Money Market Yields
„
Bank Discount Yield
10,000 − P 360
rBDY =
×
n
10,000
„
Bond Equivalent Yield
rBEY =
„
10,000 − P 365
×
P
n
Yields, not prices are quoted.
Maturity
P=
=
3,650,000
365 + n × rBEY
3,650,000
= $9,799.66
365 + 164 × .0455
Municipal Bonds
„
Asked
Ask Yield
31
4.33
4.29
4.37
Feb 25 ‘99
87
4.42
4.41
4.52
May 13 ‘99
164
4.42
4.40
4.55
Treasury Bond Quotations
„
„
„
Bid
Dec 31 ‘98
Solving the BEY formula for P, we have:
„
Days to
Maturity
BEY
BDY
What is the relation between BDY and BEY?
Treasury Bills
„
Treasury Bills
Tax Considerations
Example
Suppose that a municipal bond has a yield
of 4%, while comparable taxable bonds pay
5%. Which gives you the higher after-tax
yield if your tax bracket is 30%?
5% X (1-30%) = 3.5% < 4%
Rate
4
3/4
Yield to maturity
Price denominated in 32nds of par
Maturity
Bid
Asked
Yield to
Maturity
Ask Yield
Nov 08n
100:05
100:06
4.73
105/8
Aug 15
156:31
157:05
5.38
51/4
Nov 28
101:17
101:18
5.15
Construction of Indexes
„
„
Market capitalization-weighted
Price-weighted
1
Construction of Indexes:
Example
Construction of Indexes:
Example
„
Based on the following information, construct
price and value-weighted indexes and
compute the percentage change in each.
„
„
Price per Share
Company
Shares
Beginning
Outstanding of Year
„
End of
Year
ABC
200 million
$30
$39
XYZ
50 million
$80
$140
„
„
„
Construction of Indexes:
Example
Note the average initial price was $110/2 =
$55. The average ending price was $179/2 =
$89.5.
The percent change is (89.5 - 55)/55 = 62.7%
„
„
Construction of Indexes:
Example
Beginning Market Value
Ending Market Value
ABC
200 x $30 = $6,000
200 x $39 = $7,800
XYZ
50 x $80 = $4,000
50 x $140 = $7,000
Average
$5,000
$7,400
„
The percent change in the index is
7,400 − 5,000
= 48%
5,000
Beginning Market Value
Ending Market Value
ABC
200 x $30 = $6,000
200 x $39 = $7,800
XYZ
50 x $80 = $4,000
50 x $140 = $7,000
Average
$5,000
$7,400
„
Construction of Indexes:
Example
What is the market cap of ABC? What is the
market cap of XYZ?
ABC: 200m ($30) = $6 billion
XYZ: 50m ($80) = $4 billion
What is the return of ABC? What is the return
of XYZ?
ABC: (39-30)/30=30%
XYZ: (140-80)/80=75%
The percent change in the index is
Construction of Indexes:
Example
„
„
„
„
„
„
„
What were the portfolio weights of ABC and
XYZ in the price-weighted index and the
value-weighted index at the beginning?
In the price weighted index,
30/(30+80)=27.27%
80/(30+80)=72.73%
In the value-weighted index,
6/(4+6)=60%
4/(4+6)=40%
2
Construction of Indexes:
Example
„
„
„
„
„
„
More Examples
Recall that the return of ABC is 30% and the
return of XYZ is 75%
The price-weighted index return is:
0.2727 (30%) + 0.7273 (75%) = 62.7%
The value-weighted index return is:
0.6 (30%) + 0.4 (75%) = 48%
The price-weighted index return is higher
because XYZ gets more weight and XYZ’s
return is higher
„
More Examples
„
„
T-bill with 90 day maturity sells at a bank
discount yield of 3%.
What is the price of the bill?
T-bill, 180 days to maturity and a price of
$9,600. BDY=8%. What is BEY?
More Examples
„
„
T-bill with 90 day maturity sells at a bank
discount yield of 3%.
What is the price of the bill?
10,000 − P 360
×
n
10,000
3,600,000 − n × rBDY × 10,000
⇒P=
360
3,600,000 − 90 × 0.03 × 10,000
⇒P=
360
⇒ P = $9.925
rBDY =
More Examples
„
What is the 90 day holding period return?
More Examples
„
What is the 90 day holding period return?
HPR =
10,000 − 9,925
= 0.7557%
9,925
3
More Examples
„
What is the 90 day holding period return?
HPR =
„
More Examples
„
10,000 − 9,925
= 0.7557%
9,925
What is the 90 day holding period return?
HPR =
What is the BEY?
„
10,000 − 9,925
= 0.7557%
9,925
What is the BEY?
rBEY = 0.7557% ×
More Examples
„
365
= 3.065%
90
More Examples
Find the price of a six month T-bill with
a par value of $100,000 and a BDY of
9.18%.
„
Find the price of a six month T-bill with
a par value of $100,000 and a BDY of
9.18%.
