Download slides - Andrei Simonov

Finance

Finance means, literally, borrowing
– The borrower thinks it is debt
– The lender thinks it is investment
– Stock has been standard for hundreds of years
– Debt is the area of huge development
» A fascinating story of how to borrow money and invest
 at the cutting edge of financial innovation
– But can be very quantitative and full of jargons
– Takes time and efforts to master
Andrei Simonov - debt and money markets
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Old stuff
 Idea
of Lending
– Villages: lending=insurance
– Cities: implicit agreement becomes
explicit, interest appear
 Interest:
Idea of lending livestock
(give 30 cattles, expect 40 next
year)
– Hammurabi explicitly limited the
rate of interest to 20% on loans of
silver, 33 1/3 % on loans of grain.
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Not the whole picture…
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Japan
Started in 70-es from 10% of GDP, reach 160%
now…
 Cost of carrying is not very high although, low
interest rates.

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Why should you
care?
Andrei Simonov - debt and money markets
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Rate of Return: Common Measures

The term interest rate is sometimes referred to the
price a borrower pays a lender for a loan. Unlike
other prices, this price of credit is expressed as the
ratio of the cost or fee for borrowing and the
amount borrowed.
– This price is typically expressed as an annual percentage of the loan
(even if the loan is for less than one year).
– Today, financial economists often refer to the yield to maturity on a
bond as the interest rate.
Yield to Maturity



In Finance, the most widely acceptable rate of return measure for
a bond is the yield to maturity, YTM.
YTM is the rate that equates the price of the bond, P0B, to the PV
of the bond’s CF; it is similar to the internal rate of return, IRR.
In our illustrative example, if the price of the 10-year, 9% annual
coupon bond were priced at $938.55, then its YTM would be
10%.
P0B 
M
C
M


t
M
(
1

YTM
)
(
1

YTM
)
t 1
10
$90
$1000
$938.55  

 YTM  .10
t
10
(1  YTM )
t 1 (1  YTM )
Yield to Maturity

The YTM is the effective rate of return. As a rate measure, it includes:
– Return from coupons
– Capital gains or losses
– Reinvestment of coupons at the calculated YTM
Bond Equivalent Yield

The rate on bonds are often quoted as a simple annual rate (with no
compounding).

For bonds with semi-annual coupon payments, this rate can be found by
solving for the YTM on a bond using 6-month CFs and then multiplying
that rate by 2. This rate is also known as the bond-equivalent yield.
C
C
M


...

(1  r ) (1  r ) 2
(1  r ) n
C
C
M


 ... 
Y
Y
Y
(1  ) (1  ) 2
(1  ) n
2
2
2
M
 C  An Y / 2 
Y
(1  ) n
2
where r is the half - year discount rate, Y is the annualized
P
semi - annual compoundin g discount rate, and n is the
number of half - year periods.
Bond Equivalent Yield

Example: 10-year, 9% bond with semi-annual
payments, and trading at 937.69 would have a YTM
for a 6-month period of 5% and a bond-equivalent
yield of 10%.

Note: The effective rate is 10.25%.

Bonds with different payment frequencies often
have their rates expressed in terms of their bondequivalent yield so that their rates can be compared
to each other on a common basis.
Yield to Call

Many bonds have a call feature that allows the issuer to buy back
the bond at a specific price known as the call price, CP.

Given a bond with a call option, the yield to call, YTC, is the rate
obtained by assuming the bond is called on the first call date, CD.

Like the YTM, the YTC is found by solving for the rate that
equates the present value of the CFs to the market price.
CD
Ct
CP
P 

t
(1  YTC ) M
t 1 (1  YTC )
B
0
Yield to Worst

Many investors calculate the YTC for each
possible call date, as well as the YTM. They then
select the lowest of the yields as their yield return
measure. The lowest yield is sometimes referred
to as the yield to worst.
Bond Portfolio Yield

The yield for a portfolio of bonds is found by solving the rate
that will make the present value of the portfolio's cash flow
equal to the market value of the portfolio.

