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Introduction to Mathematics of FinanceDr. Tsang

Chapter 6

Bonds & fixed income securities - I

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Introduction• In this Chapter, we extend the concepts and

techniques covered to date and applies them to common financial securities such as bonds and preferred stocks.

• An investment that provides a return in the form of fixed periodic payments and the eventual return of principal at maturity is called Fixed-Income Security.

• Taxes and investment expenses are to be ignored unless stated.

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What are Bonds ?• A debt investment in which an investor loans

money to an entity (corporate or governmental) that borrows the funds for a defined period of time at a fixed interest rate.

• Bonds are used by companies, municipalities, states and U.S. and foreign governments to finance a variety of projects and activities.

• Bonds and stocks are both securities, but the major difference between the two is that stockholders have an equity stake in the company (i.e., they are owners), whereas bondholders have a creditor stake in the company (i.e., they are lenders).

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Why study the bond market?• NYSE, the world's largest stock exchange by market

capitalization of its listed companies at US$28.5 trillion as of May 2008

• Amounts outstanding on the global bond market increased 6% in 2008 to $83 trillion– Average daily trading volume in the U.S. bond market is

$822 billion ($100 billion for NYSE)• Bond market is the largest of all capital markets (for

the raising of capital).– China has about $2.45 trillion (March 2010) in foreign reserves.– Most of them are invested in the US government-issued

debt securities for the safety and liquidity.

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Classification of Fixed-Income Securities

• 1. Treasury Securities– U.S. Treasury securities (bills, notes, bonds)– Bunds, JGBs, U.K. Gilts . . .

• 2. Federal Agency Securities– Securities issued by federal agencies (FHLB, FNMA . . .)

• 3. Corporate Securities– Commercial paper– Medium-term notes (MTNs)– Corporate bonds . . .

• 4. Municipal Securities• 5. Mortgage-Backed Securities

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More detail

• The US was the largest market for bonds in 2008 accounting for 43% of amounts outstanding followed by Japan with 16%.

• A quarter of amounts outstanding in the US were in mortgage backed bonds, a fifth in corporate debt and 18% in Treasury bonds with most of the remainder in Federal Agency securities and municipal bonds.

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According to the European Capital Markets Institute: There is €5 trillion ($6.5tn) of euro-zone government debt outstanding

roughly $19 trillion

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INVESTORS in bond market

1. Governments2. Pension funds3. Insurance companies4. Commercial banks5. Mutual funds6. Foreign institutions7. Individual investors

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Interesting books on the bond market by Michael Lewis

• The Big Short: Inside the Doomsday Machine, how a 32-year-old investor spotted the huge bubble in the subprime-mortgage bond market in 2004

• Liar's Poker, a semi-autobiographical book by Michael Lewis describing the author's experiences as a bond trader on Wall Street during the late 1980s, one of the books that define Wall Street during that periods.

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Terminologies

• A bond is a promise by the issuing company (issuer) to repay (redemption) an original indebtedness of a fixed amount (the face value or par value of the bond) at a given time in the future (the maturity date). The time span is called the term of the bond.

• The bond pays interest periodically at a prescribed rate, known as the coupon rate.

• Bonds are commonly referred to as fixed-income securities and are one of the three main asset classes, along with stocks and cash equivalents.

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Types of bonds

• Accumulation (zero-coupon) bond – redemption price includes the principal plus all interest.

• Coupon bond – coupons are periodic interest payments (usually every six months, at least in US) made by the issuer prior to the redemption.

• Callable bond – may be redeemed early at the discretion of the issuer to provide a degree of protection against interest rate risk.

• Putable bond – lender has an option to redeem prior to maturity date. Much less commonly used.

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Preferred stocks

• It is a type security provides a fixed rate of return (dividend) similar to bonds.

• It is supposedly an ownership security of a corporation, not a debt security.

• In reality, it is a second-class ownership security of a corporation, because preferred stockholders have very limited voting-right.

• Some of them can be converted to regular/common stock and is called convertible preferred stocks.

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Common stock

• It is a type of ownership security that does not earn a fixed dividend, set by the corporation’s board of directors.

• Stockholders have voting rights.

• They are the true owners of the corporation.

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Example6.1: zero-coupon bondsA zero-coupon bond is a corporate, Treasury, or municipal

debt instrument that pays no periodic interest. Typically, the bond is redeemed at maturity for its full face value. It will be a security issued at a discount from its face value, or it may be a coupon bond stripped of its coupons and repackaged as a zero-coupon bond.

A zero-coupon bond will pay $1000 at the end of 10 years and is selling for $400 now. Find the yield rate (yield to maturity) convertible semiannually that would be earned by a buyer.

