Discount Rate, Yield to Maturity, Value of the Share, E/P Ratio and Ke

Discount Rate: - It is the rate of return that investors expect from securities of comparable risk.

Bonds or Debentures: - These are debt instruments or securities. In case of abond/debenture the stream of cash flows consists of annual interest payments and repayment of principal. These flows are fixed and known.

The Value of the Bond: - It can be found by capitalising cash flows at a rate of return, which reflects their risk. The market interest rate or yield is used as the discount rate in case of bonds (or debentures). The basic formula for the bond value is as follows:


Yield to Maturity: - A bond’s yield to maturity or internal rate of return can be found by equating the present value of the bond’s cash outflows with its price in the above equation.

Zero-Interest Bonds (called zero-coupon bonds in USA) do not have explicit rate of interest. They are issued for a discounted price; their issue price is much less than the face value. Therefore, they are also called deep-discount bonds.
The basic discounting principles apply in determining the value or yield of these bonds.

Preference shares have a preference over ordinary shareholders with regard to dividends. The preference dividend is specified and known. Similarly, in the case of redeemable preference share the redemption or maturity value is also known. Preference share value can be determined in the same way as the bond value. Here the discount rate will be the rate expected by the preference shareholders given their risk. This risk is more than the risk of bondholders and less than the equity shareholders.

Value of the Share (General) Cash flows of an ordinary (or equity) shareconsist of the stream of dividends and terminal price of the share. Unlike the case of a bond, cash flows of a share are not known. Thus, the risk of holding a share is higher than the risk of a bond. Consequently, equity capitalisation rate will be higher than that of a bond. The general formula for the share valuation is as follows:

As the time horizon, n, becomes very large (say, extends to infinity) the present value of future price approaches zero. Thus the term Pn disappears from the formula, and we can use the following equation to find the value of a share today:

Value of the Share (Zero Growth) If dividends do not grow, then capitalisingearnings can determine the share value. Under no-growth situation, earnings per share (EPS) will be equal to dividends per share (DIV) and the present value is obtained by capitalizing earnings per share:

Value of the Share (Constant Growth) In practice, dividends do grow over years. If we assume dividends to grow at a constant rate, g, then DIV1 = DIV0 (1 + g), DIV2 = DIV1 (1 + g), DIV3 = DIV2 (1 + g)..., and the share price formula can be written as follows:
This formula is useful in calculating the equity capitalisation rate (ke) when the price of the share (P0) is known.
Under the assumption of constant growth, the share value is equal to the capitalized value of earnings plus the value of growth opportunities as shown
below:
Value of Growth Opportunities (Vg) Given a firm’s EPS, ROE, the equity capitalisation rate, retention ratio and constant growth, the growth opportunities can be valued as follows:
E/P Ratio and Ke Relationship between the earnings-price ratio and capitalisation rate as follows:
The E/P ratio will equal the capitalisation rate only when growth opportunities are zero, otherwise it will either overestimate or underestimate the capitalisation rate.

Time Value For Money, Risk Premimum, Interest Rate, Capital Recovery

Time Value for Money : - Individual investors generally prefer possession of a given amount of cash now, rather than the same amount at some future time. This time preference for money may arise because of
(a) uncertainty of cash flows,
(b) subjective preference for consumption, and
(c) availability of investment opportunities.
The last reason is the most sensible justification for the time value of money.

Risk Premium: - Interest rate demanded, over and above the risk-free rate as compensation for time, to account for the uncertainty of cash flows.

Interest Rate or Time Preference Rate: - Rate which gives money its value, and facilitates the comparison of cash flows occurring at different time periods.

Required Interest Rate: - A risk-premium rate is added to the risk- free time preference rate to derive required interest rate from risky investments.

Compounding: - It means calculating future values of cash flows at a given interest rate at the end of a given period of time.

Future Value (F) of a Lump Sum: -  Today (P) for periods at rate of interest is given by the following formula:
 
The compound value factor, CVFn ,i can be found out from Table A given at the end of the book.

Future Value of an Annuity: - that is, the same amount of cash each year for periods at rate of interest is given by the following equation



The compound value of an annuity factor (CVAFn,i) can be found out from Table B given in Annexure at the end of the book.

Sinking Fund: -  An annuity to be deposited for periods at rate of interest to accumulate to a given sum. The following equation can be used:




Sinking Fund factor (SFFn, i) is a reciprocal of CVAFn, i.


Discounting: - It means calculating the present value of cash flows at a given interest rate at the beginning of a given period of time.

 Present Value (P) of a Lump Sum (F): - It occured at the end of period at rate of interest is given by the following equation:
The present value factor (PVFn, i) can be obtained from Table C given in Annexure at the end of the book.

Present Value of an Annuity (A) occurring for periods at rate of interest can be found out as follows:

Table can be used to find out the present value of annuity factor (PVAFn, i).

Capital Recovery: - It determined annual cash flows to be earned to recover a given investment. The following equation can be used:

Notice that the capital recovery factor (CRFn,i) is a reciprocal of the present value
annuity factor, PVAFn, i.


Wealth or Net Present Value: - It is defined as the difference between the present value of cash inflows (benefits) and the present value of cash outflows (costs). Wealth maximisation principle uses interest rate to find out the present value of benefits and costs, and as such, it considers their timing and risk. The following formula can be used to calculate NPV or wealth of any pattern of cash
flows:


Multi-period Compounding:  When interest compounds for more than once in a given period of time, it is called multi-period compounding. If is the nominal interest rate for a period, the effective interest rate (EIR) will be more than the nominal rate in multi-period compounding since interest on interest within a
year will also be earned, EIR is given as follows:
where is the number of compounding in a year and is number of years.


Internal Rate of Return (IRR):  IRR is the rate, which equates the present value of cash flows to the initial investment. Thus in operational terms, in the present value equation, all variables are known except rcan be found out by trial and error method:

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