100,000 − P 360
×
100,000
n
36,000,000 − n × rBDY ×100,000
⇒P=
360
36,000,000 − 180 × 0.0918 ×100,000
⇒P=
360
⇒ P = $95,410
rBDY =
Spread
BID
30.25
Types of Orders
ASK
30.375
The spread, or
difference between a
stock’s BID and ASK
price, represents a
form of transactions
costs when buying or
selling a stock.
„
„
„
Market Order
Limit Order
Stop-loss Order
Spread = .125
4
Market Orders
„
„
„
Market orders are simply buy and sell orders
that are to be executed immediately at
current market prices.
Assume the Bid-Ask prices for Microsoft stock
are $25.20 – $25.25.
A market sell order will be executed at $25.20
and a market buy order will be executed at
$25.25.
Stop loss Orders
„
„
„
Assume the Bid-Ask prices for Microsoft stock
are $25.20 – $25.25.
A stop loss sell order at $25 will not be
executed immediately. But it will be executed
if the ask falls to $25 or below.
A stop loss buy order at $25.50 will not be
executed immediately. But it will be executed
if the bid rises to $25.50 or above.
More Examples
„
„
„
Sell short 500 shares of Intel with current
price of $40. You give your broker $15,000.
A. If $44, (-500 X 4)/15,000=-13.33%
If $40, 0%
If $36, (-500 X (-4))/15,000 = 13.33%
B. At what price you receive a margin call?
„ (500 X $40+$15,000–500P)/500P=25%
ÎP=$56
Limit Orders
„
„
„
Assume the Bid-Ask prices for Microsoft stock
are $25.20 – $25.25.
A limit buy order at $25 can not be executed
immediately. But it will be executed if the ask
falls to $25 or below.
A limit sell order at $25.50 can not be
executed immediately. But it will be executed
if the bid rises to $25.50 or above.
Margin
Margin =
Account Equity
Market Value of Position
More Examples
C. What if Intel pays a dividend of $1? You
need to pay $1 X 500 = $500 dividends.
„ (-500 ($4) - $500)/$15,000= -16.7%
„ (-500 ($0) - $500)/$15,000= -3.33%
„ (-500 (-$4) - $500)/$15,000= 10%
(500 X $40+$15,000–500P-$500)/500P=25%
ÎP=$55.20
„
5
Holding Period Return
Ending Price - Beginning Price + Intermediate Income
Return =
Beginning Price
P − P + Dt
Rt +1 = t +1 t
Pt
Variance of A Portfolio
„
„
„
„
„
„
Variance is more complicated
If there are two assets in the portfolio,
Var(w1X +w2 Y) = w12Var(X) + w22Var(Y)
+ 2w1w2Cov(X,Y)
where Cov(X,Y)=Corr(X,Y)σ(X)σ(Y)
Standard deviation is the square root of
variance
Expected Return of a Portfolio
„
wi= value of asset #i in the portfolio / total
value of the portfolio
„
E(Rp) = w1E(R1) + w2E(R2) + … wnE(Rn)
Annualize Return
„
„
„
„
„
Sometimes you need to convert a monthly
return to an annual return
There are two ways you can do that
Suppose R is the per period return and T is
the number of periods per year.
APR = R × T
EAR = ( 1 + R ) T - 1
Annualize Variance
Historical Performance
Variance is proportional to time
Annualized Variance = σ2 × T
Series
Standard deviation is proportional to the
square root of time
Annualized Standard Deviation = σ × T0.5
Average
Annual Return
Standard
Deviation
Large Company Stocks
12.7%
20.3%
Small Company Stocks
17.7
34.1
Corporate Bonds
6.0
8.7
Government Bonds
5.4
9.2
U.S. Treasury Bills
3.8
3.3
Inflation
3.2
4.5
Distribution
– 90%
0%
+ 90%
6
Mean-Variance Utility
Function
„
Sharpe Ratio
Mean-Variance Utility Function
„
U = E ( r ) - 0.5 A σ
You’d like to maximize your utility, U.
Everything else being equal, you like higher
expected return, E (r).
Everything else being equal, you like lower
variance, σ2
You can’t compare the values of U’s across
individuals.
2
„
„
„
„
Capital Allocation Line
„
Capital Allocation Line
(CAL)
E(Rp)=15%
Rf=7%
E ( RP ) − R f
Sharpe Ratio =
„
σP
The portfolio of a risk-free asset and a
risky asset has the same Sharpe Ratio as
the risky asset
Sharpe Ratio =
E (RP ) − R f
σP
=
y (E ( R ) − R f )
yσ
=
E ( R) − R f
σ
Optimal Allocation
We can represent combinations of a risky
asset and the risk-free asset on a graph (this
is also the investment opportunity set):
Expected
Return
E(Ri)
Reward to risk ratio
• Risky Asset
„
Taking the first order derivatives of U with
respect to y and set it to zero.
E (r ) −r f − Aσ 2 y = 0
⇒ y* =
E (r ) − rf
Aσ 2
• Risk-free Asset
σ = 22%
σ
7