cfa level 1 quantitative analysis e book part 1

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Quantitative Analysis E-Book

Part 1 of 8


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The Time Value of Money


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1. The Time Value of Money and Interest Rates

The time value of money (TVM) refers to the fact that $1 today is worth more than $1 in the future. This is

because the $1 today can be invested to earn interest immediately. The TVM reflects the relationship

between present value, future value, time, and interest rate. The time value of money underlies rates of

return, interest rates, re quires rates of return, discount rates, opportunity costs, inflation, and risk. It

reflects the relationship between time, cash flow, and an interest rate.

There are three ways to interpret interest rate:

1. Required rate of return is the return required by investors or lenders to postpone their current

consumption.

2. Discount rate is the rate used to discount the future cash flows to allow for the time value of

money (that is, to bring future value equivalent to present value).

3. Opportunity cost is the most valuable alternative investors give up by choosing what they could

do with the money.

In a certain world, the interest rate is called the risk-free rate. For investors preferring current to future

consumption, the risk-free interest rate is the rate of compensation they require to postpone current

consumption. For example, the interest rate paid by T-bills is a risk-free rate of interest.


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In an uncertain world, there are two factors that complicate interest rates:

1. Inflation: When prices are expected to increase, lenders charge not only an opportunity cost of

postponing consumption but also an inflation premium that takes into account the expected

increase in prices. The nominal cost of money consists of the real rate (a pure rate of interest) and

an inflation rate (a pure rate of interest) and an inflation premium.

2. Risk: Companies exhibit varying degrees of uncertainty concerning their ability to repay lenders.

Lenders therefore charge interest rate to incorporate default risk. The return that borrowers pay

thus comprises the nominal risk-free rate (real rate + an inflation premium) and a default risk

premium.

Compounding is the process of accumulating interest over some period of time. Compounding period is the

number of times per year in which interest is paid. Continuous compounding occurs when the number of

compounding periods becomes infinite, that is, interest is added continuously.

Discounting is the calculation of the present value of some known future value. Discount rate is the rate

used to calculate the present value of some future cash flow. Discounted Cash Flow is the present value

today of some future cash flow.


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2. Calculate The Effective Annual Rate.

There are three ways to quote interest rates for investments paying interest more than once a year.

1. Periodic interest rate is the rate of interest earned over a single compounding period. For

example, a bank may state that a particular CD pays a periodic quarterly interest rate of 3% that

compounds 4 times a year.

2. Stated annual interest rate, also called quoted interest rate, is the annual rate of interest that does

not account for compounding within the year. It is the annual interest rate quoted by financial

institutions, and equal to the periodic interest rate times the number of compounding periods per

year. For example, the stated annual interest rate of the above CD is 3% x 4 = 12%. It is strictly a

quoting convention, and it does not give a future value directly.

3. Effective annual rate (EAR) is the annual rate of interest that takes full account of compounding

within the year. The periodic interest rate is the stated annual interest rate divided by m, where m

is the number of compounding periods in one year: EAR = (1 +periodic interest rate)^m-1 note

that the higher the compounding frequency, the higher the EAC.

For example, a $1 investment earning 8%compounded semiannually actually earns8.16%: (1 + 0.08/2)^2 -

1 = 8.16. The annual interest rate is 8%, and the effective annual rate is 8.16%


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Example

If the nominal interest rate is 8%, find the effective annual rate with quarterly compounding.

Method 1 - By Formula

m = 4, EAR = - 1 = 0.0824. The effective interest rate with quarterly compounding is 8.24%.

Method 2 - Texas Instruments:

You will use the Interest Conversion (ICONV) worksheet

1. Press 2nd ICONV to select the worksheet

2. NOM will be displayed with the previous value

3. Press 2nd [CLR WORK] to clear the worksheet


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Proceed as shown below

Keystrokes: Display

2nd ICONV: NOM = previous value

2nd CLRWORK: NOM = 0.00

8 ENTER: NOM = 8.00

Down Arrow: EFF = 0.00

Down Arrow: C/Y = previous value

4 ENTER: C/Y = 4.00

Down Arrow: EFF = 0.00

CPT: EFF = 8.24


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Method 3 - HP 12C

After you have set the calculator to END of period and cleared the financial registers, key in the nominal

interest rate as a percentage.

Proceed as shown below

Keystrokes: Display

g END: Previous value

f CLEAR FIN: 0.00000000

f CLEAR REG: 0.00000000

8 ENTER: 8.00000000

4n i 0.66666667

100 CHS PV: -100.00000000

FV: 108.2432160

100-: 8.24321600.

We can also calculate the periodic interest rate given the effective annual interest rate.


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3. The Future Value And Present Value of a Single Cash Flow.

