Chapter 2. Concepts in Valuation.

To make itself as valuable as possible to shareholders, a firm must choose the best combination of decisions on investment, financing, and dividends. That is, a firm must determine the company's return-risk character and the firm's value in the eyes of suppliers of capital. Risk can be defined as the possibility that the actual return will deviate from that which was expected.

THE TIME VALUE OF MONEY.
$1 now is more valuable than $1 in the future.

Compound Interest and Terminal Values.
The term itself implies that interest paid on a loan or an investment is added to the principal. As a result, interest is earned on interest.

TV_n = X₀ (1+r)ⁿ
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TV = Terminal value.
X₀ = Amount invested at the beginning.
r = Interest rate.

Compounding More Than Once a Year.
More compounding in a year results in a higher terminal value.

TV_n = X₀ (1+r/m)^m•n
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m = Number of times interest is paid during the year.

Quarterly Compounding. See illustration (page 14).

Infinite compounding. m approaches infinity, so this is the formula:

TV_n = X₀ • e^r•n

PRESENT VALUE OF AN ANNUITY.
Present value is a future amount discounted to present by some required rate.

A_n = PV • (1+k)ⁿ
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PV = Present value
A_n = Cash flow at the end of the year.
k = Annual interest rate (discount rate).

Beyond One Period.
In solving present-value problems, it is useful to express the interest factor separately from the amount to be received in the future. In such calculations, the interest rate is known as the discount rate, and henceforth, we will refer to it as such.

PV = A_n / (1+k)ⁿ

Present Value of an Annuity.
An annuity is an even series of future cash flows.

Relationship between PV and k. Value decreases as the required return increases, but at a decreasing rate.

Amortizing a Loan.
Amortization is the reduction of a loan’s principal amount through equal payments, which embrace both interest and principal.

Present Value When Interest Is Compounded More Than Once a Year.
More compounding in a year results in a lower present value.

PV = A_n / (1+k/m)^m•n
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m = Number of times a year interest is compounded.

INTERNAL RATE OF RETURN OR YIELD.
Internal rate of return is the rate of discount which equates the present value of cash inflows with the present value of cash outflows.

See formula (page 21).

BOND RETURNS.
A bond calls for a stated amount of money to be paid to the investor either at a single future date, maturity, or at a series of future dates, including final maturity.

Pure Discount (Zero Coupon) Bonds.
A pure discount bond is one where the issuer promises to make a single payment at a specified future date. This single payment is the same as the face value of the instrument, usually expressed as $100.

Important:
The normal pricing convention is to use semiannual compounding

A bond’s price is the present value of future coupon payments and face value, discounted by the bond’s yield.

Bond yield is simply a bond’s internal rate of return.

Coupon Bonds.
Most bonds are not of a pure discount variety, but rather pay a semiannual interest payment along with a final principal payment of $100 at maturity. See formula (page 24).

Relationship between Price and Yield.
1. When a bond’s market price is less than its face value of $100 so that it sells at a discount, the yield to maturity exceeds the coupon rate.
2. When a bond sells at a premium, its yield to maturity is less than the coupon rate.
3. When the market price equals the face value, the yield to maturity equals the coupon rate.

Holding-Period Return. The yield to maturity may differ from the holding-period yield if the security is sold prior to maturity.