fm topic 5 time value of money

Topic 5

Time Value of Money

Learning Objectives To elaborate the concept of time value of money and time line. To explain the differences between simple interest and compound

interest. To elucidate the future value of single amount. To explain the effects of frequent compounding towards future value

amount. To clarify the differences between effective and nominal interest rate. To explain the present value for an amount in the future To illustrate the calculation of future value and present value for both

ordinary annuity and annuity due. To calculate the value of uneven cash flows. To discuss on perpetuity. To elaborate on the application of time value of money concept in

loan amortization.

Generally, receiving $1 today is worth more than $1 in the future. This is due to opportunity costs.

The opportunity cost of receiving $1 in the future is the interest we could have earned if we had received the $1 sooner.

Today Future

If we can measure this opportunity cost, we can:

Translate $1 today into its equivalent in the future (compounding).

Translate $1 in the future into its equivalent today (discounting).

?Today Future

Today

?Future

Significance of the time value of money

Time value of money is important in understanding financial management.

It should be considered for making financial decisions.

It can be used to compare investment alternatives and to solve problems involving loans, mortgages, leases, savings, and annuities.

Simple Interest

Interest is earned only on principal.

Example: Compute simple interest on $100 invested at 6% per year for three years. 1st year interest is $6.00

2nd year interest is $6.00

3rd year interest is $6.00

Total interest earned: $18.00

Compound Interest

Compounding is when interest paid on an investment during the first period is added to the principal; then, during the second period, interest is earned on the new sum (that includes the principal and interest earned so far).

Is the amount a sum will grow to in a certain number of years when compounded at a specific rate.

Compounding : process of determining the Future Value (FV) of cash flow.

Compounded amount = Future Value (beginning amount plus interest earned. )

Compound Interest

Example: Compute compound interest on $100 invested at 6% for three years with annual compounding. 1st year interest is $6.00 Principal now is $106.00

2nd year interest is $6.36 Principal now is $112.36

3rd year interest is $6.74 Principal now is $119.11

Total interest earned: $19.10

• Suppose you invest $100 for one year at 5% per year. What is the future value in one year?– Interest = 100(.05) = 5– Value in one year = principal + interest = 100 + 5 = 105– Future Value (FV) = 100(1 + .05) = 105

• Suppose you leave the money in for another year. How much will you have two years from now?– FV = 100(1.05)(1.05) = 100(1.05)2 = 110.25

• FV = PV(1 + r)t

– FV = future value– PV = present value– r = period interest rate, expressed as a decimal– t = number of periods

• Future value interest factor = (1 + r)t

Future Value Future Value is the amount a sum will grow to in a certain number of years

when compounded at a specific rate.

Two ways to calculate Future Value (FV): by using Manual Formula or Using Table.

Manual Formula Table

FVn = PV (1 + r)n FVn = PV (FVIFi,n)n

Where :

FVn = the future of the investment at the end of “n” years

r = the annual interest (or discount) rate

n = number of years

PV= the present value, or original amount invested at the beginning of the first year

FVIF=Futurevalueinterestfactororthecompoundsum$1

Future Value - single sums

If you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?

Mathematical Solution:FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = $106

0 1

PV = -100 FV = ???

Future Value - single sums

If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .06, 5 ) (use FVIF table, or)FV = PV (1 + i)n

FV = 100 (1.06)5 = $133.82

0 5

PV = -100 FV = ???

Compound Interest With Non-annual Periods

Non-annual periods : not annual compounding but occur semiannually, quarterly, monthly or daily…

If semiannually compounding : FV = PV (1 + i/2)n x 2 or FVn= PV (FVIFi/2,nx2)

If quarterly compounding : FV = PV (1 + i/4)n x 4 or FVn= PV (FVIFi/4,nx4)

If monthly compounding : FV = PV (1 + i/12)n x 12 or FVn= PV (FVIFi/12,nx12)

If daily compounding : FV = PV (1 + i/365)n x 365 or FVn= PV (FVIFi/365,nx365)

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)FV = PV (1 + i/m) m x n

FV = 100 (1.015)20 = $134.68

0 20

PV = -100 FV = 134.68

Future Value - single sumsIf you deposit $100 in an account earning 6% with

quarterly compounding, how much would you have in the account after 5 years?

