Stock and Bond Valuation

Question 1

What is the time value of money (TVM)?

Answer 1

The concept that a dollar received in the present is not worth the same as a dollar received in the future; future amounts must include a premium to be equal in value

Question 2

What are the two different kinds of TVM problems?

Answer 2

Lump sums (paid all at once) and annuities (paid periodically)

Question 3

What are two different kinds of annuities?

Answer 3

(1) ordinary annuities (annuities in arrears) – paid at the end of a period
(2) annuities due (annuities in advance) – paid at the beginning of a period

Question 4

How is the future value (FV) of an amount calculated?

Answer 4

The present value (PV) of the amount is multiplied as if it were earning interest until the point in the future: FV = PV x (1+r)^n

Thus, if the interest rate (also called the “discount rate”) is deemed to be 10%, then the FV of $100 for two years in the future would be $100 x (1.10)^2 = $121 – but these problems tend to be much more complicated, and the CPA exam usually provides the factors you will need to multiply or divide the amounts by

Question 5

How is the PV of an amount calculated?

Answer 5

It is the exact opposite as finding the FV – the FV amount is divided by the interest rate (discount rate) to arrive at the PV: PV = FV / (1+r)^n

Thus, the PV of $121 received two years in the future, with a 10% discount rate, is $121 / (1.10)^2 = $121 / 1.21 = $100

Question 6

How are FV or PV factors normally determined?

Answer 6

Not by formulas – there are tables that list the PV or FV of $1 at given interest rates and given durations of time

On the CPA exam, though, usually the test will provide a few factors and require you to choose which one applies to the problem

Question 7

How do PV and FV interest factors relate to each other?

Answer 7

Future value interest factors (FVIFs) and present value interest factors (PVIFs) are reciprocals – for any given discount rate and duration of time, FVIF x PVIF = 1

Question 8

How do annuities complicate TVM problems?

Answer 8

Since they involve a number of payments in the future, their calculation is more complicated, but it can generally be accomplished through the factors provided on the test questions – knowing the formulas is helpful but not necessary

Question 9

What are the formulas for PV and FV of annuities?

Answer 9

FV = [(1+r)^n - 1] / r

PV = [1 - (1+r)^(-n)] / r

Question 10

What is the most important thing to remember for TVM problems involving annuities?

Answer 10

Keeping straight the exact timing of payments, so that the proper interest factor (PVIF or FVIF) is correctly selected and not off by one time period

Question 11

Do all TVM problems involve yearly compounding periods?

Answer 11

No, some can involve interest paid every six months, every quarter, every month, or some other period

In problems like these, the interest rate must be proportionately reduced (e.g. divided by 12 if payments are made monthly) and the number of compounding periods (“n” in the formulas) must be increased accordingly (e.g. multipled by 12 if payments are made monthly) – otherwise the calculation is exactly the same

Question 12

For a bond, what is the par value, maturity date, and coupon rate?

Answer 12

(1) par value = face amount of bond (usually $1,000)
(2) maturity date = date at which the principal of the bond (its par value) is paid back to the bondholder
(3) coupon rate = interest rate; determines amount of periodic interest payments to the bondholder

Question 13

For a bond, what is the difference between a new issue and an outstanding issue?

Answer 13

Right after it is issued (approximately two weeks), a bond is called a new issue; otherwise it is termed an outstanding issue, or a seasoned issue

Question 14

What is the general model for valuating a bond?

Answer 14

The PV of all its future payments

Bond value = PV of principal + PV of coupon payments

Question 15

What are bond discounts and premiums?

Answer 15

Given that bond coupon rates can differ from the prevailing market interest rate, different bonds can sell on the market at a price higher than their face value or lower than their face value

• If market price > face value, it sells at a premium
• If market price < face value, it sells at a discount

Question 16

What is the difference between the coupon rate and the effective interest rate?

Answer 16

Coupon rate = rate at which actual interest payments are made

Effective rate = the market rate when the bond was obtained – basically, the true interest rate of the bond after the discount or premium is taken into account (and other factors, e.g. risk premiums)

Question 17

How is the effective interest rate relevant when valuating bonds?

Answer 17

Effective rate determines the discount rate at which PV is calculated

The coupon rate determines the amount of periodic interest payments which will then be discounted to PV

Question 18

How do you calculate a return on investment for a bond?

Answer 18

These two things must be added:

(1) the total interest earned to date
(2) the capital gain or loss on the bond – i.e. its PV compared to the amount initially paid for it (e.g. if it was initially purchased at a discount, then it may have increased in value, which is a capital gain)

(1) + (2) will be the total return, and the total return divided by the initial price paid for the bond will be the return on investment

Question 19

What is a bond’s yield to maturity (YTM)?

Answer 19

The rate of return for a bond if it is held to maturity

Assumes that periodic interest payments will be reinvested at the same rate

Question 20

What is another way to understand yield to maturity (YTM)?

