8 Appendix: Modeling Stock Returns with and without Dividends
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FORWARD CONTRACTS AND FUTURES CONTRACTS
There are several questions here that focus on continuous time:
1. What does the underlying stock process look like when we think in
continuous time?
2. What does the underlying stock process without dividends look like when
we think in continuous time?
3. What does the underlying stock process with dividends look like when
we think in continuous time?
The total stock process would look like this when we think in continuous
time:
dSt
= dt
St
(Total Stock Process with Dividends
(before dividends are paid))
Here, is the stock’s total expected rate of return on an annualized basis.
It is the sum of the expected dividend yield (if the stock pays dividends) and
the expected capital gains yield. In general, it incorporates a risk premium
which we are initially going to ignore. In a competitive market, with no
adjustment for risk, would be equal to the annualized risk-free rate r.
Also, forward prices are determined under the risk-neutral measure, so any
risk premium is irrelevant to pricing the forward contract (see Chapters 13
and 15).
EXERCISE A1
Look at the total stock process just given.
a. What would be the ‘stock price’ at time t, St, if it were a bond?
b. What would be the (expected) return, , if the stock were a bond?
c. Write down the solution to the Total Stock Process for the bond price
B(T,T).
d. Translate back to the stock price process to get the solution for the stock
price at time T, ST.
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Solution
a.
b.
c.
d.
The stock price at time t would be the bond price St=B(t,T).
The expected return would be =r, the risk-free rate.
We did this earlier B(T,T)=B(t,T)*er.
Translating back ST=St*er.
As we showed in this chapter in section 4.3, the dividend payout process is,
− DIVt = − * Stdt
(Dividend Payout Process)
To get the stock price process after the dividend yield has been paid,
just replace the original stock price process (with expected return=) by the
stock price process with expected return=–. This produces the Capital
Gains (CG) process which is deﬁned as the stock process after the dividend
yield is paid.
dSt′
= ( − )dt
St′
(Capital Gains Process)
Thus, there are three ways to look at a stock process, the effect of the
dividend stream on the stock price process, and the stock process after the
dividend stream is paid,
1.
dSt
= dt
St
(Total Stock Process
before dividends are paid)
2. DIVt = * Stdt
(Dividend Payout Process)
dSt′
= ( − )dt
St′
(Capital Gains Process)
3.
The Connection between 1. and 3.
Next, we want to explore the relationship between the Total Stock Process
before dividends are paid and the Capital Gains Process. Speciﬁcally, we want
to see how we can go from 1. to 3. Then, we want to see an easy way to go
from the equilibrium forward price on 1. to the equilibrium forward price
on 3. There is an extremely easy way to do so.
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Under risk-neutrality, the original process deﬁned by 1. grows at the
continuously compounded rate r. That is =r. The terminal (time T) stock
price is just ST =erSt, as indicated in the time line below.
St
ST=e rτ St
t
T
The terminal stock price under the Total Stock Return Process will grow
based upon the Capital Gains (CG) component and the Dividend Yield
component. The CG component of the terminal stock price is e(r–)*St . This
is just a deﬁnition which says that the Capital Gains component of the stock’s
return is just that component that grows the stock price at the Capital Gains
rate of r–.
To explore this further, suppose that we split up the current, time t stock
price St into two parts. The ﬁrst part is St minus the present value of the
dividends paid out over the period [t,T]. We will denote this dividend amount
by by PV{DIV[t,T]}. The second part is just the dividend component,
PV{DIV[t,T]}. Then,
St = ⎡⎣St − PV { DIV[t ,T ]} ⎤⎦ + PV { DIV[t ,T ]}
Every dollar invested in the stock grows at the same rate, r,
e r * St = e r *⎡⎣St − PV { DIV[t ,T ]} ⎤⎦ + e r * PV { DIV[t ,T ]}
The second term is the future value of the PV of the dividends paid out over
[0,T] re-invested at the continuously compounded rate r. It is the amount
that would be in the ‘re-invested dividend account’ at time T. We write it
as TV{DIV[t,T]}, the TV stands for terminal value.