Mathematical Solution:FV = PV (FVIF i, n )FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)FV = PV (1 + i/m) m x n

FV = 100 (1.005)60 = $134.89

0 60

PV = -100 FV = 134.89

Future Value - single sumsIf you deposit $100 in an account earning 6% with

monthly compounding, how much would you have in the account after 5 years?

Present Value Present value reflects the current value of a future payment or

receipt. How much do I have to invest today to have some amount in the

future? Finding Present Values(PVs)= discountingManual Formula Table PVn = FV/ (1 + r)n PVn = FV (PVIFi,n)n

Where :FVn = the future of the investment at the end of “n” yearsr = the annual interest (or discount) rate n = number of yearsPV= the present value, or original amount invested at the beginning of

the first yearPVIF=Present Value Interest Factor or the discount sum$1

Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 1 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34

Present Value - single sumsIf you receive $100 one year from now, what is the PV

of that $100 if your opportunity cost is 6%?

Mathematical Solution:PV = FV (PVIF i, n )PV = 100 (PVIF .06, 5 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73

Present Value - single sumsIf you receive $100 five years from now, what is the

PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:PV = FV (PVIF i, n )PV = 1000 (PVIF .07, 15 ) (use PVIF table, or)PV = FV / (1 + i)n

PV = 1000 / (1.07)15 = $362.45

Present Value - single sumsWhat is the PV of $1,000 to be received 15 years from

now if your opportunity cost is 7%?

• Suppose you need $10,000 in one year for the down payment on a new car. If you can earn 7% annually, how much do you need to invest today?– PV = 10,000 / (1.07)1 = 9,345.79

• You want to begin saving for your daughter’s college education and you estimate that she will need $150,000 in 17 years. If you feel confident that you can earn 8% per year, how much do you need to invest today?– PV = 150,000 / (1.08)17 = 40,540.34

• Your parents set up a trust fund for you 10 years ago that is now worth $19,671.51. If the fund earned 7% per year, how much did your parents invest?– PV = 19,671.51 / (1.07)10 = 9,999.998 = 10,000

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Finding i1. At what annual rate would the following have to be invested; $500 to grow to RM1183.70 in 10 years.FVn = PV (FVIF i,n )1183.70 = 500 (FVIF i,10 )1183.70/500 = (FVIF i,10 )2.3674 = (FVIF i,10 ) refer to FVIF table

i = 9%2. If you sold land for $11,439 that you bought 5 years ago for $5,000,

what is your annual rate of return? FV = PV (FVIF i, n )11,439 = 5,000 (FVIF ?, 5 ) 11,439/ 5,000= (FVIF ?, 5 ) 2.3866 = (FVIF ?, 5 )i = .18

Finding n1. How many years will the following investment takes? $100 togrow to $672.75 if invested at 10% compounded annuallyFVn = PV (FVIF i,n )672.75 = 100 (FVIF 10%,n )672.75/100 = (FVIF 10%,n )6.7272 = (FVIF 10%,n ) refer to FVIF table

n = 20 years2. Suppose you placed $100 in an account that pays 9% interest,

compounded annually. How long will it take for your account to grow to $514?

FV = PV (1 + i)n

514 = 100 (1+ .09)N

514/100 = (FVIF 9%,n )5.14 = (FVIF 9%,n ) refer to FVIF table

n = 19 years

FV = PV (FVIF i, n )

11,933 = 5,000 (FVIF ?, 5 )

2.3866 = (FVIF ?, 5 ) can’t find

FV = PV(1 + r)tr = (FV / PV)1/t – 1FV = PV (1 + i)n

11,933 = 5,000 (1+ i)5 11,933 / 5,000 = (1+i)5

2.3866 = (1+i)5

(2.3866)1/5 = (1+i) 1.19 = 1+i i = .19

Finding i and nIf you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

Finding i and n Suppose you placed $100 in an account that pays

9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

FV = PV (1 + i)n

500 = 100 (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)1.60944 = .007968 NN = 202 months

– FV = PV(1 + r)t

– t = ln(FV / PV) / ln(1 + r)

• You are looking at an investment that will pay $1,200 in 5 years if you invest $1,000 today. What is the implied rate of interest?– r = (1,200 / 1,000)1/5 – 1 = .03714 = 3.714%

• Suppose you are offered an investment that will allow you to double your money in 6 years. You have $10,000 to invest. What is the implied rate of interest?– r = (20,000 / 10,000)1/6 – 1 = .1225 = 12.25%

• You want to purchase a new car, and you are willing to pay $20,000. If you can invest at 10% per year and you currently have $15,000, how long will it be before you have enough money to pay cash for the car?

