CFA-Level-1-QM-Lec-1-R6-Time-Value-of-Money.pdf - Infinity ...

Quantitative MethodsTime Value of Money

Infinity Space| Ultimate Access Education Institute | CFA Level I 1

Introduction

• Time value of money

• Interest rates

• Present value

• Future value

2Infinity Space| Ultimate Access Education Institute | CFA Level I

Interest Rates: Interpretation

Interest rates can be interpreted as:

1. Required rate of return

2. Discount rate

3. Opportunity cost

Say you lend USD 1000 today and receive USD 1100 after one year

3

Required Rate of Return Discount Rate Opportunity Cost

We someone is delaying their present consumption of money, they should be rewarded for that. That reward would be the interest.

Infinity Space| Ultimate Access Education Institute | CFA Level I

Interest Rates: Investor Perspective

4

As investors, we can view an interest rate as:

Real risk-free interest rate

+Inflation premium

+Default risk premium

+Liquidity premium

+Maturity premium

NominalRisk-freeRate

The risk that a borrower will not make the contractual payments in a timely manner or altogether has toget compensated

If we are converting the investment to cash quickly, we may receive less than the fair value of the investment. Therefore, the risk bearing investor should get compensated

Longer maturity period, the more uncertainty that the investor has to face as he/she lock in their money for a longer period. This risk has to be compensated.


The theoretical rate on a single-period investment when there is no inflation.

Simple Interest vs Compounding Interest

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A is investing 90,000 in a 1-year certificate of deposit at 8% interest per annum. However, the interest would be added every quarter. What is the investment value after one year under the following two scenarios?

• Simple Interest provided quarterly (No re-investment of interest)

• Compounded quarterly


Answer = 97,418.89Answer = 97,200

Future Value of a Single Cash Flow

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FV = PV (1 + r)N

0 1 2

PV = 100 and r = 10%1. What is the FV after one year?2. What is the FV after five years?

3 4 5


Practice Question 1

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If you invest USD 3 million in a bank, which promises to pay 4% annually, however compounded on daily basis. Which of thefollowing is closest to the amount you would receive at the end of the first year? Assume 365 days in a year.A. USD3.003 millionB. USD3.122 millionC. USD3.562 million


Concept Building Exercise

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FrequencyFuture value of

USD100Return

Annual 112 12.00%

Semiannual 112.36 12.36%

Quarterly 112.55 12.55%

Monthly 112.68 12.68%

Daily 112.7475 12.7475%

Continuous 112.7497 12.7497%

Assume the stated annual interest rate is 12%. What is the future value of USD100 at different compounding frequencies?

𝐹𝑉 = 𝑃𝑉𝑒𝑟𝑡FV calculation with continuous compounding


Stated and Effective Rates

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

*m is the number of compounding periods per year

Periodic interest rate = Stated Annual Interest Rate/m

With continuous compounding:EAR = er – 1

Use the Calculator Interest Conversion Function to do this easily


Series of Cash Flows

Annuity: Finite set of level sequential cash flows

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

o 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

10

0 1 2

0 1 2


Future Value of an Ordinary Annuity

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0 1 2 3 4 5

Ordinary annuity with A = 100 r = 5% and N = 5

FVN = A x {[(1+r)N – 1]/r}

FVN = A x {Future Value Annuity Factor}

Use the TVM Function of the Calculator


Answer = 552.56

Future Value of an Ordinary Annuity – Practice Question 1

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

Calculate the annuity factor relevant for the calculation

i) What is the future value of the investment

ii) Use the calculator to come up with the same answer


Answer = 894,713.15

= 37.28

Future Value of an Ordinary Annuity – Practice Question 2

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If you wishes to compute the future value of an annuity worth USD 100,000. You knows the relevant 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


Future Value of a Set of Unequal Cash Flows

Time Cash Flow (USD) FV of CF

1 1,000

2 2,000

3 3,000

4 4,000

5 5,000

FV of the Cash Flows

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What is the future value at year 5 @ 10%?

