Investments: Lecture 6
Philip H. Dybvig
Some cutting edge topics
Washington University in Saint Louis
- Transaction costs
- Spending rules
Copyright © Philip H. Dybvig 1997, 2000
The optimal portfolio strategy to follow in the presence of
taxes is one of the most challenging aspects of investments.
It is also one of the most rewarding, since the potential
gain from saving taxes is very large and often without
risk. Earning 200 basis points on average above the market
is not so easy; good decisions on taxes may earn 300 to 500
basis points without any accompanying risk.
The purest form of tax strategy is a tax arbitrage. It is
the job of the finance professional to identify tax
arbitrage; it is the job of the legislature and the IRS to
eliminate arbitrages that have been found! As an example,
suppose two riskless investments have different amounts of
taxable income associated with them. Then a high tax rate
individual will go long the less taxed instrument and short
the other, while a low tax rate individual will take the
opposite position. This has the net effect of transferring
taxable income from that high bracket individual to the
low bracket individual, which results in a net savings of
tax for the two. Given that the two riskless assets
have different tax treatment, there is always a range of
relative prices for the two assets which makes the trade
profitable to both individuals.
In-class exercise: tax arbitrage
Consider two riskless investments over a year, ``A'' which
is taxed at the ordinary income rate and bears interest of
10%, and ``B'' which is taxed at 1/2 the ordinary income
rate and bears interest of 8%. Assume short sales are
allowed with symmetric tax treatment. Is there a tax
arbitrage for an investor in a 30% tax bracket? for an
investor in a 40% tax bracket?
Impediments to tax arbitrage
Naturally, tax arbitrage is not popular with the IRS and
legislators! One way they limit the availability of tax
arbitrage is by giving short positions a different tax
treatment than long positions. For example, a taxable
individual cannot short a municipal bond (``muni'').
And, capital losses can typically offset capital gains
but not ordinary gains.
Even when arbitrage is not available because of the tax
treatment of short sales, we can talk about one
investment dominating another from the perspective of
a particular investor.
For a par coupon bond, the after-tax cash flows usually
look like a par coupon bond. For an investor with
tax rate t, the after-tax yield is the stated
coupon yield times (1-t). For discount or
premium bonds, matters are more complicated and the
exact tax rules for amortizing gains come into play.
In the US, the code for amortizing gains is generally
consistent with a constant interest rate equal to the
yield of the bond.
Break-even tax rate
It is often useful to think about the critical
``break-even'' tax rate at which two
investments are equally attractive. For example,
consider an investor choosing to buy one of two bonds,
a Treasury bond or an insured municipal bond (which we will
treat as riskless), to be
held to maturity. They are both being issued at par. The
Treasury bond is taxable and bears interest at the rate
of 5%, and the muni bears interest at the rate of
4%. For an investor with tax rate t, the
after-tax rate-of-return on the Treasury is 5%(1-t)
and the after-tax rate-of-return on the muni is 4%.
In this example 20% (the solution to 4%=5%(1-t)
) is the break-even tax rate.
In practice, the break-even tax rate for munis is
surprisingly low. Perhaps that is because few individuals
want to invest in such an illiquid investment; part of the
illiquidity is due to the fact that while interest on munis
is not taxable, capital gains are.
In-class exercise: break-even tax rate
Suppose at a maturity of 30 years, the par Treasury
rate is 8% and the par muni rate is 6%.
What is the break-even tax rate?
Tax timing options
In the presence of taxes, holding an asset to maturity may
be much less valuable than selectively choosing whether to
the sell the asset depending on whether its value has risen
or fallen. The value of the tax savings from careful timing
of sales has been referred to as the value of a ``tax timing
option.'' Option pricing theory has been used to evaluate
the tax timing option in some special examples, although so
far we have needed to make strong and unreasonable assumptions
to perform the analysis. In practice, valuation is complicated
by the fact that the value of realizing losses depends on the
tax status of the rest of the portfolio (e.g., whether there are
capital gains that would not otherwise be offset).