– t = ln(20,000 / 15,000) / ln(1 + 0.1) =3.02 years

5C-25

Hint for single sum problems: In every single sum present value and future

value problem, there are four variables:FV, PV, i and n.

When doing problems, you will be given three variables and you will solve for the fourth variable.

Keeping this in mind makes solving time value problems much easier!

Handy Rule of Thumb

• Rule of 72 can estimate how long it takes to double a sum of money– Time to double money = 72 / (interest rate per year)

• If interest rate = 9% per year, it will take 8 years to double the money– Time to double money = 72 / 9% = 8 years

• If the time taken to double the money is 8 years, the interest rate is 9% per year– Interest rate per year = 72 / 8 years = 9%

5C-27

5C-28

5-29

Future Value of a Mixed Stream

If the firm expects to earn at least 8% on its investments, how much will it accumulate by the end of year 5 if it immediately invests these cash flows when they are received?This situation is depicted on the following time line.

• Suppose you invest $500 in a mutual fund today and $600 in one year. If the fund pays 9% annually, how much will you have in two years?

– FV2 = 500(1.09)2 + 600(1.09)1 = 594.05 + 654.00 = 1,248.05

How much will you have in 5 years if you make no further deposits?

– FV5 = 500(1.09)5 + 600(1.09)4 = 769.31 + 846.95 = 1,616.26

• Suppose you plan to deposit $100 into an account in one year and $300 into the account in three years. How much will be in the account in five years if the interest rate is 8%?

– FV5 = 100 (1.08)4 + 300(1.08)2 = 136.05 + 349.92 = 485.97

6C-30

5-31

Present Value of a Mixed Stream

If the firm must earn at least 9% on its investments, what is the most it should pay for this opportunity?This situation is depicted on the following time line.

• You are considering an investment that will pay you $1,000 in one year, $2,000 in two years and $3,000 in three years. If you want to earn 10% on your money, how much would you be willing to pay?

– PV = 1,000 / (1.10)1 + 2,000 / (1.10)2 +3,000 / (1.10)3 = 4,815.93• Your broker calls you and tells you that he has this great

investment opportunity. If you invest $100 today, you will receive $40 in one year and $75 in two years. If you require a 15% return on investments of this risk, should you take the investment?

– PV = 40 / (1.15)1 + 75 / (1.15)2 = 91.49, reject this investment.

• You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%? PV = 25,000 / (1.12)40 + 25,000 / (1.12)41 +25,000 / (1.12)42 + 25,000 / (1.12)43 +25,000 / (1.12)44 = 1,084.71

6C-32

Compounding and DiscountingCash Flow Streams

Two types of annuity: ordinary annuity and annuity due.ordinary annuity: a sequence of equal cash flows, occurring

at the end of each period. Annuity due: annuity payment occurs at the beginning of

the period rather than at the end of the period.

0 1 2 3 4

Annuities

Mathematical Solution:FV = PMT (FVIFA i, n )FV = 1,000 (FVIFA .08, 3 ) (use FVIFA table, or)

FV = PMT (1 + i)n - 1 i

FV = 1,000 (1.08)3 - 1 = $3246.40 .08

Future Value - annuityIf you invest $1,000 each year at 8%, how much

would you have after 3 years?

Mathematical Solution:PV = PMT (PVIFA i, n )PV = 1,000 (PVIFA .08, 3 ) (use PVIFA table, or)

1PV = PMT 1 - (1 + i)n

i

1PV = 1000 1 - (1.08 )3 = $2,577.10

.08

Present Value - annuityWhat is the PV of $1,000 at the end of each of the

next 3 years, if the opportunity cost is 8%?

Perpetuities

Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

You can think of a perpetuity as an annuity that goes on forever.

So, the PV of a perpetuity is very simple to find:

Present Value of a Perpetuity

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?

PMT $10,000 i .08

= $125,000

PV = =

Ordinary Annuity vs.

Annuity Due

$1000 $1000 $1000

4 5 6 7 8

Earlier, we examined this “ordinary” annuity:

Using an interest rate of 8%, we find that:

The Future Value (at 3) is $3,246.40.The Present Value (at 0) is $2,577.10.

1000 1000 1000

What about this annuity?

