Pricing of Securities

Pricing of Securities

Pricing of Securities

Pricing of securities is important for two reasons. First, it enables the buyer to make an informed decision as to the market value (the market price) of the securities. Price contains all the available information of a company or institution may it be public or private in an efficient market context. Therefore, a market price reflects the current market value of a firm at any one time. For example, if a company XYZ has one million shares of common stock outstanding in its capital structure and its current market value is $50 per share, then the company is worth $50 million despite the fact that it may have $100 million of the total assets (book value). Second, price is used as a benchmark or a means of comparison of values of similar or different assets. If, for instance, for a given period, the average returns of Standard and Poor's 500[1](S&P's 500) is 12 percent and the average returns of a company XYZ is 16 percent, one might conclude that XYZ is performing better than the S&P's 500 for that year.

The price of an asset is really nothing more than the sum of the present values of all its expected cashflows at a given rate of return. In other words, if we can forecast the expected cashflows of a company's security and the required rate of return (which is usually determined by the market) for that security we will, therefore, be able to calculate its price. In this chapter we will show some valuation techniques and examples of certain securities that a bank is usually involved in. The present value concept has been fully discussed in chapter 2.

Pricing of Discount Securities

Discount securities are priced below par or face values to reflect a certain yield to maturity, the current market rate of returns. They do not pay periodic interest payments. The security is redeemed at the full face value at maturity. A good example of a discounted security is a T-bill. A T-bill=s yield is calculated using equation 6.1:

where N is the number of days toward maturity. As an example, a 91-days T-bill with the face value of $10,000 was priced at

$9,873.61. Its discounted yield should therefore be 5%,

T-bill yield=[($10,000 - $9,873.61)/$10,000](360/91)

=5%

The above pricing equation is unique for T-bills. Notice that the first expression on the right of equation 5.1 is not the usual way a rate of return (ROR) is calculated:

ROR=(P1 - P0)/P0(6.2)

where P0 is the first price and P1 is the last price. This rate of return is calculated based on the amount invested. In the T-bill calculation the denominator is P1 instead of P0. The T-bill rate is based on percentage of the face value (par) instead of the amount invested. Second, the T-bill calculation is based on a 360-days year, as appeared in the annualized expression (360/N), instead of a normal 365-days calender year. This fact makes it tough to compare yields of T-bills to those of other securities, for example, a bond. As matter of fact, the T-bill's discount yield will always understate the bond-equivalent yield and the effective yield.

The bond-equivalent yield, the uncompound ROR, recognizes the conventional way of calculating ROR based on the 365-days year:

It represents the usual method of calculating ROR for all

discount instruments (securities that pay no periodic interest

payments). Using the same example as above, the T-bill will have the bond-equivalent yield of 9.65%, which is 0.13% higher than its discounted yield:

Bond-equivalent yield = [($10,000 - $9,873.61)/$9,873.61](365/91)

= 5.13%

However, the bond-equivalent yield ignores the principal of compounding. In order to account for compounding, we have to consider its effective yield. It reflects the true rate of return because it explicitly acknowledges the time value of money. Equation 6.4 describes the effective yield:

Given the above example, the effective yield of the T-bill is 5.23%:

T-bill's discount yield can be converted to bond-equivalent yield directly by using equation 5.5:

Bond-equivalent yield = 365d/(360 - dN)(6.5)

where d is the T-bill's discounted yield. From the above example, the discounted yield is 5%; therefore, the bond-equivalent yield should be:

Bond-equivalent yield = 365(.05)/[(360 - (.05)(91)] = 5.13%

which is the same as the one obtained earlier from equation 6.3.

The bond-equivalent yield and the effective yield calculations are applicable to other discounted securities such as commercial paper, acceptances, and negotiable certificates of deposit.

Pricing of Capital Market Instruments

Capital market securities are long-term securities. They can have fixed periodic interest payments with a definite maturity such as a bond. However, stocks such as preferred stocks and common stocks generally do not have maturities: they are perpetual instruments. We are going to take a look at the pricing of a bond, preferred stock, and a common stock.

Bond

A bond pays fixed coupon (interest payments) periodically. At maturity there is also a return of the principal, the face value of the bond. Graphically, the cashflow structure of a bond looks like the followings:

Time012...... N

│ │ │ │

─┼────┼────┼───────────────────┤

│ │ │ │

CashflowsPb$$...... $

principal

where Pb is the current market price of the bond, N is the remaining periods towards maturity, and $ is a coupon interest payment.

The appropriate pricing equations for a bond is:

where i is the yield to maturity (YTM), which is the discounted rate (market or required rate of return) of the bond. As an illustration, there is an issue of ten-years corporate bonds, $1,000 par, and a 12% coupon interest rate paid semiannually. Two years have passed after the date of issuance and the current market rate of interest is 8%. The price or value of the bond is:

Notice that since two years have passed the remaining time period to maturity is therefore 8 years or 16 periods (8 x 2). The YTM is 4% per period (8% / 2). Finally, the coupon payment is $120 or $60 per period ($120 / 2).

Often, we are asked to find the YTM of a bond. The "precise" method calls for a "trial-and-error" approach in which we equate the price with the bond equation 6.6 above and solve for the YTM[2]. For example, a $1,000 par corporate bond has 3 years to maturity and a 7% coupon interest rate. The current price is $980, what is the YTM? We will assume semiannual payments of the coupon. We can set up the equation in the following way:

We can solve for i by putting in the various values of i in the equation to give $980 as the price. The answer is 3.88% per period or 7.76% per year.