Next, consider the ﬁrst term. It represents the capital gains component of
the stock’s return. That is, if we remove PV{DIV[t,T]} from St, then the
compounded difference represents the capital gains component of the terminal
stock price. One way to think about this is to ask yourself how much you
would be willing to pay for the capital gains component alone, without the
dividends. The answer is St–PV{DIV[t,T]}.
But this capital gains component can also be written as just e(r–)*St .
Therefore e(r–)*St=er *[St–PV{DIV[t,T]}]. Multiplying both sides by e–r the
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113
equation simpliﬁes to, e–*St=[St–PV{DIV[t,T]}]. This says that the stock price
minus the PV of its dividend stream is equally well represented by e–*St .
This is a useful result as we now shall see.
We have just shown that the current stock price minus the PV of its
dividends is just equal to the normal stock price St multiplied by the factor
e–. But this is the appropriate adjusted stock price for valuing the forward
contract, because a forward contract is written only on the capital gains
component of the stock price.
So we should be able to simply substitute e–*St into the zero dividend
forward pricing model to obtain the dividend yield form of the forward price,
F′t,T=er (e–St )=e(r–)*St . This is indeed the correct equilibrium forward price,
also obtained by no-arbitrage arguments in section 4.6.2.
This argument shows that it makes sense to re-invest the dividends, when
received, either into a continuously compounded risk-free account or into
the risk-neutral stock.
We assumed in this chapter that we stripped dividends from the stock as
we continuously received them, and re-invested them in a riskless account
growing at the continuously compounded risk-free rate. Then, these cumulated dividends were used to reduce our borrowing cost from r to r–.
Re-investing the dividends into the stock produces the same net result.
Therefore, both alternatives will result in the same terminal value in the
dividend account.
There are equivalent standard arbitrage proofs of the result we obtained
intuitively. In our currency forward hedging example, the problem that arises
in the normal (zero dividend) arbitrage is that there are re-invested payouts
or ‘interest’. The problem, as we saw in the currency forward example, is that
it won’t do to just sell one forward contract for every unit of the spot
commodity you hold. This is due to the fact that by the time the forward
contract matures you have a position in the spot that is greater than 1 unit.
This extra amount is called a tail and the procedure for hedging it is called
‘tailing the hedge’. This problem arises in futures hedging as well. There are
ways of dealing with this problem, as illustrated below.
Four Methods for Dealing with Dividends
There are four alternatives available when dividends on the stock are involved.
We can either hold fewer than 1 unit of the stock, borrow the funds needed
to ﬁnance it, and sell it forward using 1 forward contract. We just have to
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ensure that the future value of our stock position constitutes one unit of stock
which can be delivered into the short forward position.
The other alternative is to hold 1 unit of stock, borrow to ﬁnance it, and
sell forward more than 1 forward contract in order to cover the future value
of the stock with dividends re-invested in it.
Alternative 1
To pursue the ﬁrst alternative, note that 1 unit of stock currently priced at St
will appreciate to eSt with dividends re-invested in the stock by maturity
date T. Therefore, e– units of stock will appreciate to 1 unit of stock, with
dividends re-invested, by time T.
The strategy is to borrow the current amount needed to ﬁnance e– units
of stock, e–St , and sell it forward with one forward contract at the forward
price F′t,T . This costs nothing today and produces a cash outﬂow consisting
of the principal plus the interest on the loan equal to er (e–St ), and a cash
inﬂow consisting of the proceeds from the short forward sale of F′t,T . The net
cash ﬂow is equal to F′t,T–er (e–St ). If this were different from zero, this would
be an arbitrage strategy at time t.
Therefore, F ′t,T=er (e–St )=e(r–)*St which is our equilibrium forward price
in the presence of dividends.
Alternative 2
Pursuing the second alternative method we will obtain exactly the same result.