Answer = 17,156.10

1,464.10

2,662.00

3,630.00

4,400.00

5,000.00


Finding the Present Value of a Single Cash Flow

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

If we are looking at the same investment horizon, the larger the discount rate, the smaller the present value of a future amount.

𝑷𝑽 =𝑪𝑭

(𝟏 + 𝒓)𝑵


Present Value of a Single Cash Flow – Practice Question 1

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A purchases an insurance plan from an insurance company. The plan promises to pay USD 600,000 after 8 years with a 5% return. What amount of money should A most likely invest?

i) Solve using the formula

ii) TVM functions on the calculator.


Answer = 406,103.62

Present Value of a Single Cash Flow – Practice Question 2

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A is a manager of a pension fund. 5 years from today he wants a lump sum amount of USD 40,000. Given that the current interest rate is 6% a year, compounded monthly, how much A should invest today?


Answer = 29,654.89

Present Value of a Series of Cash Flows

• Present value of a series of equal cash flows (annuity)o Ordinary annuityo Annuity due

• Present value of a perpetuity

• Present value of a series of unequal cash flows

• Present value indexed at times other than zero

18Infinity Space| Ultimate Access Education Institute | CFA Level I

Present Value of an Ordinary Annuity

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0 1 2 3 4 5


𝑃𝑉 = 𝐴 ×[1 − 1 + 𝑟 −𝑁]

𝑟


Answer = 432.95

= 4.3295

Present Value of an Annuity Due

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0 1 2 3 4 5

Annuity due with A = 100 r = 5% and N = 5


Answer = 454.60

Present Value of a Perpetuity

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If you are receiving USD100 every year till perpetuity, what is the PV of Cash Flows? Assuming a discount rate of 10%

𝑃𝑉 =𝐶𝐹

𝑟


Answer = 1000.00

Present Values Indexed at Times Other Than T=0

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

John is willing to pay for a perpetual preferred stock that pays dividends worth USD 100 per year indefinitely. The first payment will be received at t = 3. Given that the required rate of return is 8%, how much should John should invest today?


Answer = 1071.67

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Find the present value of each individual cash and sum it up

0 1 2 3

100 200 300

@ 5% discount rate

The Present Value of a Series of Unequal Cash Flows


Answer = 535.80

Present Value of a Series of Unequal Cash Flows – Practice Question

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An investment yields the following expected cash flow shown in the table below. Assuming a discount rate of 9% what is the present value of this investment?

Time Period Cash Flow(USD) PV of individual cash flow

1 100

2 150

3 200

4 250

5 300

PV of all the cash flows

Use the CF function of your financial calculator


Answer = 744.52

Practice Questions - Growth Rates and Interest Rates

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The population of a small town is 1,000,000 on t=0. After two years, the population is 1,210,000. What is the growth rate?

You invest USD 900 in a debenture which gives USD 100 coupon payment at the end of every year for 6 years. In addition, you receive face value of the debenture of USD1,000 and the end of year 6. What is the interest rate?


Answer = 10%

Answer = 12.46

Practice Questions - Number of Periods

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How many years will it take to triple the amount given that the interest rate is 6% per annum compounded annually?

Formula Method Calculator Method


Answer = 18.85 years

Practice Questions - Size of Annuity Payments

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A bought a car worth USD 42,000 today. A 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 12% per annum compounded monthly, what is the approximate monthly payment?


Answer = 3171.90

The Cash Flow Additivity Principle

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0 1 2 3 4 5

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


You have USD 300 expense allocation for save up for a camera. You can save at ainterest rate of 8% per annum and borrow at a rate of 8% per annum. What is the amount that you can afford for an camera at the end of 3rd year?

Answer = 1508.90

Review of Present and Future Value Equivalence

29

A lump sum (present value) can be considered equivalent to an annuity

An annuity can be considered equivalent to a future value

A lump sum (present value) can be considered equivalent to a future value

0 1 2 3 4 5


FV = 552.560 1 2 3 4 5


PV = 432.95

0 1 2 3 4 5

PV = 432.95 FV = 552.56


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