400 (1 + j)^20 = 1000 , j=0.04688

The yield rate convertible semiannually is 2j.

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Example6.2• A 13-week Treasury bill matures for

$10000 and is bought at a discount to yield an annual rate of 7.5%. What is the price to purchase it?

Using actual/360, the price is

p * (1+ 13*7*0.075/360) = 10000

p = 10000/ (1+ 13*7*0.075/360) = 9813.94

If the discount rate is 7.5%, the purchase price is

10000* (1- 13*7*0.075/360) = 9810.42

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Zero coupon bonds• Zero coupon bonds are the simplest fixed-income

securities.• Prices of zero coupon bonds provide information about spot interest

rates and vise versa.

• On 2001.08.01, STRIPS (Separate Trading of Registered Interest and Principal of Securities, which are the securities obtained when trading the coupons and principal of bonds separately) are traded at the following prices:

For the 5-year STRIPS, we have

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Coupon bonds

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Pricing the Bond coupons

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Pricing the Cash Flows of a Bond

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Example6.3: How to value a coupon bond?1

Suppose a bond has a face value of $1000, will pay semi-

annual interest at the (coupon) rate of 5%, and will be

due 10 years after issue.

This means that the owner of one such bond will receive

$25 (=1000*5%/2) every six months for a total of 20

payments, together with a payment of $1000 10 years

after the date of issue.

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If the purchaser wishes to receive a 6% return, compounded

semiannually, then the present value at issue is

A purchaser content with a return of 4% compounded semi-

annually would value the bond at

Example6.3: How to value a coupon bond?2

Price of a bond is equal to the present value of future coupons plus that of the redemption

value.

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For a typical bond that promises to pay a fixed coupon payment of C/2 every six months and repay the par amount FV (face value) at maturity, current price P is:

Price of a bond is equal to the sum of present values of future coupons plus that of the redemption value.

where r /2 is the appropriate semiannual yield rate and n is the remaininglife of the bond measured in the unit of the coupon payment period (six months).

Coupon Bond valuation equation

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The five key bond variables

• Price, par value, coupon rate, remaining life, and the rate (r)used in the discount factor—are tied together.

• Any four of them determine the fifth.• “r” is also called yield to maturity (YTM) or the

internal rate of return (IRR).• There is no easy way to calculate the IRR or

YTM. Usually a computer will solve the equation numerically. [Excel has an IRR function which solves the equation numerically.]

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Three years after issue, the bonds are priced to yield 7%

(compounded semiannually). The price per $1000 bond is

Example6.3: How to value a coupon bond?3

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MATLAB Financial Toolbox bndprice -- Price a fixed income security from yield to maturity (SIA compliant)

Syntax

[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity)

[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity,

Period, Basis, EndMonthRule, IssueDate, FirstCouponDate,

LastCouponDate, StartDate, Face)

Period--Coupons per year of the bond. Default: 2

Face--Face or par value. Default: 100

Parameters with default values

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Using the “bndprice” function - example

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Yield

Price

Coupon rate=5% maturity=5yr period=2

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What are the factors that determine the price of a bond?1

• The price of a bond is a function of prevailing interest rates. Interest rate risk

• As rates go up, the price of the bond goes down, because that particular bond becomes less attractive (i.e., pays less interest) when compared to current offerings.

• As rates go down, the price of the bond goes up, because that particular bond becomes more attractive (i.e., pays more interest) when compared to current offerings.

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• When you buy a bond, you may pay a premium. In other words, you may pay more than the face (par) value. For example, a bond with a face value of $1,000 might sell for $1050, meaning at a $50 premium. Or, depending on the markets and such, you might buy a bond for less than face value, which means you bought it at a discount.

• Bond price also fluctuates in response to the risk perceived for the debt of the particular organization. Credit risk

• For example, if a company is in bankruptcy, the price of that company's bonds will be low because there may be considerable doubt that the company will ever be able to redeem the bonds.

What are the factors that determine the price of a bond?2

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Credit rating agency - credit risk

• The credit worthiness (i.e., the ability to pay back a loan) of a bond issuer (from companies to national governments) is rated by credit rating agency, a company that assigns credit ratings to provide independent, easy-to-use measurements of relative credit risk to increases the efficiency of the bond market.

• Moody’s Investors Service, Standard & Poor’s, Fitch Ratings (US), Japan Credit Rating Agency, Dominion Bond Rating Service (Canada)

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As of today, the British government must pay a higher interest rate to borrow money for ten years than either the

Italian or the Spanish governments, despite the extraordinary ructions going on within the eurozone.

