Exact matches only

Search in title

Search in content

Filter by Categories

101 concepts level I

101 concepts level II

2021 Level I Corporate Finance Full Videos

2021 Level I Economics Full Videos

2021 Level I Ethics Full Videos

2021 Level I FRA Full Videos

2021 Level I Portfolio Management Full Videos

2021 Level I Quantitative Methods Full Videos

Advice and How to Study Videos

All-Levels

Alternative Investments

Alternative Investments (AI)

BookLet Top Level

Corporate Issuers

Corporate Issuers (CI)

Demystified Videos

Derivatives

Derivatives (DV)

Economics

Economics (EC)

Equity

Equity Investments (EI)

Ethical and Professional Standards (ES)

Ethics

featured

Financial Reporting and Analysis

Financial Statement Analysis (FSA)

Fixed Income

Fixed Income (FI)

Level I

Level II

Level III

LM01 Alternative Investment Features, Methods, and Structures

LM01 Categories, Characteristics, and Compensation Structures of Alternative Investments

LM01 Corporate Structures and Ownership

LM01 Derivative Instrument and Derivative Market Features

LM01 Ethics and Trust in the Investment Profession

LM01 Fixed-Income Instrument Features

LM01 Fixed-Income Securities: Defining Elements

LM01 Introduction to Financial Statement Analysis

LM01 Market Organization & Structure

LM01 Market Organization and Structure

LM01 Organizational Forms, Corporate Issuer Features, and Ownership

LM01 Portfolio Management Overview

LM01 Portfolio Management: An Overview

LM01 Rates and Returns

LM01 The Firm & Market Structures

LM01 Time Value of Money

LM01 Topics in Demand and Supply Analysis

LM02 Alternative Investment Performance and Returns

LM02 Analyzing Income Statements

LM02 Code of Ethics and Standards of Professional Conduct

LM02 Code of Ethics and Standards of Professional Conduct Profession

LM02 Financial Reporting Standards

LM02 Fixed Income Markets - Issuance Trading and Funding

LM02 Fixed-Income Cash Flows and Types

LM02 Forward Commitment and Contingent Claim Features and Instruments

LM02 Introduction to Corporate Governance and Other ESG Considerations

LM02 Investors and Other Stakeholders

LM02 Organizing, Visualizing, and Describing Data

LM02 Performance Calculation and Appraisal of Alternative Investments

LM02 Portfolio Risk & Return: Part I

LM02 Portfolio Risk and Return Part I

LM02 Security Market Indexes

LM02 The Firm and Market Structures

LM02 Time Value of Money in Finance

LM02 Understanding Business Cycles

LM03 Aggregate Output, Prices and Economic Growth

LM03 Analyzing Balance Sheets

LM03 Business Models & Risks

LM03 Corporate Governance: Conflicts, Mechanisms, Risks, and Benefits

LM03 Derivative Benefits, Risks, and Issuer and Investor Uses

LM03 Fiscal Policy

LM03 Fixed-Income Issuance and Trading

LM03 Guidance for Standards I-VII

LM03 Guidance for Standards I–VII

LM03 Introduction to Fixed Income Valuation

LM03 Investments in Private Capital: Equity and Debt

LM03 Market Efficiency

LM03 Portfolio Risk & Return: Part II

LM03 Portfolio Risk and Return Part II

LM03 Private Capital, Real Estate, Infrastructure, Natural Resources, and Hedge Funds

LM03 Probability Concepts

LM03 Statistical Measures of Asset Returns

LM03 Understanding Income Statements

LM04 An Introduction to Asset-Backed Securities

LM04 Analyzing Statements of Cash Flows I

LM04 Arbitrage, Replication, and the Cost of Carry in Pricing Derivatives

LM04 Basics of Portfolio Planning & Construction

LM04 Basics of Portfolio Planning and Construction

LM04 Capital Investments

LM04 Common Probability Distributions

LM04 Fixed-Income Markets for Corporate Issuers

LM04 Introduction to the Global Investment Performance Standards (GIPS)

LM04 Monetary Policy

LM04 Overview of Equity Securities

LM04 Probability Trees and Conditional Expectations

LM04 Real Estate and Infrastructure

LM04 Understanding Balance Sheets

LM04 Understanding Business Cycles

LM04 Working Capital and Liquidity.

