Time Value of Money Lecture

Time Value of Money

JQY

Agenda

• Time Lines

• Future Values

• Present Values

• Solving for Interest Rate and Time

• Future Value of an Annuity

• Present Value of an Annuity

• Perpetuities

• Uneven Cash Flow Streams

• Semiannual and Other Compounding Periods

• Comparison of Different types of interest rates

• Fractional Time Periods

• Amortized Loans

• Amortization

What is Time Value?

• We say that money has a time value because that money can be invested with the expectation of earning a positive rate of return

• In other words, “a dollar received today is worth more than a dollar to be received tomorrow”

• That is because today’s dollar can be invested so that we have more than one dollar tomorrow

If you have P10,000 today, and you deposit it in the bank, how much will you most likely receive in 10 years?

a. P9,500

b. P14,000

c. P10,000

a. P9,500

b.P14,000 c. P10,000

Timelines

0 1 2 3 4 5

PV FV

Today

An important tool used in the time value of money analysis

A graphical representation used to show the timing of cash flows

A timeline is a graphical device used to clarify the timing of the cash flows for an investment

Each tick represents one time period

Future Value

• The value of an asset or cash at a specified date in the future that is equivalent in value to a specified sum today

• Terms: PV – present value, or beginning amount in your account

i – interest rate the bank pays on the account per year

INT – amount of interest you earn during the year (aka discount rate, opportunity cost rate)

FV – future value or ending amount of your account at the end of n years

n – number of periods involved in the analysis

Future Value

• Simple Annual Interest

Q: Today, Peter invested P1,000 for 5 years with simple annual interest of 10%. How much is its future value?

A: FV = PV x [1 + (i*n)]

FV = P1,000 x [1 + (10%*5)]

FV = P1,000 x 1.5

FV = P1,500

Future Value • Interest compounded

Q: Today, Peter invested P100 for 3 years at 10%, compounded annually. How much is its future value?

A: FV = PV (1+i)ⁿ 0 1 2 3

100 FV = ?

After 1 year : FV = PV (1+i)^1 = 100 (1+10%)^1 = 110

After 2 years: FV = PV (1+i)^2 = 100 (1+10%)^2 = 121

After 3 years: FV = PV (1+i)^3 = 100 (1+10%)^3 = 133.10

The Magic of Compounding

• In 1898, USA bought the Philippines from Spain for $20 million

• This happened about 114 years ago, so 5% per year could be earned, the value of the Philippines now (in 2012) would be approximately:

If they could have earned 10% per year, the Philippines would

have been worth:

20m (1.05)114 = 5,207,269,916

20m (1.10)114 = 1,046,637,049,000

Agenda

• Future Value

• Present Value • Annuities

• Rates of Return

• Amortization

If you need to have P10,000 in 10 years, how much will you likely

have to invest today?

a. P7,000

b. P10,000

c. P12,000

a.P7,000 b. P10,000

c. P12,000

Present Value

• The value today of a future cash flow or series of cash flows.

• Represents the amount that needs to be invested to achieve some desired future value.

PV

FV

i

N

N

1

Present Value: An Example

• Suppose that your five-year old daughter has just announced her desire to attend college. After some research, you determine that you will need about $100,000 on her 18th birthday to pay for four years of college. If you can earn 8% per year on your investments, how much do you need to invest today to achieve your goal?

• N = 13; I = 8%; PV = ?; PMT = 0; FV = 100,000

PV

100 000

108769 79

13

,

.$36, .

Solving for Interest and Time

• I = (FV/PV)^1/n – 1

• Sample problem

You can buy a security at a price of $78.35, and it will pay you $100 after 5 years. How much is the interest rate you’d earn if you bought the security?

I = (100/78.35)^(1/5) – 1 = 5%

Solving for Interest and Time

• N = ln (FV/PV) / ln (1+i)

• Sample problem

Mr. Amos invested P60,000 in stocks at a 10% interest rate compounded semi-annually. How many years did it take Mr. Amos for his investment to reach P100,000?

N = ln (100,000/60,000) / ln (1 + 5%)

N = 10.47 / 2 = 5.24 years

Annuities • An annuity is a series of payments of an equal amount at

fixed intervals for a specified number of periods.

