary annuity) x (1 + r)FV of annuity due

= (FV of ordinary annuity) x (1 + r)

Note: The terms ordinary annuity and regular annuity mean the samething.

ExampleProblem: You have a rental property that you want to rentfor 10 years. Prospective tenant A promises to pay you a rentof $12,000 per year with the payments made at the end ofeach year. Prospective tenant B promises to pay $12,000 peryear with payments made at the beginning of each year.Which is a better deal for you if the appropriate discountrate is 10 percent?


Problem: You have a rental property that you want to rentfor 10 years. Prospective tenant A promises to pay you a rentof $12,000 per year with the payments made at the end ofeach year. Prospective tenant B promises to pay $12,000 peryear with payments made at the beginning of each year.Which is a better deal for you if the appropriate discountrate is 10 percent?

Unequal cash flows 2An asset promises to produce the following series of cashflows. At the end of each of the first three years, $5,000. Atthe end of each of the following four years, $7,000. And,$9,000 each year subsequently. If your required rate ofreturn is 10 percent, how much is this asset worth to you?

Find PV of this series of cash flows.


An asset promises to produce the following series of cashflows. At the end of each of the first three years, $5,000. Atthe end of each of the following four years, $7,000. And,$9,000 each year subsequently. If your required rate ofreturn is 10 percent, how much is this asset worth to you?

Find PV of this series of cash flows.

Special topics Compounding period is less than 1 year Continuous compounding Loan amortization


Compounding period is less than 1 year

Saying that compounding period is less than 1 year isequivalent to saying thatfrequency of compounding is more than once per year


Common examplesCompounding period Compounding frequency

Six-months /semiannual

2

Quarter 4


Quarter 4

Month 12

Day 365

Example (1)Suppose that your bank states that the interest onyour account is eight percent p.a.. However, interestis paid semi-annually, that is every six months ortwice a year.

The 8% is called the stated interest rate.(also called the nominal interest rate)

But, the bank will pay you 4% interest every 6months.


Suppose that your bank states that the interest onyour account is eight percent p.a.. However, interestis paid semi-annually, that is every six months ortwice a year.

The 8% is called the stated interest rate.(also called the nominal interest rate)

But, the bank will pay you 4% interest every 6months.

Example (2)Ok, so we know how much interest is paid every 6months. Over a year, what is the percentage interest youactually earn?

In other words,You want to know the effective annual rate , EAR


Ok, so we know how much interest is paid every 6months. Over a year, what is the percentage interest youactually earn?

In other words,You want to know the effective annual rate , EAR

Example (3)Suppose you deposit $100 into the account today.Account balance at end of 6 months:

100 x 1.04 = 104Account balance at end of 1 year:

104 x 1.04 =108.16Effective interest rate, EAR= (108.16 100)/100 = 0.0816 or 8.16%In other words, the effective annual rate (EAR) is the rate ifcompounded annually will produce the same future value asa stated or nominal rate of r% compounded m times eachyear.


Suppose you deposit $100 into the account today.Account balance at end of 6 months:

100 x 1.04 = 104Account balance at end of 1 year:

104 x 1.04 =108.16Effective interest rate, EAR= (108.16 100)/100 = 0.0816 or 8.16%In other words, the effective annual rate (EAR) is the rate ifcompounded annually will produce the same future value asa stated or nominal rate of r% compounded m times eachyear.

When frequency of compounding is morethan once a year

11rateannualEffective

1

1

m

nm

nm

m

r

m

rPVFV

m

r

FVPVn = number ofyearsm = frequency ofcompounding peryearr = stated interestrate


11rateannualEffective

1

1

m

nm

nm

m

r

m

rPVFV

m

r

FVPVn = number ofyearsm = frequency ofcompounding peryearr = stated interestrate

Annuity with semi-annual compoundingYou would like to accumulate $16,500 over the next 8years. How much must you deposit every six months,starting six months from now, given a 4 percent perannum rate with semiannual compounding?


You would like to accumulate $16,500 over the next 8years. How much must you deposit every six months,starting six months from now, given a 4 percent perannum rate with semiannual compounding?

Problem:stated rate, effective rateYou have decided to buy a car whose price is $45,000.The dealer offers to finance the entire amount andrequires 60 monthly payments of $950 per month.What are the yearly stated and effective interest ratesfor this financing?

Answer:stated = 9.723 % p.a.effective = 10.168 % p.a.


You have decided to buy a car whose price is $45,000.The dealer offers to finance the entire amount andrequires 60 monthly payments of $950 per month.What are the yearly stated and effective interest ratesfor this financing?

Answer:stated = 9.723 % p.a.effective = 10.168 % p.a.

Problem:The stated interest rate for a bank account is 7 percentand interest is paid semi-annually. How many yearswill it take you to double your money in this account?


