Principles of Finance with Excel (Основы финансов c Excel)
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PFE, Chapter 17, Efficiency page 10
Here’s an example. Suppose there are two bonds in the financial market, Bond A and
Bond B:
• Bond A sells today for $100 and pays off $110 in one year. The bond’s IRR is
110
10.00% 1
100
=−
.
• Bond B also sells today for $100. This bond has a payoff only at the end of 2 years, at
which point it pays $125. The IRR of the bond is
1/2
125
11.80% 1
100
⎛⎞
=
−
⎜⎟
⎝⎠
.
Now suppose that you’re trying to price a Bond C, which has a payoff of $23 in one year
and $1023 in 2 years. The price-additivity principle says that the way to price this bond is to
apply the IRRs calculated above separately to each year’s bond payment. In a formula:
()
2
23 1023
839.31
1.10
1.1180
Bond price =+ =
In this formula we’ve discounted the first bond payment of $23 by the market interest rate on the
one-year bonds, and we’ve discounted the second bond payment of $1023 by the IRR derived
from Bond B. Here’s the spreadsheet:
PFE, Chapter 17, Efficiency page 11
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ABCD
Bond A: maturity in one year
Price today 100
Payoff in one year 110
IRR 10.00% <-- =B4/B3-1
Bond B: maturity in two years
Price today 100
Payoff in one year 0
Payoff in two years 125
IRR 11.80% <-- =(B10/B8)^(1/2)-1
Bond C: A bond with payments at end of year 1 and year 2
Date
Payment
Present value of
payment
1
23 20.91 <-- =B16/(1+B5)
2
1023 818.40 <-- =B17/(1+B11)^2
Bond price?
839.31 <-- =SUM(C16:C17)
PRICE ADDITIVITY IN BONDS
In a picture:
PFE, Chapter 17, Efficiency page 12
Pricing a bond by additivity--schematic
Year 0 Year 1 Year 2
Bond A -100.00 110.00
IRR 10.00%
Bond B -100.00 125.00
IRR 11.80%
Bond C 23.00 1,023.00
Present value
of payment
-839.31 20.91 818.40
Use Bond A's IRR to
price the first payment
from Bond C.
Use Bond B's IRR to
price the second
payment from bond C
Bond C price=value of
year-1 plus value of
year 2 payments.
Figure 17.1
To sum up: We’ve used market discount rates derived from bonds with only one
payment to additively price a bond with multi-year payments.
Additivity, example 2: Open-end mutual funds
The webpage of the United States Securities and Exchange Commission (SEC) defines a
mutual fund as:
A mutual fund is a company that brings together money from many people and
invests it in stocks, bonds or other assets. The combined holdings of stocks, bonds or
other assets the fund owns are known as its portfolio. Each investor in the fund owns
shares, which represent a part of these holdings.
http://www.sec.gov/investor/tools/mfcc/mutual-fund-help.htm
Figure 17.2 gives some more information from the SEC about mutual funds.
PFE, Chapter 17, Efficiency page 13
Mutual Funds
A mutual fund is a company that pools money from many investors and invests the money
in stocks, bonds, short-term money-market instruments, or other securities. Legally known
as an "open-end company," a mutual fund is one of three basic types of investment
company. The two other basic types are closed-end funds and Unit Investment Trusts
(UITs).
Here are some of the traditional and distinguishing characteristics of mutual funds:
• Investors purchase mutual fund shares from the fund itself (or through a broker for
the fund), but are not able to purchase the shares from other investors on a
secondary market, such as the New York Stock Exchange or Nasdaq Stock Market.
The price investors pay for mutual fund shares is the fund’s per share net asset value
(NAV) plus any shareholder fees that the fund imposes at purchase (such as sales
loads).
• Mutual fund shares are "redeemable." This means that when mutual fund investors
want to sell their fund shares, they sell them back to the fund (or to a broker acting
for the fund) at their approximate NAV, minus any fees the fund imposes at that
time (such as deferred sales loads or redemption fees).
• Mutual funds generally sell their shares on a continuous basis, although some funds
will stop selling when, for example, they become too large.
• The investment portfolios of mutual funds typically are managed by separate entities
known as "investment advisers
" that are registered with the SEC.
Mutual funds come in many varieties. For example, there are index funds
, stock funds, bond
funds, money market funds, and more. Each of these may have a different investment
objective and strategy and a different investment portfolio. Different mutual funds may also
be subject to different risks, volatility, and fees and expenses
.