• Annuities are very common: – Rent – Mortgage payments – Car payment – Pension income

• The timeline shows an example of a 5-year, $100 annuity

• Annuity = equal PMT

0 1 2 3 4 5

100 100 100 100 100

Annuities

• Ordinary (Deferred) Annuity

– An annuity whose payments occur at the end of each period.

• Annuity Due

– An annuity whose payments occur at the beginning of each period.

0 1 2 3 4 5

100 100 100 100 100

100 100 100 100 100 5-period Annuity Due

5-period Regular Annuity

Future Value of an Ordinary Annuity

• Mary deposited P100 at the end of each year for 3 years in a savings account that pays 5% interest per year. How much will she have at the end of three years?

• Fva = 100 {[(1+5%)^3 – 1] / 5%}

• Fva = 315.25

Future Value of an Annuity Due

• Mary deposited P100 at the beginning of each year for 3 years in a savings account that pays 5% interest per year. How much will she have at the end of three years?

• Fvad = 100 {[(1+5%)^3 – 1] / 5%} (1+5%) = 331.01

Present Value of an Ordinary Annuity

• Mary deposited P100 at the end of each year for 3 years in a savings account that pays 5% interest per year. How much is the present value of her payments?

• Pva = 100 [1 – (1/(1+5%)^3) / 5%]

• Pva = 272.32

Present Value of an Annuity Due

• Mary deposited P100 at the beginning of each year for 3 years in a savings account that pays 5% interest per year. How much is the present value of her payments?

• Pvad = {100 [1 – [1/(1+5%)^(3-1)] / 5%} + 100

• Pvad = 285.94

Ordinary Annuity – Solving for Payment when FV is known

• Harold wants to accumulate P50,000 at the end of 5 years. How much should he pay every year assuming that the interest is 5%, and the first payment will be made at the end of the year?

• PMT = 50,000 / [(1+5%)^5 – 1] / 5% = 9,048.74

Annuity Due – Solving for Payment when FV is known

• Harold wants to accumulate P50,000 at the end of 5 years. How much should he pay every year assuming that the interest is 5%, and the first payment will be made at the beginning of the year?

• PMTad = 50,000 / {[(1+5%)^5 – 1] / 5%} (1+5%)

• PMTad = 8,617.85

Ordinary Annuity – Solving for Payment when PV is known

• How much should Sharon pay every year for 5 years assuming that the interest is 5%, and the first payment will be made at the end of the year, if the present value is 10,000?

• PMT = 50,000 / [(1 – (1/(1+5%)^5 )/5%] / 5% = 2,309.75

Annuity Due – Solving for Payment when PV is known

• How much should Sharon pay every year for 5 years assuming that the interest is 5%, and the first payment will be made at the beginning of the year, if the present value is 10,000?

• PMTad = 10,000 / {[(1 – (1/(1+5%)^(5-1) )/5%] / 5%} + 1

• PMTad = 2,199.76

Ordinary Annuity – Solving for N when FV is known

• Sharon plans to save P100 per year (first payment at end of the year). Assuming that the interest is 5%, how many years does it take for Sharon to accumulate 1,000?

• N = ln[1 – (1,000/-100)5%] / ln (1+5%) = 8.31 years

-

Annuity Due – Solving for N when FV is known

• Sharon plans to save P100 per year (first payment at beginning of the year). Assuming that the interest is 5%, how many years does it take for Sharon to accumulate 1,000?

• Nad = {ln[(1,000 x 5%) / (100 x 1.05)] + 1} / ln (1+5%)

• Nad = 7.98 years

Ordinary Annuity – Solving for N when PV is known

• Sharon plans to save P100 per year (first payment at end of the year), and the present value if P1,000 Assuming that the interest is 5%, solve for N:

• N = - ln[1 – (1000/100) 5%] / ln (1+5%) = 14.207 years

Annuity Due – Solving for N when PV is known

• Sharon plans to save P100 per year (first payment at the beginning of the year), and the present value if P1,000 Assuming that the interest is 5%, solve for N:

• Nad = {- ln[1 + 5% (1 – (1000/100)] / ln (1+5%)} + 1

• Nad = 13.25 years

Ordinary Annuity – Solving for I when FV and PV are known

• Can only be calculated through a trial and error process unless financial calculator is used.