For example, a portfolio consisting of a two-year, 5% annual
coupon bond priced at par (100) and a three-year, 10% annual
coupon bond priced at 107.87 to yield 7% (YTM) would
generate a three-year cash flow of $15, $115, and $110 and
would have a portfolio market value of $207.87. The rate that
equates this portfolio's cash flow to its portfolio value is 6.2%:
$15
$115 $110
$207.87 

1
(1  y)
(1  y) 2 (1  y)3
 y  .062
Bond Portfolio Yield
 Note:
The bond portfolio yield is not the weighted
average of the YTM of the bonds comprising the
portfolio. In this example, the weighted average (Rp) is
6.04%:
R P  w1 (YTM 1 )  w 2 (YTM 2 )
 $100 
 $107.87 
Rp  
(.05)  
(.07)  .0604


 $207.87 
 $207.87 
 Thus,
the yield for a portfolio of bonds is not simply the
average of the YTMs of the bonds making up the
portfolio.
Federal Marketable Debt: July 31, 12
Type
Marketable Debt
Bills
Notes
Bonds
Inflation-Indexed
(TIPS)
Total Marketable
Amount
2010($
billions)
1785
4978
816
577
___
8,156
Amount
2011($
billions)
Amount
2012($
billions)
1,490
6,199
987
681
1,578
7,061
1,166
783
------9,357
------10,588
Source: Board of Governors of the Federal Reserve System,
to
http://www.treasurydirect.gov/govt/reports/pd/mspd/2012/opds072012.pdf
Types of Treasury Securities
– Market Series:
» Treasury Bills
» Treasury Bonds
» Treasury Notes
» Treasury-Inflation Index Bonds
– Non-Market Series: Securities that cannot be traded.
» Government Agency Series
» Foreign Series
» U.S. Savings Bonds
Initial offering



Initial offering: the auction process
– Treasury securities are sold by a single-price auction technique.
– Bidders can enter competitive or noncompetitive bid.
– Competitive bid states amount and yield required.
– Competitive bids accepted based on lowest yield (known as the stop
yield)
– All competitive bids offering to receive a yield equal to or lower than the
stop yield are accepted and priced on the basis of the stop yield (the
highest yield of all accepted offers).
– Noncompetitive bids receive the stop yield
Initial maturity from 13 weeks to 30 years
Some securities:
– Treasury bills : No more than 1 year, no coupon
– Treasury notes; 1 – 10 years
– Treasury bonds: > 10 years
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List of primary dealers as of Sept 2011

BNP Paribas Securities Corp.
Barclays Capital Inc.
Cantor Fitzgerald & Co.
Federal Reserve Bank of New York:
Citigroup Global Markets Inc.
www.newyorkfed.org/markets/pridealers_listing.html
Credit Suisse Securities (USA) LLC
Daiwa Capital Markets America Inc.
Deutsche Bank Securities Inc.
Goldman, Sachs & Co.
HSBC Securities (USA) Inc.
Jefferies & Company, Inc.
J.P. Morgan Securities LLC
MF Global Inc.
Merrill Lynch, Pierce, Fenner & Smith Incorporated
Mizuho Securities USA Inc.
Morgan Stanley & Co. LLC
Nomura Securities International, Inc.
RBC Capital Markets, LLC
RBS Securities Inc.
SG Americas Securities, LLC
UBS Securities LLC.
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Auction example
$10 billion issue of 10-year notes, $1 billion of non-competitive bids, $2 billion purchase
from Federal Reserve, $7 billion left for competitive bids
Offers:
$0.5 billion @ 3.10%
$0.6 billion @ 3.11%
$1.4 billion @ 3.12%
$2.6 billion @ 3.13%
$3.2 billion at 3.14%
 Accepted yield is 3.14%, the rate necessary to sell the whole issue.
 3.13% would have been sufficient for $5.1 billion of competitive bids.
 3.14% is sufficient for $8.3 billion.
 All bidders offering to accept a rate lower than 3.14% are accepted.
 Only $1.9 billion of $3.2 billion bid at 3.14% is needed, so competitive bidders at this
rate receive 1.9/3.2 = 59.375% of the amount bid for
 Noncompetitive bids, nonpublic purchases, and all accepted competitive bids will
receive notes with a yield of 3.14%.
 The coupon rate is likely to be 3.125% (in units of 1/8 of 1%, and makes the price
closest to, but no higher than par)
 What is the price of the bond?
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Salomon Squeeze






Primary dealers routinely submitted huge bids
– On June 27, 1990, Salomon submitted a bid that exceeded the total value
of the issue. Two weeks later when Salomon submitted a massive bid of
$30 billion for $10 billion in 30-year notes
Rules prevents any participants from obtaining more than 35%
Rules require dealer to obtain prior authorization from the clients when
bidding
Salomon’ Mozer admitted that on February 21, 1991, he submitted an
unauthorized bid for 35% of the $9 billion 5-year note auction in the name of
Warburg, a Salomon customer, in addition to a bid for 35% in Salomon's
name. Salomon's two bids turned out to be at the stop-out yield and the Fed
awarded $1.7 billion in notes each to Salomon and Warburg.
And then he did it again on May 22nd getting $bln10.6 out of $bln12 of 2 year
notes (together with Quantum and Tiger funds).
What is so bad about it? One player has a control over the secondary market
of T-notes, it also affects significantly REPO market.
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Price yield relationship