Alternatively, there is an approximate method of calculating the YTM. This method is accurate enough to give you a "ball park" YTM. Equation 6.7 shows the approximate YTM:

Given the previous example, through the approximate method the YTM is 3.87% per period or 7.74% per year, which is very close to that found through the more accurate "trial-and-error" approach:

ferred Stock

Preferred stocks pay fixed dividend for perpetuity. The cashflow from one period to the next is identical. Therefore, the pricing equation takes the form of a perpetuity with no growth (zero growth model):

Pp = D/i(6.8)

where Pp is the price of preferred stock and D is the fixed dividend. If we have a preferred stock that pay $10 per share and the current yield of 10%, its price will therefore be $100 per share:

Pp =$10/.10 =$100.

Common Stock

Common Stock usually exhibit certain growth pattern in the dividend payments. If we assume a constant growth rate for perpetuity we can use the Gordon's Model to obtain the price:

Ps=D0(1 + g) / (i - g)=D1 / (i - g)(6.9)

where Ps is the price of a common stock and g is the estimated growth rate of dividends. D0 is the most current or last known dividend paid. Note that D0 compounded by the growth rate, g, is the expected dividend of the next period or D1.

As an example, the last known dividend for a stock XYZ is $2 and the approximate growth rate of its dividend is 10%. The market yield for this stock is 15%. The price of this stock is:

Ps=$2(1 + .10) / (.15 - .10)=$44.

It is important to note that Gordon's Model mathematically assumes the yield, i, to be greater than the dividend growth rate, g. Otherwise, the price will be negative when i < g, and undefined when i = g, all of which are impossible conditions.

We can obtain the yield, i, directly given the growth rate, g, and the current price:

i=(D1 / Ps) + g(6.10)

The Capital Asset Pricing Model (CAPM) provides an alternative way to obtain the yield. A distinct advantage over the Gordon's Model is the fact that CAPM recognizes risky assets explicitely by 1) dividing assets into two groups: risk-free and risky assets, and 2) assigning a measure of risk. Essentially, the higher the risk of assets the higher the expected returns of the assets - investors require greater compensations for risk taking. CAPM is expressed as:

E(i)=Rf + (Rm - Rf)(6.11)

where Rf is the risk-free rate of return, Rm is the return of the risky assets, and  is the volatility factor or the measure of risk[3]. Rm-Rf is the risk premium. The expression, (Rm - Rf), is also known as the "price of risk".

What is the effective rate of return, i, of a common stock if there is a definite holding period? The effective return can be solved by setting the sum of the present value of the dividend stream and the present value of the expected selling price at the end of the holding period to be equal to the current price:

The summation is from t to N, where t is the time period and N is the holding period.

As an example, an investor plans to hold a common stock, whose current price is $40, for the next 5 years. During that five years he estimates the dividend growth to be 8% per year and the expected selling price at the end of year 5 is $80. The last known dividend was $1.50 per share. What is the effective rate of return of his stock? First, a dividend stream has to be determined:

Dyr.1 = $1.50(1.08) = $1.62

Dyr.2 = $1.62(1.08) = $1.75

Dyr.3 = $1.75(1.08) = $1.89

Dyr.4 = $1.89(1.08) = $2.04

Dyr.5 = $2.04(1.08) = $2.20

Lastly, we apply set up the problem according to equation 6.12 and solve for the effective rate of return, i, through trial-and-error:

where t is from 1 to 5. The answer is 17.52%.

Summary

The price of an asset is really nothing more than the sum total of the present value of all its expected (future) cashflow given a certain rate of return. Follow this tenet price can easily be determined if we know 1) the expected cashflow structure, and 2) the required rate of return we must make in this investment. The latter is usually determined by the market. In this chapter examples of pricings of T-bills, bonds, preferred stocks and common stocks are shown.

Questions and Problems

1. What is pricing of a security so important?

1. Given:initial investment = $960

par value = $1,000

Days to maturity =120

1. What is the annual discount yield (assuming no compounding)?
2. What is the effective annual yield?

1. National Bank issued a new series of bonds on 1/1/62. The bonds carried $1,000 par, a 12% coupon interest rate, and had a 30 year maturity. Interest is paid semi-annually.

1. What is the YTM on 1/1/62?
2. What is the value of a bond on 1/1/67, t years later, if the market rate of interest then is 10%?
3. Based on the value calculated in part b, what would be the current yield on 1/1/67?

1. Calculate the expected return for Super Company which has a beta of .73 when the risk free rate is .08 and you expect the market return to be .15.

1. If a stock paid dividend of $3 at the end of the year, had and expected growth rate of 8% and a discount rate of 18%, How much is the stock worth?

1. What is the value of a preferred stock that pays fixed dividend of 12% on the face value of $100 when the current yield is 7%?

1

[1]Standard & Poors's 500, or S&P's 500, is a popular stock index. It is an average value-weighed composite index of share prices of the top 500 companies in the U.S. and weighted according to the relative values of each company's outstanding shares. It is often used as a benchmark to compare the relative performance between it and those of portfolios of stocks and mutual funds.

[2] The "trial-and-error" approach is also employed in the calculation of internal rate of return (IRR). IRR is defined as the rate of return which makes the net present value (NPV), the difference between the present value of a cashflow stream and the initial cost of a project, equal to zero.

[3] Beta, , is defined as the covariance between the market returns and a company's stock returns divided by the variance of the returns of the market. It is a proxy for risk by measuring the volatility of the company's stock returns relative to the market returns (market's beta is one). If the company's beta is greater than one then it is more volatile (more "risky") than the market. If the company's beta is less than one then it is less volatile (less "risky") than the market.