Under this alternative we simply purchase one unit of the spot commodity by
borrowing its entire current value. We simultaneously sell forward the number
of units of stock expected to be in the stock account at time T. The dividends
continuously received on the stock over the period [t,T] are re-invested in
the stock. This re-investment strategy would result in a terminal stock account
consisting of e units of stock. Just as in the forward currency example, we
would have to sell forward this expected amount by selling e forward
contracts.
The strategy costs nothing today and produces a cash outﬂow consisting of
the principal plus the interest on the loan equal to er St and a cash inﬂow
consisting of the proceeds from the short forward sale of eF ′t,T . The net cash
ﬂow is equal to eF′t,T –er St .
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If this were different from zero, there would be an arbitrage strategy at time
t. Thus F′t,T=e–er St=e(r–)*St which is again our equilibrium forward price
in the presence of dividends.
Alternative 3
The third approach is to hold one unit of the spot commodity and fully ﬁnance
it by riskless borrowing at the rate r, sell forward one forward contract, and
strip the continuously received dividends as received, placing them into a riskless
savings account. When the debt comes due, one uses the proceeds in this
‘dividend re-invested account’ to pay off some of the debt. The net effect is
that the effective borrowing cost has been reduced to r–. To obtain the
equilibrium forward price, one then just substitutes r– for r in the no-dividend
formula to obtain F′t,T=e(r–)*St.
Alternative 4
There is a fourth approach. That procedure is to note that the capital gains
process is obtained from the total return process by the transformation
St′=e–St . This is how to transform the Total Stock Process before dividends
are paid (the process in 1.) into the Capital Gains Process in 3. It is a nice
little trick that will allow us to go freely back and forth between processes
with dividends and processes without dividends. Note that this gets you the
underlying process and not the forward price. That is more work.
But it is useful to know how to do this. Why? When we get to options,
there are transformation rules that allow us to generate option pricing formulas
from existing ones, by just knowing how to transform one underlying process
into another. Knowing how to transform the original process 1. into the
stripped-of-dividends (CG) process 3., combined with the transformation rules,
essentially tells us how to go from the forward price for 1. (where we assume
the process pays no dividends) to the forward price for 3. where the process
does pay dividends. We illustrated this in this chapter.
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FORWARD CONTRACTS AND FUTURES CONTRACTS
KEY CONCEPTS
1. Dividends (payouts) on the underlying.
2. Modeling Continuous Yields for zero-coupon bonds and for common
stocks.
3. How Dividends Affect Stock Prices.
4. How Capital Gains Affect Stock Prices.
5. Pricing Forward Contracts on ‘Stocks’ paying a Continuous Dividend
Yield using the Net Interest Model.
6. Pricing Forward Contracts on ‘Stocks’ paying a Continuous Dividend
Yield using No-Arbitrage.
7. Three Deﬁnitions of an Arbitrage Opportunity.
8. Riskless Arbitrage.
9. Risky Arbitrage.
10. Forward Pricing using No-Arbitrage.
11. Currency Spot and Currency Forwards.
12. Price Quotes in the FX Market.
13. Pricing Currency Forwards.
14. Pricing FX Forward Contracts using No-Arbitrage.
15. An Example of Pricing FX Forward Contracts.
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END OF CHAPTER EXERCISES FOR CHAPTER 4
1. Suppose that the underlying asset is the risk-free zero-coupon bond with
price given by B(t,T)=e–r .
a. Using the forward pricing formula at the end of Chapter 3, what would
be the forward price today, Ft,T , for a forward contract on the zerocoupon bond?
b. Does your answer to a. make economic sense? Why? or why not?
2. In the same scenario we just discussed in exercise 1, suppose that Ft,T >PT=
B(T,T)=1.0. Construct a risk-free arbitrage.
3. In the same scenario we just discussed in exercise 2, suppose that Ft,T
B(T,T). Construct a risk-free arbitrage. Make sure that you check all three
properties of a risk-free arbitrage opportunity.
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4. Suppose that the stock price increases to $110 in the next instant t+⌬t.
That is St+⌬t=$110. The dividend yield is equal to 3%. What is the dollar
amount of the dividend payable at time t+⌬t if ⌬t=1 day?