LM05 Analyzing Statements of Cash Flows II

LM05 Capital Investments and Capital Allocation

LM05 Company Analysis: Past and Present

LM05 Fixed-Income Markets for Government Issuers

LM05 Introduction to Geopolitics

LM05 Introduction to Industry and Company Analysis

LM05 Monetary and Fiscal Policy

LM05 Natural Resources

LM05 Portfolio Mathematics

LM05 Pricing and Valuation of Forward Contracts and for an Underlying with Varying Maturities

LM05 Pricing and Valuation of Forward Contracts.

LM05 Sampling and Estimation

LM05 The Behavioral Biases of Individuals

LM05 Understanding Cash Flow Statements

LM05 Understanding Fixed-Income Risk and Return

LM05 Working Capital & Liquidity

LM06 Analysis of Inventories

LM06 Capital Structure

LM06 Cost of Capital-Foundational Topics

LM06 Equity Valuation: Concepts and Basic Tools

LM06 Financial Analysis Techniques

LM06 Fixed-Income Bond Valuation: Prices and Yields

LM06 Fundamentals of Credit Analysis

LM06 Hedge Funds

LM06 Hypothesis Testing

LM06 Industry and Competitive Analysis

LM06 International Trade

LM06 Introduction to Geopolitics

LM06 Introduction to Risk Management

LM06 Pricing and Valuation of Futures Contracts

LM06 Simulation Methods

LM07 Analysis of Long-Term Assets

LM07 Business Models

LM07 Capital Flows and the FX Market

LM07 Capital Structure

LM07 Company Analysis: Forecasting

LM07 Estimation and Inference

LM07 International Trade and Capital Flows

LM07 Introduction to Digital Assets

LM07 Introduction to Linear Regression

LM07 Inventories

LM07 Pricing and Valuation of Interest Rate and Other Swaps

LM07 Pricing and Valuation of Interest Rates and Other Swaps

LM07 Technical Analysis

LM07 Yield and Yield Spread Measures for Fixed-Rate Bonds.

LM08 Currency Exchange Rates

LM08 Equity Valuation: Concepts and Basic Tools

LM08 Exchange Rate Calculations

LM08 Fintech in Investment Management

LM08 Hypothesis Testing

LM08 Long Lived Assets

LM08 Measures of Leverage

LM08 Pricing and Valuation of Options

LM08 Topics in Long-Term Liabilities and Equity

LM08 Yield and Yield Spread Measures for Floating-Rate Instruments

LM09 Analysis of Income Taxes

LM09 Income Taxes

LM09 Option Replication Using Put-Call Parity

LM09 Option Replication Using Put–Call Parity

LM09 Parametric and Non-Parametric Tests of Independence

LM09 The Term Structure of Interest Rates: Spot, Par, and Forward Curves

LM10 Financial Reporting Quality

LM10 Interest Rate Risk and Return

LM10 Non-current (Long-Term) Liabilities

LM10 Simple Linear Regression

LM10 Valuing a Derivative Using a One-Period Binomial Model

LM11 Financial Analysis Techniques

LM11 Financial Reporting Quality

LM11 Introduction to Big Data Techniques

LM11 Yield-Based Bond Duration Measures and Properties

LM12 Applications of Financial Statement Analysis

LM12 Introduction to Financial Statement Modeling

LM12 Yield-Based Bond Convexity and Portfolio Properties

LM13 Curve-Based and Empirical Fixed-Income Risk Measures

LM14 Credit Risk

LM15 Credit Analysis for Government Issuers

LM16 Credit Analysis for Corporate Issuers

LM17 Fixed-Income Securitization

LM18 Asset-Backed Security (ABS) Instrument and Market Features

LM19 Mortgage-Backed Security (MBS) Instrument and Market Features

New Booklet Top level

Portfolio Management

Portfolio Management (PM)

Quantitative Methods

Quantitative Methods (QM)

Uncategorized

Please select your exam.

IFT Notes for Level I CFA^® Program

LM01 Time Value of Money

Part 2

3. Future Value of a Single Cash Flow

Let’s understand this concept with a simple example.

Say present value (PV) = $100 and interest rate (r) = 10%.

What is the future value (FV) after one year?

What is the future value (FV) after two years?