Inverse relationship between yields and prices
If YTM = coupon rate, price = par value
- Bond sells at par
If YTM > coupon rate, price < par value
- Bond sells at a discount
If YTM < coupon rate, price > par value
– Bond sells at a premium
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Price yield relationship
Assume the coupon rate is a constant 10 percent.
maturity
Yield : 12%
Yield: 7.8%
20
$849.54
$1,221.00
15
$862.35
$1,192.54
10
$882.36
$1,155.75
9
$891.72
$1,140.39
7
$907.05
$1,116.97
5
$926.40
$1,089.67
3
$950.83
$1,057.85
2
$965.35
$1,040.82
1
$981.67
$1,020.78
0
1000.00
1000.00
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Yield price relationship

Even if the discount rate is a constant
– Price of discount bonds will increase with time
– Price of premium bonds will decrease with time
– Par bonds will remain at par
– All bond prices converge to the face value at maturity
– Discount bonds have a built-in capital gain.
– Premium bonds have a built-in capital loss.
– Don’t be surprised to find that bond prices change everyday
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Price Path of Premium or Discount Bonds over Time
Price
Premium Bond
Par Bond
Par value
Discount Bond
0
Time to Maturity
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Bond Risk


Investment risk is the uncertainty that the actual
rate of return realized from an investment will
differ from the expected rate.
There are three types of risk associated with
bonds and fixed income securities:
1. Default Risk
2. Call Risk
3. Market Risk
Default Risk Premium

Because there is a default risk on corporate, municipal, and
other non-U.S. Treasury bonds, they trade with a default
risk premium (also called a quality or credit spread).

This premium is often measured as the spread between the
rates on a non-Treasury security and a U.S. Treasury
security that are the same in all respects except for their
default risk.

Another measure of default risk – CDS (Credit default
spreads)
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In 2014 (Q1)

https://www.capitaliq.com/media/179529-Sov_Report_Q1_2014.pdf
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Best…
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Call Risk
Call Risk is the uncertainty that the realized return will
deviate from the expected return because the issuer calls
the bond, forcing the investor to reinvest in a market with
lower rates.
Note:
 When a bond is called the holder receives the call price
(CP). Since the CP usually exceeds the principal, the
return the investor receives over the call period is often
greater than the initial YTM.


The investor, though, usually has to reinvest in a market
with lower rates that often causes his return for the
investment period to be less than the initial YTM.
WSJ, April 15, 2009
WASHINGTON -- In effort to save billions of dollars, the U.S. Treasury
Department Wednesday announced "a bond call" of the 12.5% Treasury bonds of
2009-2014 originally issued Aug. 15, 1984, and due Aug. 15, 2014.
According to Wednesday's announcement, securities not redeemed on Aug. 15,
will stop earning interest.
In general, when Treasury "calls" a bond, it stops paying interest on the date of
the call -- before the maturity date.
The Treasury said there are $4.4 billion of the 12.5% bonds issued in August 1984
outstanding -- $3.4 billion of which are held by private investors. The cusip
number on the bonds is 912810DL9.
"These bonds are being called to reduce the cost of debt financing," said Treasury.
It noted that the 12.5% interest rate is significantly above the current cost of
securing financing for the five years remaining to their maturity. Using current
market projections, Treasury estimates gross savings from the call to be about $2
billion.
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
US Treasury bonds
calls since 2000
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Call Risk Premium
Call Risk Premium:
Call RP  YTM Callable  YTM Non Callable
RP greater in higher interest rate periods
Valuation of Callable Bonds

Theoretically, the price of a callable bond, PC,
should be equal to the price of an identical, but
noncallable bond, PNC, minus the value of the call
feature or call premium, VC.
P P
C

NC
V
C
The value of the call feature can be estimated using
the option pricing model developed by Black and
Scholes.
Market Risk



Market Risk is the uncertainty that the realized return will deviate from
the expected return because of interest rate changes.
Recall, the return on a bond comes from:
– Coupons
– Interest earned from reinvesting coupons: interest on interest
– Capital gains or losses
A change in rates affects interest on interest and capital gains or losses.
Can yields become negative?
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Market Risk
A
change in interest rates has two effects on a bond's
return: price effect and interest-on-interest effect.
Price Effect: Interest rate changes affect the price of a bond;
this is referred to as price risk. If the investor's horizon date,
HD, is different from the bond's maturity date, then the investor
will be uncertain about the price he will receive from selling the
bond (if HD < M), or the price he will have to pay for a new
bond (if HD > M).
Interest-on-Interest Effect: Interest rate changes affect the return
the investor expects from reinvesting the coupon -- reinvestment
risk. Thus, if an investor buys a coupon bond, he automatically is
subject to market risk.
Market Risk
(INT on INT) 
 (INT on INT)
r
PtB 
r
PtB
(INT on INT) 
r
PtB 
Market Risk

One obvious way an investor can eliminate market risk is to purchase
a pure discount bond with a maturity that is equal to the investor's
horizon date.

