r5 time value of money

Quantitative Methods

Time Value of Money

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Graphs, charts, tables, examples, and figures are copyright 2012, CFA

Institute. Reproduced and republished with permission from CFA Institute.

All rights reserved.

Contents

1. Introduction

2. Interest Rates: Interpretation

3. The Future Value of a Single Cash Flow

4. The Future Value of a Series of Cash Flows

5. The Present Value of a Single Cash Flows

6. The Present Value of a Series of Cash Flows

7. Solving for Rates, Number of Periods, or Size of Annuity Payments

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Video Lecture 1 39 minutes

Video Lecture 2 40 minutes

1. Introduction

• Time value of money

• Interest rates

• Present value

• Future value

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2. Interest Rates: Interpretation

Interest rates can be interpreted as:

1. Required rate of return

2. Discount rate

3. Opportunity cost

Say you lend $900 today and receive $990 after one year

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Required Rate of Return Discount Rate Opportunity Cost

Interest Rates: Investor Perspective

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As investors, we can view an interest rate as:

Real risk-free interest rate +

Inflation premium +

Default risk premium +

Liquidity premium +

Maturity premium

Nominal Risk-free Rate

Practice Question 1

Jill Smith wishes to compute the required rate of return. Which of the following premiums is she least likely to include?

A. Inflation premium

B. Maturity premium

C. Nominal premium

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Answer: C Required rate of return includes inflation premium, maturity premium, default risk premium, and liquidity premium. There is no such component as a nominal premium.

Practice Question 2

Which of the following is least likely true?

A. Discount rate is the rate needed to calculate present value

B. Opportunity cost represents the value an investor forgoes

C. Required rate of return is the maximum rate of return an investor must receive to accept an investment

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Answer: C Required rate of return is the minimum rate of return an investor must receive to accept an investment. Therefore, option C is least likely to be the interpretation of interest rates.

Investments Maturity (in years)

Liquidity Default risk Interest Rates (%)

A 1 High Low 2.0

B 1 Low Low 2.5

C 2 Low Low r

D 3 High Low 3.0

E 3 Low High 4.0

Practice Question 3

1. Explain the difference between the interest rates on Investment A and Investment B.

2. Estimate the default risk premium.

3. Calculate upper and lower limits for the interest rate on Investment C, r.

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3. Future Value of a Single Cash Flow

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FVN = PV (1 + r)N

0 1 2

PV = 100 and r = 10% What is the FV after one year? What is the FV after two years?

Practice Question 4 Cyndia Rojers deposits $5 million in her savings account. The account holders are entitled to a 5% interest. If Cyndia withdraws cash after 2.5 years, how much cash would she most likely be able to withdraw?

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FV Calculation Using a Financial Calculator


Keystrokes Explanation Display

[2nd] [FORMAT] [ ENTER ] Get into format mode DEC = 9

[2nd] [QUIT] Return to standard calc mode 0

You invest $100 today at 10% compounded annually. How much will you have in 5 years?

Keystrokes Explanation Display

[2nd] [QUIT] Return to standard calc mode 0

[2nd] [CLR TVM] Clears TVM Worksheet 0

5 [N] Five years/periods N = 5

10 [I/Y] Set interest rate I/Y = 10

100 [PV] Set present value PV = 100

0 [PMT] Set payment PMT = 0

[CPT] [FV] Compute future value FV = -161.05

Set to floating decimal

3.1 Frequency of Compounding

You invest 80,000 in a 3-year certificate of deposit. This CD offers a stated annual interest rate of 10% compounded quarterly. How much will you have at the end of three years?


Multiple Compounding Periods - Calculator


You invest 80,000 in a 3-year certificate of deposit. This CD offers a stated annual interest rate of 10% compounded quarterly. How much will you have at the end of three years?

Practice Question 5


Donald invested $3 million in an American bank that promises to pay 4% compounded daily. Which of the following is closest to the amount Donald receives at the end of the first year? Assume 365 days in a year. A. $3.003 million B. $3.122 million C. $3.562 million

3.2 Continuous Compounding

Infinite compounding periods per year continuous compounding


FVN = PV e r N

An investment worth $50,000 earns interest that is compounded continuously. The stated annual interest is 3.6%. What is the future value of the investment after 3 years?

Concept Building Exercise


Frequency Future value of $100 Return

Annual 112 12.00%

Semiannual 112.36 12.36%

Quarterly

Monthly

Daily

Continuous

Assume the stated annual interest rate is 12%. What is the future value of $100 at different compounding frequencies?

3.3 Stated and Effective Rates


With a discrete number of compounding periods: EAR = (1 + Periodic interest rate)m – 1 With continuous compounding: EAR = er

– 1

4. The Future Value of a Series of Cash Flows

• Annuity: finite set of level sequential cash flows

Ordinary annuity: an annuity where the first cash flow occurs one period from today

Annuity due: an annuity where the first cash flow occurs immediately

• Perpetuity: set of level never-ending sequential cash flows with the first cash flow occurring one period from today


0 1 2

0 1 2

4.1 Future Cash Flows – Ordinary Annuity


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Ordinary annuity with A = 1,000 r = 5% and N = 5

Ordinary Annuity - Formula


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FVN = A {[(1+r)N – 1]/r}

FVN = A {Future Value Annuity Factor}

Ordinary Annuity - Calculator



N = 5 I/Y = 5 PV = 0 PMT = 1,000 CPT FV

Practice Question 6


Haley deposits $24,000 in her bank account at the end of every year. The account earns 12% per annum. If she continues this practice, how much money will she have at the end of 15 years?