The future value of a single cash flow can be computed using the following formula:

FV_N = PV(1 + r)^N

where:

FV_N = future value of the investment

N = number of periods

PV = present value of the investment

r = rate of interest

Therefore,

FV₁ = 100(1 + 0.1)¹ = $110
FV₂ = 100(1 + 0.1)² = $121

Notice that with compound interest, after two years we have $121. Whereas, with simple interest, after two years we would have $120. The difference between the two values ($1) represents the interest-on-interest component. In Year 2, we not only receive interest on the $100 principal, but we also receive interest on the $10 interest earned in Year 1 that has been reinvested.

Example

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?

Solution:

FV_N = PV(1 + r)^N

FV_2.5 = 5(1 + 0.05)^2.5 = $5.649 million

FV Calculation using a Financial Calculator

You will often use the following keys on your TI BA II Plus calculator:

N = number of periods

I/Y = rate per period

PV = present value

FV = future value

PMT = payment

CPT = compute

One important point to note is the signs used for PV and FV. If the value for PV is negative “-”, then the value for FV is positive “+”. An inflow is often represented as a positive number, while outflows are denoted by negative numbers.

Before you begin, set the number of decimal points on your calculator to 9 to increase accuracy.

Keystrokes	Explanation	Display
[2nd] [FORMAT] [ ENTER ]	Get into format mode	DEC = 9
[2nd] [QUIT]	Return to standard calc mode	0

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

The key strokes to compute the future value of a single cash flow are illustrated below.

Keystrokes	Explanation	Display
[2nd] [QUIT]	Return to standard calc mode	0
[2^nd] [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

4. Non-Annual Compounding (Future Value)

When our compounding frequency is not annual, we use the following formula to compute future value:

$FV_N=PV{\left(1+\frac{r_s}{m}\right)}^{mN}$

where:

r_s = the stated annual interest rate in decimal format

m = the number of compounding periods per year

N = the number of years

Let’s understand this concept using an example.

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?

Solution:

There are two methods to solve this question.

Formula Method

PV is $80,000.

The stated annual rate is 10%.

The number of compounding periods per year is 4. The total number of periods is 4 x 3 = 12.

Therefore future value after 12 quarters (3 years) is

FV₁₂ = $80,000(1 + 0.025)¹² = $107,591

Calculator Method

You can also solve this problem using a financial calculator; the key strokes are given below:

N = 12, I/Y = 2.5%, PV = $80,000, PMT = 0, CPT FV = -$107,591

PMT is 0 because there are no intermediate payments in this example.

Example

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

Solution
The correct answer is B.

Formula Method

$FV_{N}=PV(1+\frac{r_{s}}{m})^{mN}$

$FV_{1 year}=$ \ 3 million$(1+\frac{0.04}{365})^{365} = \ ${\textdollar3.122 million}$

Calculator Method

N = 365, I/Y = 4/365%, PV = $3 million, PMT = 0; CPT FV = -$3.122 million

5. Continuous Compounding

We saw examples with annual compounding. Then we discussed quarterly compounding and in the above example we looked at daily compounding. If we keep increasing the number of compounding periods until we have infinite number of compounding periods per year, then we can say that we have continuous compounding.

The formula for computing future values with continuous compounding is:

${FV}_N = PVe^{rN}$

where:

r = continuously compounded rate

N = the number of years

Let’s look at an example.

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?

Solution:

PV = $50,000; r = 0.036; N = 3

$FV = \ $50,000 \times \ e^{0.036*3} = \ $55,702$

Concept Building Exercise

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

Frequency	Future value of $100	Return
Annual	112.00	12.00%
Semiannual	112.36	12.36%
Quarterly	112.55	12.55%
Monthly	112.68	12.68%
Daily	112.75	12.75%
Continuous	112.75	12.75%

Instructor’s Note:

1. For the same stated annual rate, the returns keep getting better as we compound more often.
2. If you have two banks that offer the following rates
A: 12.5% compounded annually
B: 12% compounded daily

Which offer is better?

Even though A’s 12.5% looks better, B’s 12% compounded daily will effectively give you a return of 12.75%. Therefore the offer from bank B is better.

5.1. Stated and Effective Rates

Now we come to the related concept of stated versus effective rates. In the above concept-building exercise, the stated rate was 12% across the board, but the effective rate that an investor actually earns depends on the compounding frequency. The effective rates were different for different compounding frequencies.