5. Here is an example of the forward pricing formula for stocks paying
dividends. Suppose that r=5% and that =3%. The current stock price St
is $100. Time to expiration of the forward contract is 3 months.
a. Calculate the forward price.
b. Compare the forward price in a. to the forward price on the stock if
it did not pay dividends.
6. Suppose that the 3-month risk-free borrowing rate in the US is rDE=3%
annualized and that the 3-month risk-free lending rate in the UK is rFE=5%
annually. The current spot rate is St=$1.6605/BP.
a. Calculate the 3-month forward rate.
b. What do you conclude from this example about the relationship
between the forward rate and the spot rate?
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SELECTED CONCEPT CHECK SOLUTIONS
Concept Check 1
a. Direct costs of storing inventory are things like storage facilities (e.g.
warehouses), and insurance. Direct costs appear in ﬁnancial statements.
Indirect costs are things like the opportunity cost of tying up one’s money
in commodities. This includes ﬁnancing costs to purchase the inventory.
The full opportunity cost does not typically appear in ﬁnancial statements,
although interest is usually reported.
b. The convenience yield is like a dividend yield in the sense that it is a beneﬁt.
It is unlike dividends in the sense that it is not directly paid. It is therefore
an indirect beneﬁt.
c. Convenience yield incorporates a risk-management component. The ﬁrm
faces demand in two forms: the ﬁrst is expected demand, and the second
is unexpected demand. Expected demand can be planned for by holding
enough inventory to cover it.
Unexpected demand cannot be planned for in the same way because
it is unexpected. But it can be hedged by holding excess inventory (above
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that needed to meet expected demand). The convenience yield of holding
excess inventory is this indirect beneﬁt it provides by allowing the ﬁrm to
hedge the unexpected demand for its products.
Concept Check 3
First translate 1 day into its fraction of a year. A year consists of roughly 250
trading days due to holidays, weekends and other market closings. So a trading
day is equal to 1/250 of a year, or about .004 of a year. This becomes our
⌬t. Then,
DIVt=St⌬t
=.03*$100*.004
=$0.012, or around a penny.
Concept Check 5
S′T=Ste(–)*
=$150*e(10–.03)*0.5
=$155.343
Concept Check 6
a. A long or a short position in an FX forward contract meets only one
condition of an arbitrage opportunity, namely zero cost today (see
Deﬁnition 1, part 3., section 4.6.1). Unfortunately, we know that the payoff
can be positive or negative at expiration so there is no guarantee of a positive
proﬁt in any state of the world. Therefore, such positions are clearly not
riskless arbitrage opportunities (see parts 1. and 2. of Deﬁnition 1).
Neither are they risky arbitrage opportunities even though they have a
chance of a positive proﬁt at expiration (see Deﬁnition 3, part 3., section
4.6.1). The reason is that part 2. of Deﬁnition 3 is violated, negative proﬁts
(costs) could arise at expiration.
b. An unexpired lottery ticket that someone lost and that you found is not
a riskless arbitrage because winning is not a certainty. However, it is a
risky arbitrage because there is a state of the world in which there is
a (large) positive payoff, that in which you have the winning number. If
you had to pay for it, it would involve a cost. Such investments are not
arbitrages.
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c. It’s nice that options, calls or puts, do have non-negative payoffs in all
states of the world. This means that part 2. of deﬁnition 3 is satisﬁed. Part
2. is also satisﬁed because of the optional character of options.
If you could get options for free (like the lottery ticket), then they would
be risky arbitrage opportunities. This is why options cost money, the market
acts to eliminate riskless and risky arbs. Therefore, options are related to
forward contracts, but they are not forward contracts.
What must be true about options in order to prevent them from being
arbitrage opportunities is that they have to sell for positive prices.
Concept Check 7
The number of BP you can buy with $US100,000,000 is the number of BP
‘in’ $US 100,000,000. Since each BP costs $1.6605, this is,
$100,000,000/$1.6605=60,222,824.45BP.
The veriﬁcation is easy,
60,222,824.45BP *$1.6605=$US100,000,000.
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