General tax strategies
In general, optimal investment strategies in the presence of taxes
are very complicated and difficult to compute. (For example, see
my paper with Koo on my research page.) Here are two rules of thumb
for taxable investors:
- It is a good idea to realize capital losses whenever you can
use them. If you are a type of investor that has unlimited loss
carryforward, it may be a good idea to realize losses for future
- It is a bad idea to realize capital gains unless it is necessary
to realize gains to avoid an unacceptably high risk exposure.
Performance measurement for taxable accounts
Performance measurement for taxable accounts is a tricky matter,
especially when different managers are being evaluated in different
time periods. It is typical of optimal tax strategies that taxes
are deferred, not avoided, and that makes it difficult to choose
how to allocate gains to different periods. Several methods of
accounting for performance of taxable accounts are used in practice,
but to my mind none are completely satisfactory. I would like to
work on the problem of devising improved performance measures for
One issue in taxable investments is the so-called ``wash sale''
rule that says that you cannot deduct a loss if the position is
reinstated within a month, in which case the two events are netted
out for tax purposes. This rule is intended to minimize sales
motivated primarily for tax timing. The rule is not very effective
since there are many close substitutes in financial markets and
therefore it is easy to reestablish an economically similar situation
without invoking the wash sale rule.
The favorable tax treatment of some individual retirement plans is in
the form of allowing the funds to accumulate tax-free with taxation
of gains only at maturity. Another favorable treatment is if the
income generating the initial investment is not taxed going in but is
only taxed at maturity. Assuming a constant tax rate t (over time
and for income as well as gains), we have that an investment of $100
of pre-tax income at a riskless interest rate r for T years
will grow to $100(1-t)(1+r-tr)^T without special tax treatment,
to $100(1-t) + ($100(1-t)(1+r)^T-$100(1-t))(1-t) with tax-free
accumulation, and to $100((1+r)^T)(1-t) if taxes are paid only
at the end.
Tax-free accumulation: numbers
The following table illustrates the advantage of tax-free accumulation assuming
an interest rate of r=5% and a tax rate of t=30%. The differences
become more dramatic the higher the interest rate and the higher the tax rate.
# years | pay as you go | pay in and out | pay out only
1 | 72.45 | 72.45 | 73.50
5 | 83.14 | 83.54 | 89.34
10 | 98.74 | 100.82 | 114.02
30 | 196.476 | 232.77 | 302.54
Transaction costs represent an important but incompletely understood component
of investment performance. Here are some thoughts:
- Transaction costs imply that we will stray somewhat from the ideal mix of
assets (either across or within asset classes) before we will trade.
- As costs increase from zero, it is optimal to make the no-trade range large
to avoid transaction costs which are of first order of importance rather than
maintain risk exposure which is of second order.
- It is useful to think of costs as being spread over the holding period of
an asset. A bid-ask spread of $0.25 on a stock costing $50 and
held for 2 years can be thought of as reducing the return by about
$0.25/($50*2) = .0025 or 25 basis points. This approximation
makes the most sense over horizons much shorter than the duration of the
- Most investment managers trade more than can be justified by benefits of
trading to the clients, in part because they have to meet benchmarks.
In-class exercise: transaction costs
Consider two stocks each priced around $50/share. One has a spread of
$1/8 and the other has a spread of $3. What is the spread's
adjustment to expected return when the investment horizon is 1 year? when the
investment horizon is 20 years?
In each exchange, there is a minimum number of shares, called a round lot, that
can be purchased without facing larger commissions and spreads. In general, we
prefer to trade in round lots to avoid the higher costs. For smaller amounts
of money under investment, this effectively limits the amount of diversification
we can do (unless we buy some sort of mutual fund or index product).