All funds charge management fees for operating the fund. Some also charge for their
distribution and service costs, commonly referred to as "12b-1" fees
. Some funds may also
impose sales charge or loads
when you purchase or sell fund shares. In this regard, a fund
may offer different "classes
" of shares in the same portfolio, with each class having different
fees and expenses.
Source: http://www.sec.gov/answers/mutfund.htm
Figure 17.2. Description of mutual funds from the SEC website.
PFE, Chapter 17, Efficiency page 14
Does it matter if you bundle securities together in a mutual fund? How should the price
of such a fund be determined? The principle of pricing additivity gives us a way to handle this
problem—it suggests that the price of a mutual fund should be determined by the market prices
of all the fund’s assets.
As a simple example, suppose you start a new company, the Super-Duper Fund, which
sells a mutual fund of a very specific type.
•
Super-Duper currently has 10,000 shareholders, each of who has invested $100—so that
the total assets of the company are $1,000,000.
•
Super-Duper’s money is currently invested 50% in shares of IBM (currently trading at
$100 and in shares of Intel (currently trading at $50). Thus the company currently owns
5,000 shares of IBM and 10,000 shares of Intel.
•
The number of shares in the fund is flexible.
4
Right now, there are 10,000 shares, but this
number can go up or down:
o If a shareholder wants to sell, you promise to liquidate his proportional part of the
fund’s assets. So if Uncle Joe from Winona, who own one share worth $100,
wants to sell his share in the company, Super-Duper will sell ½ share of IBM and
1 share of Intel and repay him his $100. Now the fund will have $999,900 in
assets, still invested 50% in IBM and 50% in Intel.
o If any new shareholders want to join, Super-Duper will buy—per $100 of new
funds which come into the company—$50 of IBM and $50 of Intel.
5
4
In the jargon of mutual funds, this makes it an open-end fund. Our next example considers a closed-end fund.
PFE, Chapter 17, Efficiency page 15
Suppose that today no one sells or buys shares in the fund. The asset value of the Super-
Duper fund today is $1,000,000. Now suppose that tomorrow the price of IBM is $110 and the
price of Intel is $48. Then the value of a fund share is $103 (cell C14 below):
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ABCDE
Today
Tomorrow
before new
fundholders
Number of shareholders 10,000 10,000
Portfolio
Price of IBM 100 110
Price of Intel 50 48
Portfolio composition
Shares of IBM 5,000 5,000
Shares of Intel 10,000 10,000
Total fund value 1,000,000 1,030,000 <-- =C10*C6+C11*C7
Value of 1 fund share 100 103 <-- =C13/C3
Number of shareholders 10,500
Total fund value 1,081,500 <-- =B17*C14
Portfolio composition
Shares of IBM 4,915.91 <-- =B18*50%/C6
Shares of Intel 11,265.63 <-- =B18*50%/C7
SUPER-DUPER OPEN END MUTUAL FUND
Tomorrow:
after new fundholders
New shares are created at the
current fund share price, so the
fund is now worth
10,500*$103=$1,081,500 .
Now suppose that at the close of the day tomorrow another 500 individuals buy shares of
the fund. This means that they pay 500*$103 = $51,500 to buy shares in the fund. Assuming
that the fund sticks to its current policy of splitting its investment equally between IBM and
Intel, the total fund value of $1,081,500 (cell B18 above) will now be invested in 4,915.91 shares
of IBM and 11,265.63 shares of Intel.
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Actually, Super-Duper Fund does all this at the end of the day. So if Uncle Joe wants to sell his share and Aunt
Maude wants to invest an additional $100, Super-Duper has a wash, and it can save on the transactions costs of
buying and selling. Every penny helps!
PFE, Chapter 17, Efficiency page 16
In an open-end mutual fund the number of shares is flexible. New shareholders buy into
the fund at the per-share value of the fund, and shareholders in the fund who want to cash out
cash out at the per-share value of the fund. At any point in time, the per-share value of the fund
is given by the formula:
()
--
'
fund net asset value NAV
open end fund per share value
number of shares in fund
market value of fund s portfolio fund expenses
number of shares in fund
=
−
=
.
Notice that we’ve introduced a new bit of jargon: A mutual fund’s net asset value (NAV) is the
market value of the fund’s portfolio minus fund expenses.
Mutual fund costs
The fund has some expenses which are charged to the fund holders and deducted from
the value of the fund. These include the costs of buying and selling shares. Another fund cost is
the cost of paying the managers: Typically fund managers charge their clients a percentage cost.
