Lecture Notes 2 Concepts and Tools for Portfolio, Equity Valuation ...

Foundations of Finance: Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses

Professor Alex Shapiro

Lecture Notes 2

Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses:

Applications to Savings/CDs, Loans/Mortgages

I. Readings II. What is an Interest Rate? III. Time Line IV. Investing for a Single Period V. Single Cash Flow, Multiple Periods, and Future Value VI. Single Cash Flow, Multiple Periods, and Present Value VII. Equivalent Effective Interest Rates Over Different

Compounding Periods.

VIII. Alternate Interest Rate Concepts IX. Multiple Cash Flows. X. A Particular Cash Flow Pattern: Annuity XI. A Particular Cash Flow Pattern: Perpetuity

XII. Appendix & Additional Readings

Buzz Words:

Time Value of Money, Equilibrium, Arbitrage, Perfect Creditworthiness (No Credit Risk, No Default Risk), Locked-in Rates, Discounting, Compounding, EAR, APY, APR, Continuous Compounding, Basis Point, Zeros.

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Foundations of Finance: Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses: Applications to Savings/CDs, Loans/Mortgages

I. Readings

RWJ Chapters 4 and 5.

Web: Investment Growth Calculator: treat the "Rate of Return" as an interest rate, and enter values to visualize the power of compounding.

II. What is an Interest Rate?

A. The time value of any commodity reflects:

1. Preferences for consumption sooner than later. 2. Physical productivity. 3. "Convenience yield" (inventories).

B. The time value of money reflects that:

1. Money can be converted into consumption. 2. Money can be converted into physical capital for production. 3. Money's convenience yield stems from the value of cash in

facilitating transactions (it is a medium of exchange).

C. In "equilibrium," these characteristics of money are valued equally, and hence they define the real rate of interest.

D. Inflation

1. Inflation measures the rate of depreciation of money.

Inflation implies a distinction between real interest rate (measured in commodities) and nominal interest rate (measured in currency). Example: If the productivity of physical capital is 4% (a real interest rate), and inflation is 3%, the nominal interest rate is (approximately) 7%.

2. All the interest rates in this course are nominal.

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Foundations of Finance: Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses: Applications to Savings/CDs, Loans/Mortgages

III. Time Line

$1 received today is not the same as a $1 received in the future.

The timing of a cash flow affects its value.

$1 today is never worth less than $1 in the future

When valuing cash-flow streams, the timing of the cash flows is crucial; a good idea is to draw a time line:

Money today:

t=0 $100

is NOT the same as Money in the future.

t=0 $100

The latter time line represent the contractual payment of a debt contract. Specifically, this can be a bond, that promises to pay its owner a terminal single cash flow. Since there are no intermediate cash flows (called coupon payments), this is a zero-coupon bond, or simply a "zero." If $1 in the future, is worth more than $1 today, you can arbitrage ("free lunch"). In well functioning markets, any arbitrage is quickly eliminated, and so for our purposes, we assume there are no arbitrage opportunities.

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Foundations of Finance: Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses: Applications to Savings/CDs, Loans/Mortgages

IV. Investing for a Single Period

A. Definition

The effective interest rate r (expressed as a decimal) over any period tells what $x will be worth at the end of the period using the following formula:

$x (1+r)

B. Example

Suppose you can invest $100 at an effective annual interest rate of 12%, by buying a CD (certificate of deposit). What is your $100 worth at the end of the year?

Answer: First, we draw the Time Line, using the notation C for cash flow, and using subscripts for the timing of the cash flow on the time line:

0

end of year 1

$100

C1

Next, we perform the calculations:

C0 = $100 Interest = $100 ? 0.12 = $12 C1 = C0 + Interest = $100+$12 = $100(1+0.12) = $112

0

end of year 1

$100

$112

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Foundations of Finance: Concepts and Tools for Portfolio, Equity Valuation, Fixed Income, and Derivative Analyses: Applications to Savings/CDs, Loans/Mortgages

V. Single Cash Flow, Multiple Periods, and Future Value

A. Example

Suppose you can invest $100 at an effective annual interest rate of 12%, by investing in your offshore savings account. What is your $100 worth at the end of 3 years?

0

1

2

3

$100

C3

1. One Way to Answer: Obtain C3 in 3 steps

C0 = $100

C1 = C0 (1+r) = $100(1+0.12) = $112

C2 = C1 (1+r) = $112(1+0.12) = $125.44

( = 100+12+12+12?0.12 )

C3 = C2 (1+r) = $125.44(1+0.12) = $140.4928

0

1

2

3

$100

$112

$125.44

$140.49

2. Another Way to Answer: The 3 steps could be combined into 1 step

C0 = $100 C 3 = C 0 (1+r)(1+r)(1+r) = $100(1+0.12)(1+0.12)(1+0.12) = $140.49

More conveniently:

C 3 = C 0 (1+r)3 = $100(1+0.12)3 = $140.49

3. First, notice that we reinvest interest to earn more interest. This is referred to as compounding.

Second, note how the answer above is the initial investment C0 times a multiplier that only depends on the effective interest rate and the investment interval. This multiplier is known as the future value interest factor (FVIF).

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