When you make a single investment today, its future value that will be received an N year from now is as

follows:

FVN = PV X (1 + r)N

Where

• FV = future value at time n

• PV = present value

• r = interest rate per period

• N = number of years

A key assumption of the future value formula is that interim interest earned is reinvested at the given

interest rate (r). This is known as compounding.


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In order to receive a single future cash flow N years from now, you must make an investment today in the

following amount:

Notice that the future cash flow is discounted back to the present. Therefore the interest rate is called the

discount rate.

You should be able to calculate PVs and FVs using your calculator, where

• N = number of periods

• I/Year = yield in market place or the Required Rate of Return

• PV = present value

• PMT = payment amount per period

• FV = the future value of the investment

One can solve for any of the above variables. Just input the other variables and solve for the unknown.

Using the calculator on the test will prove to be a very time efficient manner of calculating present values

and future values.


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Example 1

An analyst invests $5million in a 5-year certificate of deposit (CD) at a local financial institution. The CD

promises to pay 7% per year compounded annually. The institution also allows him to reinvest the interest

at the same CD rate for the duration of the CD. How much will the analyst have at the end of five years if

his money remains invested at 7% for five years with no withdrawals of interest?

Before using the Texas Instruments BAII PLUS and HP 12C calculator, it is essential to ensure that your

settings are correct. The default settings on the calculator are not necessarily the settings you need

calculator are not necessarily the settings you need when making the calculations. Follow these steps to

ensure your calculator is correctly set.


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Texas Instruments BAII PLUS Settings

1. Press 2nd QUIT 2nd [CLR TVM] to clear the worksheet.

2. Press 2nd [P/Y] to enter payments per year and/or compounding periods per year.

3. The P/Y label and current value are displayed. The default value is 12. You must now key in 1 and

then ENTER since you want 1 payment per year.

4. If the question says there are 12 payments per year you would then change this to 12.

HP 12C Settings

1. Turn the calculator on by pressing the ON key.

2. Clear the memory and set decimals to 2 places by pressing the following keys:

3. CLEAR REG f 2 - 0.00 will display.

4. CLEAR FIN - this clears all the data in the financial mode.


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The calculator keys to press are

Texas instruments hp 12c

key stokes display key stokes display

5000000 pv pv=5,000,000 5 enter n 5.000

5n n=5.00 7 enter i 7.000

7 i/y 1/y=7.00 5000000 chs pv -5,000,000

cpt fv fv=7,012,758.654 fv 7,012,758,653


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If you are given the FV and need to solve for PV, the calculator keys to press are

Texas Instruments HP 12C

Key Stokes Display Key Stokes Display

5 N N = 5.00 5 ENTER n 5.000

7 I/Y I/Y = 7.00 7 ENTER i 7.000

7,012,758.654 FV = 7,012,752.654 7,012,758,653 CHS FV -7,012,758,653

CPT PV 5,000,000 PV 5,000,000


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When compounding periods are not annual Some investments pay interest more than once a year. When

you solve these problems, make sure that periodic interest rates must correspond to the number of

compounding periods in the year. For example, if time periods are quoted in quarters, quarterly interest

rates should be used.

Whenever the compounding periods are other than annual

Where

• = the quoted annual interest rate

• m = the number of compounding periods per year

• N = the number of years.

Example 2

An analyst invests $5million in a 5-year certificate of deposit (CD) at a local financial institution. The CD

promises to pay 7% per year, compounded semiannually. The institution also allows him to reinvest the

interest at the same CD rate for the duration of the CD. . How much will the analyst have at the end of five

years if his money remains invested at 7% for five years with no withdrawals of interest?


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The calculator keys to press are:

Note that the answer is greater than when the compounding was annual. This is because interest is earned

twice a year instead of only once.

If the number of compounding periods becomes infinite, interest is compounding continuously.

Accordingly, the future value N years from now is computed as follows:



5000000 PV= 5,000,000 5 ENTER 2xn 10.00

5 x 2 = 10 N N = 10.00 7 ENTER 2/i 3.500

7/2 = 3.5 I/Y I/Y = 3.50 5000000 CHS PV -5,000,000

CPT FV FV= 7,052,993.803 FV 7,052,9993803


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4. The Future Value and Present Value of a Series of Equal Cash Flows (Ordinary

Annuities, Annuity Dues, and Perpetuities).

Annuity is a finite set of sequential cash flows, all with the same value. Ordinary annuity has a first cash

flow that occurs one period from now (indexed at t = 1). In another word, the payments occur at the end of

each period.

• Future value of a regular annuity

FV

Where

• A = annuity amount

• N = number of regular annuity payments

• r = interest rate per period.

• Present value of a regular annuity


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Annuity due has a first cash flow that is paid immediately (indexed at t = 0). In another word, the payments

occur at the beginning of each period.