The yields on 10-year British Gilts have risen to 4.06pc, compared to 4.05pc and 4.01pc for

Spain. So if international bond markets are turning wary of Club Med sovereign bonds, they seem

even more distrustful of British bonds.

… … …

While Britain went in to this crisis with a much lower public debt than Greece or Italy (though higher total debt

than either), it now has the highest budget deficit in the OECD (Organisation for Economic Co-operation and

Development) rich club — and perhaps the world — at 13pc of GDP.

Britain and the PIGS

By Ambrose Evans-Pritchard Economics Last updated: February 15th, 2010

http://blogs.telegraph.co.uk/finance/ambroseevans-pritchard/100003763/britain-and-the-pigs/

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Greece’s fortunes were dealt yet another blow as Standard & Poor’s slashed its credit rating to junk status - BB+ - the first time that has happened to a euro member since the single currency was created, pushing yields on 10-year Greek bonds up to a record 9.73pc.

The credit-rating agency also cut Portugal’s sovereign debt ratings by two notches to A-, as the swirling storm hit the country with full-force. … Yields on 10-year Portuguese bonds spiked 48 basis points to 5.67pc, replicating the pattern seen as the Greek crisis started.

Portugal’s public debt will be just 84pc of GDP by the end of this year, far lower than that of Greece, at 124pc. However, its private debt is much higher and data from the IMF shows that its external debt position is worse.

27 Apr 2010

http://www.telegraph.co.uk/finance/economics/7640783/ECB-may-have-to-turn-to-nuclear-option-to-prevent-Southern-European-debt-collapse.html

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When the bond market effectively forced former US president Bill Clinton to balance the US budget deficit in the 1990s with a massive sell-off of US Treasury bonds, his adviser James Carville said: "I used to think that if there was reincarnation, I wanted to come back as the president or the pope or as a 400 baseball hitter. But now I would like to come back as the bond market. You can intimidate everybody".

“Bond traders who could decide Portugal's fate tell of a 'crazy, fun' week”

http://www.guardian.co.uk/business/2010/apr/30/debt-crisis-bond-traders-profile

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Protestors clash with riot police during a protest outside the

Parliament in Athens

Athens in turmoil as riot police and

protesters clash in Greece

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Book value of a bond• In accounting, book value is the value of an asset

shown on a firm's balance sheet.• When a bond is purchased at a discount/premium, each

subsequent coupon payment can be broken down to interest and principal, in much the same way that a loan is amortized.

• The book value of a bond is simply the present value of the remaining coupon payments and redemption value computed using the yield rate at which the bond was purchased.

• The interest part of the current coupon is simply the book value at the previous time point multiplied by the yield rate;

• The mark up or mark down is the remainder of the coupon.

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Example: Book value of a bond1

The bond of the previous example(6.3-3) has an initial book value equal

to the purchase price of 890.79. The semi-annual coupon is 25, so the interest

part of the first coupon after purchase is 890.79(.035) = 31.18 and the mark up is

31.18-25 = 6.18. The book value after the payment of the coupon is 890.79+6.18 =

896.97.

The computation above shows that the interest payment is larger than the coupon. The

appreciation in the book value of the bond is attributable to the unpaid interest to the bond

holder.

Writing the purchase price as 890.79 = 1000 + (25 - 35)a(14, .035) shows the discount at

purchase can be viewed as a loan which is amortized by payments of 25 - 35 = -10. The

principal part of each loan payment is then the amount of markup.

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Example: Book value of a bond2

If the same bond (example6.3-3) had been purchased to yield 3% , the purchase price would have been 25 a(14, 0.015) + 1000(1.015)^(-14) = 1125.43

In this case, the interest part of the first coupon after purchase is 1125.43(.015) = 16.88 and the mark down is 25 - 16.88 = 8.12. The book value after the payment of the coupon is 1125.43 - 8.12 = 1117.31. In this case, the excess of the coupon over the interest is a return of the buyer’s capital.

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A bond with redemption value of 1000 and coupon rate of 6%

payable semi-annually for 20 years is sold 5 years after issue at a

price to yield 4%. What price was paid for the bond?

Example6.4: known yield rate

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• Suppose a $10,000 par bond has 6% semi-annual coupons and matures in 10 years. What is the yield rate if the price is $9,000?

Example6.5: unknown yield rate

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Yield rates are not necessarily unique

• Suppose payments of 100 now and 109.20 two years from now are to be made in return for receiving 209 one year from now.

• The yield rate i then satisfies

100(1 + i)^2 + 109.20 = 209(1 + i)

from which i = .04 or i = .05.

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Price between coupon payments dates 1

• Preceding valuation assumed the price (book value) is calculated just after a coupon has been paid.