If your fund charges 1% (in the U.S. this is typical), then this cost ($10,000 per year in our
example) has to be taken out of the value of the fund.
Our Super-Duper Fund doesn’t charge shareholders to buy or sell shares in the fund.
However, there are also mutual funds which charge to buy shares. These so-called front-end
load mutual funds are more expensive than no-load funds. Suppose, for example, that Super-
Super-Duper were to charge a 7% front-end load. Then you would pay $107 (=$100 + 7% front-
end load) to buy a share of the fund. Front-end loads are obviously expensive; mutual fund
PFE, Chapter 17, Efficiency page 17
salespeople sometimes justify these extra charges as an appropriate price to pay for the expertise
of better fund management, but there is almost no evidence to show that this is true.
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Example 3: Closed-end mutual funds—when additivity fails
Value-additivity doesn’t always work. In this subsection we give an example of closed-
end mutual funds. These are investment companies for which value additivity usually fails. A
closed-end mutual fund is an investment company with a fixed number of shares. Like open-end
mutual funds, closed-end mutual funds invest in a portfolio of stocks. As opposed to an open-
end mutual fund, however, where the number of shares can be expanded or contracted, a closed-
end mutual fund has a fixed number of shares which are sold on the stock market. The company
issues no more new shares, and the market price fluctuates with supply and demand for the
fund’s shares.
Here’s an example. The Chippewa Fund is a closed-end fund which looks a lot like the
Super-Duper Fund. Like Super-Duper, Chippewa has 10,000 shares. Chippewa’s share portfolio
currently consists of $500,000 of IBM stock and $500,000 of Intel stock, and its shares are
registered on the Chippewa Stock Exchange. The fund has no other assets.
What should be the price of a Chippewa Fund share? It seems that it should be equal to
the per-share value of the fund’s assets—in our case $100 per share (as you saw in our
discussion of open-end mutual funds, the finance jargon is the net asset value of the Chippewa
Fund is $100 per share). But checking the newspapers, you find that the share price of the
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Recall that in Chapter 12 we discussed a technique for judging mutual fund performance using the capital asset
pricing model (CAPM). Finance researchers employing this and more sophisticated techniques find little evidence
that front-load mutual funds outperform no-load mutual funds.
PFE, Chapter 17, Efficiency page 18
Chippewa Fund is $90, below its net asset value. A back-check of the prices of the Chippewa
Fund shows you that Chippewa almost always sells for less than its net asset value. In fact a
finance-knowledgeable friend has told you that almost all closed-end funds sell for less than their
net asset value.
The reasons why closed-end funds sell at a discount are not well-understood.
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What is
well-understood, however, is that it is difficult to arbitrage a closed-end fund discount—meaning
that it is difficult for investors to make money out of the discount and, by making money, cause
the discount to disappear. Suppose, for example, that shares of Chippewa fund trade below $100
net asset value, say at $90. Then both existing and potential fund shareholders have a problem:
On the one hand, the existing shareholders are holding $100 market-value shares worth only $90.
If the closed-end fund were to break up, existing shareholders would get the net asset value of
$100. So all the shareholders would in principle favor breaking up the fund, but no individual
shareholder would want to sell his individual shares before such a breakup. A potential new
shareholder is faced with the same problem: He gets $100 (market value) of shares for $90, but
he has no guarantee that the value of the closed-end fund will ultimately get back to the market
value.
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A readable survey of closed-end fund discounts is a paper by Elroy Dimson and Carolina Minua-Paluello, “The
Closed-End Fund Discount,” which is available on the Web. In their introduction, they write: “Closed-end funds
are characterized by one of the most puzzling anomalies in finance: the closed-end fund discount. Shares in
American funds are issued at a premium to net asset value (NAV) of up to 10 percent, while British funds are issued
at a premium amounting to at least 5 percent. This premium represents the underwriting fees and start-up costs
associated with the flotation. Subsequently, within a matter of months, the shares trade at a discount, which persists
and fluctuates … . Upon termination (liquidation or ‘open-ending’) of the fund, share price rises and discounts
disappear.”
PFE, Chapter 17, Efficiency page 19
This whole scenario may sound somewhat improbable, but in fact there are many closed-
end mutual funds. Figure 17.3 gives an actual example: Tri-Continental Corporation is a closed-
end fund registered on the New York Stock Exchange. On 23 November 2001 the fund’s shares
were worth 11.18% less than the market value of the fund’s portfolio. This closed-end fund
discount is pervasive throughout the closed-end fund industry.