If such a bond does not exist (or does, but does not yield an adequate
rate), a bondholder will be subject (in most cases) to market risk.
Reinvestment risk





Suppose you are an investor that is willing to hold bonds for 5 years, and try to choose
among the following bonds:
bond
coupon
maturity
YTM
1
5%
3
9.0%
2
6%
20
8.6%
3
11%
15
9.2%
4
8%
5
8.0%
Which bond would you choose?
– 3: highest YTM, have to sell the bond after 5 years
– 1: have to reinvest the interest as well as the principal
– 4: no reinvestment of principal, but have to reinvest interest
YTM does not help much in choosing the bonds
– If yield changes
In order to calculate the “right” return, we have to project the future
reinvestment rates and the selling price
We assume that reinvestment opportunities will be available
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Total return




In order to calculate the “right” return, we have to project the future reinvestment rates and the
selling price
Example: An investor with a 3-year investment horizon wants to buy a 20-year 8% coupon bond
for $82.84, with YTM as 10%. He expects to be able to reinvest the coupon interests at 6%, and
3 years later he can sell the bond to offer a YTM of 7%. What is the total expected return of this
investment?
4  F 6 0.03  25.874,
Step 1: the future value of coupons 3 years later is
Step 2: the price of the bond 3 years later is
((1  r ) n  1)
4  A340.035 

100
 109.851
(1.035)34
F nr 
r
Step 3: The total return is
Y
109.851  25.874
(1  ) 6 
2
82.84
 109.851  25.874 1/ 6 
Y  2  
  1  17.16%
82.84



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Treasury STRIPS








STRIPS is an acronym for Separate Trading of Registered Interest and
Principal of Securities
Converts each coupon and principal payment into a separate zero-coupon
security.
Example: a 30-year bond can be broken down into 60 separate coupon
receipts (1 for each semiannual coupon payment with the first maturing in 6
months and the last maturing in 30 years) and 1 principal receipt.
Originated as products offered by investment banks (Merrill Lynch TIGERs
and Salomon Brothers CATS).
STRIPS are direct obligations of the U.S. Treasury and thus have no credit
risk.
All Treasury notes and bonds are eligible to be stripped.
Stripped securities can be “reconstituted”
Market for fixed-principal STRIPS is very active (liquid)
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Interest rate:
the crystal ball of financial
market
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Yield curve



Yield curve
– The plot of yield to maturity against time to maturity
– The bench mark for the cost of credit for loans of various maturities
– http://www.bloomberg.com/markets/rates/index.html
The shape of yield curve
– Normal yield curve
– Inverted yield curve
– Humped yield curve
– Exhibit 5-3
The changes of the shape of yield curve
– Parallel shifts (corresponding to the duration assumption)
– Steepening
– Flattening
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Yield curve



When we want to calculate the price of a bond, which interest rate should we
use?
– Say there are two bonds, both having the same maturity of 2 years. One
has coupon rate of 8% and the yield is 7%. The other has coupon rate of
5% and the yield is 6.5%. Should we use the 7% or 6.5% as the interest
rate to calculate the price of other bonds?
– Bond yield is a function of coupon rate, and thus is bond specific.
The bottom line: the yield curve should not be used to price a bond
The appropriate interest rate must be independent of coupon rate
– The yield on zero-coupon bond (or more precisely the bond with a single
future cash flow)
» Called spot rate
– The yield curve that consists of spot rates is called the spot rate curve
– How to create the spot rate curve?
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Bootstrapping


The method to create the spot rate curve, called bootstrapping, is to calculate
the spot rates through the present value formula.
Example of boot strapping: Suppose you observe the following list of bond
prices. Calculate the spot rates.
Maturity
annual coupon bond price
0.25
0
97.5
0.50
0
94.9
1.00
0
90.0
1.5
8%
96.0
2.00
12%
101.6
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Bootstrapping
Denote Z t  PV of $1 that will be paid after t years.
With semi - annual compoundin g,
100
1
 94.9  100 Z 0.5  94.9  Z 0.5 
 0.949, R 0.5  10.75%
(1  R 0.5 / 2)
(1  R 0.5 / 2)
100
1
 90.0  100 Z1  90.0  Z1 
 0.9, R 1  10.82%
2
(1  R 1 / 2)
(1  R 1 / 2) 2
4
4
104