Same 3-year time line,Same 3 $1000 cash flows, butThe cash flows occur at the beginning

of each year, rather than at the end of each year.

This is an “annuity due.”

1000 1000 1000

Future Value - annuity due If you invest $1,000 at the beginning of each of the next 3 years at 8%, how much would you have at the

end of year 3?

Mathematical Solution: Simply compound the FV of the ordinary annuity one more period:

FV = PMT (FVIFA i, n ) (1 + i) FV = 1,000 (FVIFA .08, 3 ) (1.08) (use FVIFA table, or)

FV = PMT (1 + i)n - 1 i

FV = 1,000 (1.08)3 - 1 = $3,506.11 .08

Present Value - annuity dueMathematical Solution: Simply compound the FV of the

ordinary annuity one more period:

PV = PMT (PVIFA i, n ) (1 + i) PV = 1,000 (PVIFA .08, 3 ) (1.08) (use PVIFA table, or)

1PV = PMT 1 - (1 + i)n

i

1PV = 1000 1 - (1.08 )3 = $2,783.26

.08

(1 + i)

(1.08)

• Suppose you win the $10 million sweepstakes. The money is paid in equal annual end-of-year installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

• Suppose you begin saving for your retirement by depositing $2,000 per year in a savings account. If the interest rate is 7.5%, how much will you have in 40 years?

6C-45

29.150.124,50.05

)05.0(111

333,333.33r

r)(111

CPV30t

454,513.040.075

10.075)(12,000r

1r)(1CFV40t

• Suppose you borrow $2,000 at 5%, and you are going to make annual payments of $734.42. How long will you take to pay off the loan?

To pay your children’s education, you wish to have accumulated RM25,000 at the end of 15 years. To do this, you plan to deposit an equal amount into the bank at the end of each year. If the bank is willing to pay 7% compounded annually, how much must you deposit each year to obtain your goal?FVn = PMT (FVIFA7%,15) RM25,000 = PMT (FVIFA7%,15) RM25,000 = PMT(25.129)

Thus, PMT = RM994.876C-46

3t2,0000.05

0.05)(111

734.42r

r)(111

CPVtt

Annual Percentage Rate (APR)• This is the annual rate that is quoted by law.

• By definition, APR = period rate times the number of periods per year.

– Period rate = APR / number of periods per year

• . What is the APR if the monthly rate is 0.5%?– 0.5(12) = 6%

• What is the APR if the semiannual rate is 0.5%?– 0.5(2) = 1%

• What is the monthly rate if the APR is 12% with monthly compounding?– 12 / 12 = 1%

6C-48

Effective Annual Rate (EAR)

Which is the better loan: 8% compounded annually, or 7.85% compounded quarterly? We can’t compare these nominal (quoted)

interest rates, because they don’t include the same number of compounding periods per year!

We need to calculate the EAR (or Annual Percentage Yield (APY)

Effective Annual Rate(EAR)

Find the APY for the quarterly loan:

The quarterly loan is more expensive than the 8% loan with annual compounding!

EAR= ( 1 + ) m - 1quoted ratem

EAR = ( 1 + ) 4 - 1

EAR = .0808, or 8.08%

.07854

Making Decisions using EAR

• You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?

– First account:• EAR = (1 + .0525/365)365 – 1 = 5.39%

– Second account:• EAR = (1 + .053/2)2 – 1 = 5.37%

• Which account should you choose and why?

Amortized Loans

Loans paid off in equal installments over time are called amortized loans.

Example: Home mortgages, auto loans. Reducing the balance of a loan via annuity payments is

called amortizing. The periodic payment is fixed. However, different

amounts of each payment are applied towards the principal and interest.

With each payment, you owe less towards principal. As a result, amount that goes toward interest declines with every payment (as seen in Figure 5-4).

If you want to finance a new machinery with a purchase price of $6,000 at an interest rate of 15% over 4 years, what will your annual payments be?

Finding Payment: Payment amount can be found by solving for PMT using PV of annuity formula.

PV of Annuity = PMT {1 – (1 + r)–4}/r

6,000 = PMT {1 – (1 + .15)–4}/.15

6,000 = PMT (2.855)

PMT = 6,000/2.855 = $2,101.58 PVIFA = 6,000=PMT (PVIFA15%,4)

6,000=PMT(2.855)

PMT = 6,000/2.855 = $2,101.58