Answer 20

It is the discount rate by which the sum of all future cash flows for the bond (interest and principal), when discounted to PV, equals the amount actually paid for the bond

In this sense it is just like the effective interest rate

Question 21

How are YTM problems typically solved?

Answer 21

Often by trial and error – using an estimated discount rate to discount the total future payments back to PV, and altering the discount rate until the PV = the actual payment made

There are also bond yield tables which can give approximate YTMs

Question 22

When would a bond issuer want to call its bonds before their maturity date?

Answer 22

If the prevailing market interest rates are lower than the coupon rates, then the bond issuer may want to call the bonds and reissue other bonds at a lower coupon rate

This can only occur if the bonds are callable, i.e. if the bond issuer has the right to call the bonds in the first place

Question 23

What is a bond’s yield to call (YTC)?

Answer 23

The same as YTM, except that the reference point is not the bond’s maturity date, but the bond’s call date

YTC is more significant for callable bonds than YTM

Question 24

If a bond issuer can call a bond on different dates (rather than the maturity date, which is singular), how is the YTC calculated?

Answer 24

The YTC is ordinarily calculated using the earliest available call date

Question 25

How do changes in market interest rates affect the prices of bonds with different maturity dates?

Answer 25

A change in market interest rates more powerfully affects the price of a longer-duration bond than for a shorter-duration one

Increase in market interest rates –> decrease in bond price, since the bonds would then be less valuable than other available interest rates

Question 26

What are the general rights implicit in common stock?

Answer 26

(1) right to assets upon liquidation

(2) right to receive dividends

Question 27

What is the difference between a realized and an unrealized capital gain or loss?

Answer 27

Realized = occurs when the stock is sold
Unrealized = occurs before the stock is sold

Capital gain/loss = difference between price when sold and price when initially purchased

Question 28

What is the total return for a stock?

Answer 28

The sum of dividends received and the stock’s capital gain or loss

Question 29

How does stock valuation differ from bond valuation?

Answer 29

In both, the PV of all future cash flows must be determined

But stocks have no maturity date, and thus have a potentially infinite stream of cash flows

Question 30

How is a stock’s value determined if the stockholder intends to hold it indefinitely and there is no prospect of dissolution?

Answer 30

The PV of all future dividend payments

This would thus be a PV calculation for an infinite future cash flow stream

Question 31

How is a stock’s value determined if the stockholder intends to resell it?

Answer 31

The valuation is still the PV of an infinite stream of future cash flows (so long as the company’s end does not loom near), and this value can change as time progresses

This is because when the stock changes hands, at that point it still has value for its future cash flows, and so at any point the infinite future cash flows (discounted to PV) are the relevant consideration in valuating the stock

Question 32

What is a stock’s growth rate?

Answer 32

The rate at which the dividends of the stock increases – thus a stock with zero growth has constant dividends

Question 33

How is the PV of a stock with a constant growth rate calculated?

Answer 33

Using the Gordon model (named after Myron Gordon):

PV = d / (k-g)

d = the dividend payment one year (or other period) in the FUTURE
k = discount rate
g = growth rate

Question 34

How is the PV of a stock with zero growth calculated?

Answer 34

PV = d / k

d can be the dividends received in any year, since by definition those amounts do not change from year to year

Question 35

How is the PV of a stock with uneven growth calculated?

Answer 35

Any stock presented with uneven growth will have specific amounts of growth for a number of years and then a growth that is assigned to all the remaining years unto infinity

Thus the normal lump-sum PV formulas will apply to all the years where the specific growth rate is known, and then the Gordon model will apply to the final available year where a growth rate is assumed to be constant from then on – this final year must be discounted twice: once to arrive at the value of all future dividends at that date, and another to take that value and discount it to its PV

Question 36

What is an example for calculating the PV of a stock with uneven growth?

Answer 36

A company pays year-end dividends of $5 in year 1, and is expected to have 2% growth in year 2, 3% growth in year 3, and 4% growth in year 4 and onward. The discount rate (k) is 8%.

dividend for year 2 = $5 x 1.02 = $5.10
yr 3 = $5.10 x 1.03 = $5.25
yr 4 = $5.25 x 1.04 = $5.46

PV of year 1 dividend = $5 x 0.9091 = $4.55
yr 2 = $5.10 x 0.8264 = $4.21
yr 3 = $5.25 x 0.7513 = $3.94
value of remaining dividends at end of year 3 = (yr 4 dividend) / (k-g) = $5.46 / (8% - 4%) = $136.50
-PV of this amount = $136.50 x .7513 = $102.55

value of stock = PV at year 0 = PV of all future dividends = $4.55 + $4.21 + $3.94 + $102.55 = $115.25

Question 37

In matters of uncertainty, how is the expected rate of return for a stock calculated?

Answer 37

The rate of return can be established for different scenarios (e.g. company X will provide a return of only 3% if the economy is in a recession), and the different scenarios can be assigned different probabilities (e.g. there is a 25% chance that the economy will be in a recession)

The expected rate of return is the weighted average of the return for all possible scenarios multiplied by their respective probabilities

Question 38

What is a probability distribution?