Practice Question 7


Iago wishes to compute the future value of an annuity worth $120,000. He is aware that the FV annuity factor is 21.664 and the interest rate is 4.5%. Which of the following is least likely to be useful for the future value computation?

A. Annuity worth

B. Future value annuity factor

C. Interest rate

Answer: C

4.2 Unequal Cash Flows

Time Cash Flow ($)

1 1,000

2 2,000

3 3,000

4 4,000

5 5,000


What is the future value at year 5?


End of Lecture 1

5. Finding the Present Value of a Single Cash Flow


PV = FVN (1+r)-N

For a given discount rate, the farther in the future the amount to be received, the small the amount’s present value.

Holding time constant, the larger the discount rate, the smaller the present value of a future amount.

Practice Question 8


Liam purchases a contract from an insurance company. The contract promises to pay $600,000 after 8 years with a 5% return. What amount of money should Liam most likely invest? Solve using the formula and TVM functions on the calculator.

Answer: 406,104

Practice Question 9


Mathews wishes to fund his son, Nathan’s, college tuition fee. He purchases a security that will pay $1,000,000 in 12 years. Nathan’s college begins 3 years from now. Given that the discount rate is 7.5%, what is the security’s value at the time of Nathan’s admission?

Answer: 521,583

Practice Question 10


Orlando is a manager at an Australian pension fund. 5 years from today he wants a lump sum amount of AUD40, 000. Given that the current interest rate is 4% a year, compounded monthly, how much should Orlando invest today?

Answer: 32,760

6. Present Value of a Series of Cash Flows

• Present value of a series of equal cash flows (annuity)

• Present value of a perpetuity

• Present value indexed at times other than zero

• Present value of a series of unequal cash flows


6.1 Present Value of a Series of Equal Cash Flows


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Ordinary annuity with A = 10 r = 5% and N = 5

PV of an Ordinary Annuity: Using the Formula


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PV = A {[1 – 1/(1+r)N]/r}

PV of an Ordinary Annuity: Using the Calculator


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Annuity Due – The Concept


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Annuity due with A = 10 r = 5% and N = 5

PV of an Annuity Due: Using the Formula


0 1 2 3 4 5


PV (annuity due) = A {[1 – 1/(1+r)N]/r} (1+r)

PV of an Annuity Due: Using the Calculator


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Key Strokes Display

[2nd] [BGN] [2nd] [SET] BGN

[2nd] [QUIT] BGN 0

[2nd] [CLR TVM] BGN 0

5 [N] BGN N = 5

5 [I/Y] BGN I/Y = 5

10 [PMT] BGN PMT = 10

0 [FV] BGN FV = 0

[CPT] [PV] BGN

[2nd] [BGN] [2nd] [SET] END

[2nd] [QUIT] 0

6.2 Present Value of a Perpetuity


PV = A/r

Present value is one period before the first cash flow Simple example to understand the formula: You invest $100 and get 5% for ever. What is the cash flow?

6.3 Present Values Indexed at Times Other Than t=0


An annuity or perpetuity beginning sometime in the future can be expressed in present value terms one period prior to the first payment

Discount back to today’s present value



Bill Graham is willing to pay for a perpetual preferred stock that pays dividends worth $100 per year indefinitely. The first payment will be received at t = 4. Given that the required rate of return is 10%, how much should Mr. Graham pay today?

Answer: 751.31

6.4 The Present Value of a Series of Unequal Cash Flows


Find the present value of each individual cash Sum the respective present values

0 1 2 3



Andy makes an investment with the expected cash flow shown in the table below. Assuming a discount rate of 9% what is the present value of this investment?

Answer: 550

Time Period Cash Flow($)

1 50

2 100

3 150

4 200

5 250


• Solving for Interest Rates and Growth

• Solving for Number of Periods

• Solving for the Size of Annuity Payments

• Review of Present Value and Future Value Equivalence

• The Cash Flow Additivity Principle


7.1 Solving for Interest Rates and Growth Rates


A $100 deposit today grows to $121 in 2 years. What is the interest rate? Use both the formula and the calculator method. The population of a small town is 100,000 on 1 Jan 2000. On 31 December 2001 the population is 121,000. What is the growth rate? You invest $900 today and receive a $100 coupon payment at the end of every year for 5 years. In addition, you receive $1,000 and the end of year 5. What is the interest rate?

7.2 Solving for the Number of Periods


You invest $2,500. How many years will it take to triple the amount given that the interest rate is 6% per annum compounded annually? Use both the formula and the calculator method.

Answer: 18.85 years

7.3 Solving for the Size of Annuity Payments


Freddie bought a car worth $42,000 today. He was required to make a 15% down payment. The remainder was to be paid as a monthly payment over the next 12 months with the first payment due at t=1. Given that the interest rate is 8% per annum compounded monthly, what is the approximate monthly payment?

Answer: 3,106

7.4 Review of Present and Future Value Equivalence

A lump sum can be considered equivalent to an annuity

An annuity can be considered equivalent to a future value

A lump sum can be considered equivalent to a future value


Ordinary annuity with A = 10 r = 5% and N = 5 PV = 43.29

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FV = 55.26

Lump sum = PV = 43.29

7.5 The Cash Flow Additivity Principle


Amounts of money indexed at the same point in time are additive

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Summary

1. Interest Rates

2. Future Value

3. Present Value



Conclusion

• Review learning objectives

• Examples and practice problems from the curriculum

• Practice questions from other sources


r5 time value of money

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