• Future value of an annuity due

It consists of two parts: the future value of one annuity payment now, and the future value of a regular

annuity of (N -1) period. Calculate the two parts and add them together. Alternatively you can use this

formula:

Note that, all else equal, the future value of an annuity due is equal to the future value of an ordinary

annuity times (1 + r)


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• Present value of an annuity due

It consists of two parts: an annuity payment now and the present value of a regular annuity of (N - 1)

period. Use the above formula to calculate the second part, and add the two parts together. The process

can also be simplified to a formula:

Note that, all else equal, the present value of an annuity due is equal to the present value of an ordinary

annuity times (1 + r).

Hint: Remember these formulas - you can use them to solve annuity-related questions direct lt, or to

double-check your answers given by your calculator

A Perpetuity is a perpetual annuity: an ordinary annuity that extends indefinitely. In another word, it is an

infinite set of sequential cash flows that have the same value, with the first cash flow occurring one period

from now.

This equation is valid for a perpetuity with level payments, positive interest rate r. The first payment occurs

one period from now (like a regular annuity). An example of perpetuity is a stock paying constant paying

constant dividends.


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Example - Future value of a regular annuity

An analyst decides to set aside $10,000 per year in a conservative portfolio projected to earn 8% per

annum. If the first payment he makes is one year from now, calculate the accumulated amount at the end of

10 years.

Method 1: Using a formula

• Identify the given variables: A = 10,000, r = 0.08, N = 10.

• Identify the appropriate formula: FV = A x {[(1 + r)N - 1] / r}

• Solve for the unknown: FV = 10,000 {[(1 + 0.08)10 - 1] / 0.08} = $144,865.


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Method 2: Using a calculator

Texas Instruments settings

1. 2nd P/Y = 1 and key in 1 ENTER.

2. SET END since this is a regular annuity. You do this by pressing 2nd BGN 2nd SET until you see

END displaying. You press 2nd SET twice if necessary. After setting to END, you must always

press 2nd QUIT and then continue.

Exploration: Change the problem to an annuity due (i.e. SET BGN) and compare the amounts (Answer is

SET BGN) and compare the amounts. (Answer is $156,454.87 - a difference of $11,589.25).



2nd Quit 0.00 fCLEAR FIN Last value

2nd CLR RVM 0.00 fCLEAR REG 0.0000000

10000 PMT PMT =

10,000.00 10 n 10.0000

10 N N= 10.00 8 i 8.0000

8 I/Y I/Y = 8.00 10000 CHS PMT -10,000.000

CPT FV 144,866.62 FV 144,865.6247


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5. The Future Value and Present Value of a Series of Uneven Cash Flows.

Series of uneven cash flows: the cash flow stream is uneven over many time periods. There is no single

formula available to compute the present or future value of a series of uneven cash flows.

Present value

• When we have unequal cash flows, we must first find the present value of each individual cash flow

and then sum the respective present values. (Usually with the help of a spreadsheet)

Future value

• Once we know the present value of the cash flows, we can easily apply time-value equivalence by

using the formula to calculate the future value of a single sum of money


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Example

John wants to pay off his student loan in three annual installments of $2,000, $4,000 and $6,000

respectively in the next three years. How much should John deposit into his bank account today if he wants

to use the account balance to pay off the loan? Assume that the bank pays 8% interest, compounded

annually.


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6. The Cash Flow Additivity Principle.

The additivity principle Dollar amounts indexed at the same point in time are additive.

Suppose we are considering two series of cash flows (A and B). The annual interest rate is 5%. We want to

know the future value of combined cash flows at t = 3.

We can calculate the future value of each series and add them up. The future value of series A is 100 x

1.052+ 100 x 1.05 + 100 = 315.25, and the future value of series B is 150 x 1.052 + 150 x 1.05 + 150 =

472.875. The future value of A + B is 788.125.

Alternatively, we can add the cash flows of each series first, and then find the future value of the combined

cash flows: 250 x 1.052 + 250 x 1.05 + 250 = 315.25 = 788.125.


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We can use this principle to solve many uneven cash flow problems if we add dollars indexed at the same

point in time. Consider a cash flow series, A, with $100 indexed at t = 1, 2, 3 and 5, and $0 at t = 4. This

series is an almost-even cash flow, flawed only by the missing $100 at t = 4. How do we find the present

value of this series?

• We can create an annuity B with $100 indexed at t = 1, 2, 3, 4, 5. It's easy to find the present at t 1,

2, 3, 4, 5. It s easy to find the present value of this series.

• Then we isolate an easily evaluated cash flow B - A: it has a single cash flow of $100 at t = 4. It's

also easy to find the present value of this single cash flow.

• We then subtract the present value of B – A from the present value of B.


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