• However, an approximate equation can be solved, provided that all inputs in the equation below is known:

i = {{Annual PMT + [(FV – PV)/Annual N]} / [(40% x FV) + (60% x PV)]}}

Ordinary Annuity – Solving for I when FV and PV are known

Belle has 1,000 today. She plans to make an investment where she pays 50 annually at the end of the year. She expects to receive 1,500 at the end of 10 years. How much is the interest rate that is required for this investment so that Belle will receive 1500 after 10 years?

i = {{50 + [(1,500 – 1,000)/10]} / [(40% x 1,500) + (60% x 1,000)]}}

Ordinary Annuity: Solving for I when FV is known (CALC)

• Frederick will be contributing P10,000 to his girlfriend’s account as a sign of his “love” every month starting next month. He calculates that in 10 years, there will be P1,500,000 in his girlfriend’s account if she does not touch any of the deposit. How much interest is the account earning if Frederick’s calculations are correct?

Annuity Due: Solving for I when FV is known (CALC)

• Frederick contributes P10,000 to his girlfriend’s account as a sign of his “love” every month starting at the beginning of this month. He calculates that in 10 years, there will be P1,500,000 in his girlfriend’s account if she does not touch any of the deposit. How much interest is the account earning if Frederick’s calculations are correct?

Summary • Rules regarding annuity, ceteris paribus:

Ordinary Annuity Annuity Due

Future Value Lower Higher

Present Value Lower Higher

Payment Higher Lower

N Higher Lower

Interest Higher Lower

Perpetuities

• A stream of equal payments expected to continue forever

• PV (Perpetuity) = Payment / Interest rate

• Suppose each consol (British government perpetual bonds) promised to pay $100 in perpetuity, if the discount rate or opportunity cost rate is 5% and 10%:

• PV (Perpetuity) = 100/5% = $2,000

• PV (Perpetuity) = 100/10% = $1,000

Uneven Cash Flow Stream

• A series of cash flows in which the amount varies from one period to the next

• PMT = equal cash flows coming at regular intervals

• CF = uneven cash flows

Uneven Cash Flows: An Example (PV)

• Assume that an investment offers the following cash flows. If your required return is 7%, what is the maximum price that you would pay for this investment?

0 1 2 3 4 5

100 200 300

PV

100

107

200

107

300

10751304

1 2 3. . .

.

Uneven Cash Flows: An Example (FV)

• Terminal Value = The future value of an uneven cash flow stream

• Suppose that you were to deposit the following amounts in an account paying 5% per year. What would the balance of the account be at the end of the third year?

0 1 2 3 4 5

300 500 700

FV 300 105 500 105 700 1555752 1

. . , .

Non-annual Compounding

• We could assume that interest is earned semi-annually, quarterly, monthly, daily, or any other length of time

• The only change that must be made is to make sure that the rate of interest is adjusted to the period length

Non-annual Compounding (cont.)

• Suppose that you have $1,000 available for investment. After investigating the local banks, you have compiled the following table for comparison. In which bank should you deposit your funds?

Bank Interest Rate Compounding

First National 10% AnnualSecond National 10% MonthlyThird National 10% Daily

Non-annual Compounding (cont.)

• We can find the FV for each bank as follows:

FV 1 000 110 11001

, . ,

FV

1 000 1

010

121104 71

12

,.

, .

FV

1 000 1

010

365110516

365

,.

, .

First National Bank:

Second National Bank:

Third National Bank:

Obviously, you should choose the Third National Bank

Continuous Compounding

• There is no reason why we need to stop increasing the compounding frequency at daily

• We could compound every hour, minute, or second

• We can also compound every instant (i.e., continuously):

F Pe rt

Here, F is the future value, P is the present value, r is the annual rate of

interest, t is the total number of years, and e is a constant equal to

about 2.718

Continuous Compounding (cont.)

• Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. What is the future value of your $1,000 investment?

F e 1 000 1105170 10 1

, , ..

This is even better than daily compounding

The basic rule of compounding is: The more frequently interest is

compounded, the higher the future value

Continuous Compounding (cont.)

• Suppose that the Fourth National Bank is offering to pay 10% per year compounded continuously. If you plan to leave the money in the account for 5 years, what is the future value of your $1,000 investment?

F e 1 000 1 648 720 10 5

, , ..