If we are given a compounding frequency, we can compute effective rates using the following formulae:

Effective annual rate for discrete compounding:

$EAR = (1+ \ periodic \ interest \ rate)^m \ - \ 1$

where:

m = number of compounding periods in one year

For daily compounding, m = 365
For monthly compounding, m = 12
For quarterly compounding, m = 4
For semiannual compounding, m = 2

For example, for a stated annual rate of 12% and quarterly compounding, the EAR will be equal to:

EAR = (1 + 0.12/4)⁴ – 1 = 1.1255 – 1 = 0.1255 = 12.55%

Instructor’s Note:

A lot of people get confused about the -1 at the end of the formula. The idea is actually fairly straight forward. Basically we have 1.03⁴ which is telling us how much $1 will become at the end of 4 periods. $1 is going to become $1.1255. But this is not a rate. To figure out the rate we have to subtract the original $1 that we invested. So we are left with 0.1255 which is our effective rate.

Effective Annual Rate for continuous compounding:

$EAR = \ e^{r_s} \ - \ 1$

where r_s= stated annual interest rate

For example, for a stated annual rate of 12% and continuous compounding, the EAR will be equal to:

EAR = e^0.12 – 1 = 1.1275 – 1 = 0.1275 = 12.75%

6. Series of Cash Flows

The different types of cash flows based on the time periods over which they occur include:

Annuity: A finite set of equal sequential cash flows. The two types are
- Ordinary annuity: The first cash flow occurs one period from now (indexed at t = 1).
- Annuity due: The first cash flow occurs immediately (indexed at t = 0).
Perpetuity: A set of equal never-ending sequential cash flows with the first cash flow occurring one period from today. (The period is finite in case of an annuity whereas in perpetuity it is infinite.)

6.1. Equal Cash Flows – Ordinary Annuity

Let’s say that we have an ordinary annuity with A = $1,000, r = 5% and N = 5, i.e. we are going to receive a payment of $1,000 at the end of each year for the next 5 years and our discount rate is 5%. We can compute the FV of this annuity using the following three methods:

Brute-Force Method

We can take each individual cash flow and see how much each of these will be worth at the end of five years. Then we can add all the values to compute the total FV of the annuity.

For instance, the first $1,000 deposit made at t = 1 will compound over four periods; the second deposit of $1,000 will compound over three periods and so on. We then add the future values of all payments to arrive at the future value of the annuity which is $5,525.63.

Formula Method

The future value of an annuity can also be computed using the following formula:

$FV_N=\ A\ \left[\frac{{\left(1+r\right)}^N-\ 1}{r}\right]$

where:

A = annuity amount

N = number of years

The term in square brackets is known as the ‘future value annuity factor’ (FVAF). This factor gives the future value of an ordinary annuity of $1 per period. Hence the formula given above can also be written as: FV = A x FVAF.

Therefore, using the formula:

$FV_5=\ 1,000\ \left[\frac{{\left(1.05\right)}^5-\ 1}{0.05}\right]=\$5,525.63$

Calculator Method

Given below are the keystrokes for computing the future value of an ordinary annuity.

Keystrokes	Explanation	Display
[2nd] [QUIT]	Return to standard calc mode	0
[2^nd] [CLR TVM]	Clears TVM Worksheet	0
5 [N]	Five years/periods	N = 5
5 [I/Y]	Set interest rate	I/Y = 5
0 [PV]	0 because there is no initial investment	PV = 0
1000 [PMT]	Set annuity payment	PMT = 1000
[CPT] [FV]	Compute future value	FV = -5525.63

On the exam you should use the calculator method, because this is the fastest method and does not require you to memorize the annuity formula.

Example

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?

Solution:

N = 15, I/Y = 12%, PV = 0, PMT = $24,000; CPT → FV = -$894,713.15

6.2. Unequal Cash Flows

If cash flow streams are unequal, the future value annuity factor cannot be used. In this case, we find the future value of a series of unequal cash flows by compounding the cash flows one at a time.

This concept is illustrated in the figure below. We need to find the future value of five cash flows: $1,000 at the end of Year 1; $2,000 at the end of Year 2; $3,000 at the end of Year 3; $4,000 at the end of Year 4; and $5,000 at the end of Year 5.