There are lots of ways to execute trades. For example, for NYSE stocks, we can
send a market order or a limit order to the NYSE. Or, we can ``work'' the order,
trying to execute our trade in pieces over time when we believe we can get the
best price. Alternatively, we can send our order to an electronic clearing
network that will execute the trade overnight or during the day in an auction
with other big institutional investors. Unfortunately, there is little hard
evidence on how we can minimize execution costs, and it is hard to infer this
information from available price and quantity data. It makes economic sense
that there should be some trade-off between immediacy (or what amounts to the
same thing, probability of execution) and price, but we do not know even this
for sure. It does seem obvious that the spread and quality of execution is
potentially the largest part of transaction costs, especially for thinly traded
Spending rules and asset allocation
The remainder of this lecture is devoted to spending rules and their connection
to asset allocation. This is connected closely to my work that was awarded
the Common Fund Prize. That prize was awarded for my paper ``Duesenberry's
Ratcheting of Consumption...'' in recognition of its important for the practice
of managing university endowments.
Repeated single-period perspective
We can usefully think of the portfolio management problem as an
alternating sequence of choices of asset allocation and spending rule.
At the start of an investment period, we have a certain amount of
endowment to allocate to investments in different assets. At the end of
the period, our assets have grown or declined in value, due to our skill
and luck. At that point, we decide how much of our current endowment to
include in this year's budget and how much to re-invest. The outcome of
that decision gives us the start-of-period endowment for the next asset
The single-period perspective treats each asset allocation or
spending decision as a separate choice, linked only through the fact
that previous decisions determine what resources are now
Single-period perspective: limitations
- asset considerations
- Commitments to illiquid assets (e.g. real estate or private placements)
must be made over many periods.
- Transaction costs are incurred in a single period but may be justified
by benefits over time.
- preference considerations
- It is difficult to assess the value of returns over a single period
without knowing the pattern of use of funds in the future.
- (my main point) We have different preferences about the maintenance of
existing programs and commitments than we do about starting new
Examples of Common Practices
- Spending rule
- Plan to preserve capital.
- Spend a percentage based on some historical return on capital.
- Using a moving average to smooth income (partial recognition of my main
point, but not the best response).
- Whether the percentage is done on a unit basis is important when new
money comes in.
- Investment policy
- Ranges for proportions are chosen.
- Investment choices depend little on the history.
- If anything, losses are followed by a more aggressive policy to try to
make up lost ground.
Features of my proposal (simple extreme case)
- Spending rule
- Plan to preserve capital to an extent dictated by time and risk
- Increase spending on essential activities only when the endowment
reaches a new maximum.
- Investment policy
- Conceptually, separate out the portfolio into a committed account and a
- The committed account is immunized.
- The discretionary account is invested in fixed proportions.
- When spending increases, the funds needed to ensure committed spending
are transferred from the discretionary account to the committed account.
- The investment policy is consistent with the CPPI analyzed by Black and
More on my Proposal
- General features within the model
- declining or rising commitments over time (e.g. attrition in faculty)
- some expenditures not committed at all in advance: funded by a separate
account with proportional spending
- General features requiring adjustment of the model
- stochastic interest rates
- commitment to an irregular pattern of commitments
- reliance on other sources of income (e.g. tuition)
What is committed expenditure?
Useful models are simplifications of the world. In practice, commitments
are not absolute. For example, we can actually eliminate valued
departments or buy off tenured faculty for less than the present value
of their future salary. Or, we can possibly sell our new building if we
are unable to meet the mortgage payments. However, these are desperate
actions and it is a good approximation to think that we want to be
certain to make these expenditures.
- Investment opportunities are the same at different points in time. A
riskless asset bears a constant interest rate, and the mean and standard
deviation per unit time of the efficient risky portfolio is constant.
- Consumption (or more generally consumption of some goods) is not
permitted to decline (more general at more than some rate).
- Preferences are the same looking forward from any point in time, given
consumption does not decrease. Risk and time preferences are scale
- The model is similar to the Merton model except for the constraint on
Performance comparison with traditional strategies
- qualitative properties similar to portfolio insurance in some ways,
but without the jerky restart every year
- much better worst-case scenario (persistent declining market)
- better best-case scenario
- worse in up-and-down markets (whipsaw effect)