Different Rates • Nominal (Quoted, Stated, Annual Percentage) Interest Rate

– The rate charged by banks and other financial institutions

– For example, 6% compounded quarterly, 5% compounded monthly

• Effective (Equivalent Annual) Rate – The annual rate of interest actually being earned

– EAR = (1+iNOM/m)^m – 1

– If payment is only once a year, EAR = nominal rate

• Periodic Rate – Rate charged by a lender or paid by a borrower each period

– iPER = iNOM/m

– If Nominal rate is quoted at 18%, payable monthly, periodic rate is 18%/12 or 1.5%

– If payment is only once a year, Nominal rate = periodic rate.

Landbank charges 10% interest rate, compounded quarterly. How much is the nominal, EAR, and periodic rate?

Nominal = 10%, EAR = 10.38%, Periodic = 2.5%

Fractional Time Periods • If you deposit $100 in a bank that uses daily

compounding and pays a nominal rate of 10% with a 365 days, how much is the FV after 9 months?

N = 365 x 9/12 ; I = 10% / 365 ; PV = 100

FV = PV x [(1 + i)^n] = 100 x [(1 + 0.000273973)^274] = 107.79

• You borrow $100 that charges 10% simple interest but you borrow only for 274 days. How much interest do you owe?

Interest owed = 100 x 10% x 274/365 = $7.51

Amortized Loans

• A loan that is repaid in equal payments over its life.

• If a firm borrows $1,000 and the loan is to be repaid in 3 equal payments at the end of each of the next three years, and the lender charges 6% on the loan balance, how much is the periodic payment, and construct the loan amortization schedule.

• N = 3; I = 6%; PV = 1000; PMT = ?; FV = 0

• PMT = 1,000 / [(1 – (1/(1+6%)^3)/6%] / 6% = 374.11

Amortization Schedule

Payment Interest Principal Repayment Balance

Beg 1,000.00

Y1 374.11 60.00 314.11 685.89

Y2 374.11 41.15 332.96 352.93

Y3 374.11 21.18 352.93 (0.00)

Balloon Loan

• A long-term loan, often a mortgage, that has one large payment due upon maturity.

• Advantage: very low interest payments, requiring very little capital outlay during the life of the loan.

• Disadvantage: An undisciplined borrower will be in trouble because he has to make a large single payment upon maturity.

Partial Amortization: Balloon Loans

• A house is worth $200,000, and a bank agrees to lend the potential home buyer $175,000 secured by a mortgage on the house. However, the buyer only has $5,000 and he is unable to make the full $25,000 downpayment. The seller may take a note of 20k, 8% interest rate and payments at the end of the year based on a 20 year amortization schedule but with loan maturing at the end of the 10th year.

• N = 20; I = 8%; PV = 20,000; PMT = ?

• Annual PMT of the note = 2,037.04

Additional Problem:

• To save money for a new house, you want to begin contributing money to a brokerage account. Your plan is to make 40 contributions to the brokerage account. Each contribution will be for $1,500. The first contribution will occur today and then every quarter, you will contribute another $1,500 to the brokerage account. Assume that the brokerage account pays a 6 percent return with annual compounding. How much money do you expect to have in the brokerage account in ten years (Quarter 40)? How much money do you expect to have in the brokerage account in Quarter 39?

Additional Problem:

• Today you opened up a local bank account. Your plan is make five $1,000 contributions to this account. The first $1,000 contribution will occur today and then every six months you will contribute another $1,000 to the account. (So your final $1,000 contribution will be made two years from today). The bank account pays a 6 percent nominal annual interest, and interest is compounded monthly. After two years, you plan to leave the money in the account earning interest, but you will not make any further contributions to the account. How much will you have in the account 8 years from today?

Problem 8-30 • Erika and Kitty, who are twins, just received $30,000 each for their 25th

birthday. They both have aspirations to become millionaires. Each plans to make a $5,000 annual contribution to her “early retirement fund” on her birthday, beginning a year from today. Erika opened an account with the Safety First Bond Fund, a mutual fund that invests in high-quality bonds whose investors have earned 6% per year in the past. Kitty invested in the New Issue Bio-Tech Fund, which invests in small, newly issued bio-tech stocks and whose investors have earned an average of 20% per year in the fund’s relatively short history.

• Requirement 1: If the two women’s funds earn the same returns in the future as in the past, how old will each be when she becomes a millionaire?

• Requirement 2: How large would Erika’s annual contributions have to be for her to become a millionaire at the same age as Kitty, assuming their expected returns are realized?

• Requirement 3: Is it rational or irrational for Erika to invest in the bond fund rather than in stocks?

Thank you for listening!