Answer 38

A listing of all possible events for an investment and the probability for each one

Question 39

What are the two different kinds of risk for stocks?

Answer 39

(1) company-specific risk = risk that the particular company will fail or worsen
(2) market risk = risk due to changes in the whole stock market (e.g. a change in interest rates)

Question 40

What is diversifiable risk?

Answer 40

Risk that can be avoided through diversifying one’s portfolio, i.e. not having all your eggs in one basket, but investing in many different stocks

By definition, company-specific risk is diversifiable (i.e. nonsystematic), but market risk is not – and is thus called systematic risk

Question 41

How is a stock’s risk generally measured?

Answer 41

After mapping out the expected rate of return with a probability distribution, the stock’s risk is the volatility of that distribution – specifically, a higher standard deviation (σ) in the distribution signifies higher risk

Question 42

How is a stock’s volatility with reference to the rest of the market measured?

Answer 42

With a beta coefficient: β

If β = 1.0 for a stock, then it moves perfectly with the rest of the market; a 7% increase in the market would be accompanied by a 7% increase in that particular stock

If β < 1.0, then the stock moves less than the rest of the market, and is thus less volatile

If β > 1.0, then the stock moves more than the rest of the market, and is thus more volatile

Question 43

How does the beta coefficient (β) relate to the measurement of a portfolio’s risk?

Answer 43

The portfolio’s β is the weighted average of the β for each of the individual stocks in the portfolio

This measures the volatility of the portfolio in relation to the market overall, and thus measures the riskiness of the portfolio

Question 44

What is the correlation coefficient (r) for various stocks?

Answer 44

r measures the way in which two stocks can move in relation to one another

They can have:

• a perfect positive correlation (+1.0), so that they move the same direction to the same degree
• a perfect negative correlation (-1.0), so that they move the opposite direction to the same degree,
• a correlation of 0, so that they do not move in relation to each other at all
• any other correlation between these amounts

Question 45

What is the correlation coefficient (r) for most stocks?

Answer 45

Most have some positive coefficient between 0 and +1.0, i.e. a partial positive correlation

Question 46

How does the correlation coefficient (r) relate to diversification?

Answer 46

Market risk is not lessened at all if a stockholder acquires stocks whose r = +1.0 in his portfolio, since they will act as a joint unit however they behave – and so market risk cannot be diversified unless the correlation is less than +1.0

Question 47

What is the capital asset pricing model (CAPM)?

Answer 47

A model that calculates the expected return of a stock by assuming that investors should be compensated for their investment in two ways:

(1) the time value of money – investors should be paid for the fact that they are giving money to a corporation
(2) the risk premium – investors should be compensated for the fact that they are placing their money in a riskier enterprise than risk-free ones (i.e. gov’t bonds)

The CAPM gives a required rate of return, i.e. how much an average investor should expect to gain on the investment in order to see it as worthwhile

Question 48

What is the CAPM formula?

Answer 48

Ra = Rf + β (Rm - Rf)

Ra = required rate of return on the asset (stock)
Rf = risk-free rate (i.e. interest rate for gov't bonds)
β = beta coefficient; measure of volatility
Rm = market rate of return

Question 49

How is the CAPM formula structured?

Answer 49

The first part of the sum (Rf) refers to how investors are compensated for the time value of money, since, if nothing else, they could always invest their money to get that risk-free rate of return

The second part of the sum [β (Rm - Rf)] refers to how investors are compensated for the risk they undertake: they should be paid for a risk premium by investing in the market rather than risk-free bonds (Rm - Rf), and this premium is multiplied by the volatility of the stock (β)

Question 50

Which government bonds typically provide the risk-free rate of return (Rf)?

Answer 50

Long-term U.S. Treasury bonds

Question 51

What is a security market line (SML)?

Answer 51

A graphical representation of the CAPM:

• x-axis = β
• y-axis = required return
• slope = market risk premium (Rm - Rf)

Question 52

What does a steeper security market line (SML) say about investors’ risk aversion?

Answer 52

A steeper SML means that it has a higher slope and thus a higher market risk premium which investors require for their investments – investors would thus be MORE averse to risk, since they would require more value (a higher premium) to compensate them for it

Question 53

How does a security market line (SML) help investors select stocks to invest in?

Answer 53

Stocks with given levels of riskiness (β) and expected returns can be plotted on the graph

If they are above the SML, then their expected return is higher than the minimum required by the investor for that given level of risk, so they would be acceptable
-if below, then the expected return would not be worth the risk

Question 54

What is the efficient markets hypothesis (EMH)?

Answer 54

Holds that stocks widely traded on the market will have their market price accurately reflect their intrinsic value – i.e. there won’t be inefficiencies leading to under- or overvalued stock – in which case investors cannot ordinarily “beat the market” in such a way