Continuous compounding (1)Continuous compounding is a special case ofcompounding more often than once a year. In thiscase, the compounding frequency is infinity.

FV = PV x erNPV = FV x e-rN

e is the exponential function (that appears on yourfinancial calculator as [ex]), r is the stated interestrate, and N is the number of years.Foundations of Finance - I 56

Continuous compounding is a special case ofcompounding more often than once a year. In thiscase, the compounding frequency is infinity.FV = PV x erNPV = FV x e-rN

e is the exponential function (that appears on yourfinancial calculator as [ex]), r is the stated interestrate, and N is the number of years.

Continuous compounding (2)If the stated interest rate is r p.a., the effectiveinterest rate with continuous compounding isgiven by

effective interest rate = er - 1

Example: Suppose the stated interest rate is 9percent p.a., then, with continuous compounding,the effective interest rate is

e0.09 - 1 = 1.09417 - 1 = 0.09417 or 9.417%Foundations of Finance - I 57

If the stated interest rate is r p.a., the effectiveinterest rate with continuous compounding isgiven by

effective interest rate = er - 1

Example: Suppose the stated interest rate is 9percent p.a., then, with continuous compounding,the effective interest rate is

e0.09 - 1 = 1.09417 - 1 = 0.09417 or 9.417%

Continuous compounding (3)Problem BNM Bank pays a stated interest of 4 percentper year and utilizes continuous compounding. Youdeposit $1,800 into this account. What will be your accountbalance in 11 years?

Verify that answer is $2,794.87

Problem ZXC bank pays a stated interest of 5 percent p.a. and utilizes continuous compounding. How long will ittake you to double your money in this account? (Youranswer can contain fractional years.)

Verify that answer is 13.863 years


Problem BNM Bank pays a stated interest of 4 percentper year and utilizes continuous compounding. Youdeposit $1,800 into this account. What will be your accountbalance in 11 years?

Verify that answer is $2,794.87

Problem ZXC bank pays a stated interest of 5 percent p.a. and utilizes continuous compounding. How long will ittake you to double your money in this account? (Youranswer can contain fractional years.)

Verify that answer is 13.863 years

Loan AmortizationAmortization is the process of separating apayment into two parts:The interest paymentThe repayment of principal

Note:Interest payment decreases over timePrincipal repayment increases over time


Amortization is the process of separating apayment into two parts:The interest paymentThe repayment of principal

Note:Interest payment decreases over timePrincipal repayment increases over time

Example of loan amortization 1

You have borrowed $8,000 from a bank and havepromised to repay the loan in five equal yearlypayments. The first payment is at the end of thefirst year. The interest rate is 10 percent. Draw upthe amortization schedule for this loan.

Amortization schedule is just a table that shows howeach payment is split into principal repayment andinterest payment.


You have borrowed $8,000 from a bank and havepromised to repay the loan in five equal yearlypayments. The first payment is at the end of thefirst year. The interest rate is 10 percent. Draw upthe amortization schedule for this loan.

Amortization schedule is just a table that shows howeach payment is split into principal repayment andinterest payment.

Example of loan amortization 21) Compute periodic payment.Verify that the payment= 2,110.38Amortization for first yearInterest payment = 8000 x 0.1 = 800Principal repayment= 2,110.38 800 = 1310.38Immediately after first payment, the principalbalance is = 8000 1310.38 = 6,689.62


1) Compute periodic payment.Verify that the payment= 2,110.38Amortization for first yearInterest payment = 8000 x 0.1 = 800Principal repayment= 2,110.38 800 = 1310.38Immediately after first payment, the principalbalance is = 8000 1310.38 = 6,689.62

Example of loan amortization 3Amortization for second yearInterest payment = 6689.62 x 0.1 = 668.96(using the new balance!)Principal repayment= 2,110.38 668.96 = 1441.42Immediately after second payment, the principalbalance is = 6,689.62 1441.42 = 5,248.20

Verify the entire schedule (on following slide)Foundations of Finance - I 62

Amortization for second yearInterest payment = 6689.62 x 0.1 = 668.96(using the new balance!)Principal repayment= 2,110.38 668.96 = 1441.42Immediately after second payment, the principalbalance is = 6,689.62 1441.42 = 5,248.20

Verify the entire schedule (on following slide)

Verify the amortization scheduleYear

Beg.Balance Payment Interest Principal

End.Balance

1 8,000.00 2,110.38 800.00 1,310.38 6,689.62

2 6689.62 2,110.38 668.96 1,441.42 5,248.20


2 6689.62 2,110.38 668.96 1,441.42 5,248.20

3 5248.20 2,110.38 524.82 1,585.56 3,662.64

4 3662.64 2,110.38 366.26 1,744.12 1,918.53

5 1918.53 2,110.38 191.85 1,918.53 0.00

Introduction & Time Value of Money

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