 96.0  4 Z 0.5  4Z1  104 Z1.5  96
(1  R 0.5 / 2) (1  R 1 / 2) 2 (1  R 1.5 / 2) 3
 Z1.5  0.8520, R1.5  10.97%.
6
6
6
106



 101.6
2
3
4
(1  R 0.5 / 2) (1  R 1 / 2) (1  R 1.5 / 2) (1  R 2 / 2)
 6 Z 0.5  6 Z1  6 Z1.5  106 Z 2  101.6
Z 2  0.8055, R2  11.11%

The zero rates
maturity
0.50
1.00
1.50
2.00
zero rates (%)
10.75
10.82
10.97
11.11
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Arbitrage free pricing of Treasuries

Example: Suppose you observe the following bond prices. Calculate the spot rates
and the price of the zero with 1-year to mature.
Bond
maturity
coupon
price
A
0.5
8.00
100.97
B
1.00
6.00
99.96
104  Z 0.5  100.97
 Z 0.5  0.9709
3  Z 0.5  103  Z1  99.96
Z1  0.9422
zero1  100  Z1  $94.22


Arbitrage opportunity: a possibility to make some positive cash flow with no
possibility of a negative cash flow
– A free lunch!
Arbitrage free pricing principle: the bond markets are efficient enough to eliminate
arbitrage opportunities
– This implies that identical cash flows must have the same price. If not, someone
could buy the cheaper flow and short sell the more expensive one. This is an
arbitrage opportunity
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Arbitrage free pricing of Treasuries

Q: What will happen if the one year zero is selling at $96.00 now?
This means there is an arbitrage opportunity that we can explore. We want to find a
combination of security Bond A and B such that we can replicate the cash flow of a
one year zero. We will then buy this synthetic security and short the zero.
w1 104  w2  3  0
w1  0  w2 103  100
w2  0.9709,
w1  0.0280
In other words we can short sell 0.028 of bond A and buy 0.9709 of bond B. In the
meantime we short sell one year zero.
Trade
short 0.028 of A
buy 0.9709 of B
short 1 of 1-year zero
total
Cash Flow now
2.83
-97.05
96
1.78
6 mo.
-2.91
2.91
0
0
Andrei Simonov - debt and money markets
12 mo.
0
100
-100
0
59
Arbitrage free pricing of Treasuries

Q: What will happen if the one year zero is selling at $90.00 now?
This means there is an arbitrage opportunity that we can explore. We want to find a
combination of security Bond A and B such that we can replicate the cash flow of a
one year zero. We will then short this synthetic security and buy the zero.
w1 104  w2  3  0
w1  0  w2 103  100
w2  0.9709,
w1  0.0280
In other words we can buy 0.028 of bond A and short 0.9709 of bond B. In the
meantime we buy one year zero.
Trade
buy 0.028 of A
short 0.9709 of B
buy 1 of 1-year zero
total
Cash Flow now
-2.83
97.05
-90
4.22
6 mo.
2.91
-2.91
0
0
Andrei Simonov - debt and money markets
12 mo.
0
-100
100
0
60
Bootstrapping



Bonds that can be used for bootstrapping are
– On-the-run Treasury issues
» Include the 3-month, 6-month, and 1-year Treasury bills, the 2-year, 5year, 10-year notes, and 20-year bonds.
» Bonds that are not on-the-run are called off-the-run.
– On-the-run Treasury issues and selected off-the-run Treasury issues
» Do not have enough on-the-run securities
» Problem of on-the-run securities
 Sometimes too expensive (yields too low because of high liquidity).
– All Treasury issues
– Treasury strips
» Easy
Par yield curve
– Easy, but cannot be used to price other securities
Only zero rate curve is usually used to price bonds.
Andrei Simonov - debt and money markets
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Term structure of interest rate


If we have the term structure of spot rates, we can calculate the price of any
treasury securities.
But why do we go through this trouble of bootstrapping? Why not just use
STRIP prices?
– STRIPs are not as liquid
» There is a liquidity spread
– The demand varies for different maturity of STRIPs
– STRIPs are taxed differently from the coupon bonds
– All of these might work to distort the relationship between STRIP and
bonds
Andrei Simonov - debt and money markets
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