• What price that a purchaser would pay for the bond a fractional time t through a coupon period?

• Assume that the purchaser will – obtain a yield rate equal to that of the current

bond holder. – receive all of the next coupon.

• The current holder would expect to receive part of this coupon as interest for the period.

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How to allocate value between accrued interest and price for the bond?

The purchase price including accrued interest is known as the "full" or "dirty price". The price excluding accrued interest is known as the "flat" or "clean price“, which is the book value and is usually quoted in the financial press. It changes smoothly through time.The full price fluctuates due to coupon accrual.

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Graph of the purchase price of a bond

which is equal to the flat price + accrued interest (assumed that the flat price remains constant over the 2 years,) The accrued interest must be calculated according to formula. Note that the purchase price steadily increases each day until reaching a peak the day before an interest payment, then drops to minimum on the day of the payment.

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Accrued interest

Between payment dates, the price of the bond will be the flat price + the accrued interest. Accrued interest is the interest that has been earned, but not paid, and is most simply calculated by the following formula:

Accrued Interest = Interest Payment x

Number of Days Since Last Payment

─────────────Number of daysbetween payments

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The theoretical method argues that the flat price should be the book value B after the preceding coupon payment, accumulated by (1+i)^t, where i is the yield rate, giving the flat price of B(1+i)^t. The accrued coupon is the coupon amount c accumulated [by S(t,i)] c((1+i)^t -1)/i. The book value is then the difference B(1+i)^t -c((1+i)^t -1)/i.The practical method argues that the flat price is the book value B after the preceding coupon accumulated at simple interest, giving B(1 + it). The accumulated coupon is tc, and the book value is B(1 + it) - tc.The semi-theoretical method is the most widely used method, and has been accepted as the standard method of calculation by the securities industry. The flat price is determined as in the theoretical method, and the accrued coupon is determined as in the practical method.

Determining the full price of a bond: Three methods

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[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity)

[Price, AccruedInt] = bndprice(Yield, CouponRate, Settle, Maturity, Period, Basis, EndMonthRule, IssueDate, FirstCouponDate, LastCouponDate, StartDate, Face)

Price - NUMBONDS-by-1 vector for the clean price of the bond. The dirty price of the bond is the clean price plus the accrued interest. It equals the present value of the bond cash flows of the yield to maturity with semiannual compounding.

AccruedInt - NUMBONDS-by-1 vector for the accrued interest payable at settlement.

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Assignment 6.1

Calculate & graph of the purchase price ("full" or "dirty price“) of the following bond as function of time from 1-Jun-2010 to 1-Jun-2012, compounded semi-annually:

coupon rate=0.04, yield rate=0.06, face value=100, maturity date=1-Jun-2016,

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Callable bond• Some corporate bonds have a call provision that gives the

issuer the right to call or redeem the bonds after a few years.

• For example, the issuer may have the right to call an issue of bonds at any time after five years at par value. Alternately, there may be a declining schedule of redemption price such that the issuer pays 105% of par value to call during the sixth year, 104% to call in the seventh year, and so forth.

• If interest rates go down sufficiently, issuer can call and replace the bonds with new ones or other forms of debt at a lower rate.

• Callable bond is a regular bond with an enbedded call option for the lender. So the pricing of callable bond is more complicated.

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Example - callable bond 1• A 100 par value 4% bond with semiannual

coupons callable at $109 starting 5 years after issue for the next 5 years, at $104.50 starting 10 years after issue for the next 5 years, and maturing at $100 at the end of 15 years. Find the highest price which an investor can pay and still be certain of a yield of 5% convertible semiannually.

• Since the bond will not be called at a yield of 5% (higher yield causes the bond price to move down), the price is100v30 + 2*a(30, 0.025) = 89.53where v=1/1.025

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Example - callable bond 2

Same problem. Find the highest price which an investor can pay and still be certain of a yield of 3% convertible semiannually.

Since lower yield causes the bond price to move up, the bond will be called at some point. So we calculate the lowest possible PV of all call schemes.

v=1/1.015, v30=0.63976

109vn + 2*a(n, 0.015) for 9<n<20 n=10 112.37

104.5vn + 2*a(n, 0.015) for 19<n<30 n=20 111.93

100vn + 2*a(n, 0.015)=112.01 for n=30

The answer is the lowest of these, $111.93, for redemption 10 years after issue.

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Assignment 6.2

A 100 par value bond with 5% semi-annual coupons is callable at $110 starting 5 years after issue for the next 5 years, then at $105 starting 10 years after issue for the next 5 years, and maturing at $100 at the end of 15 years.

Find the highest price which an investor can pay and still be certain of a yield of 3% convertible semi-annually. Verify the procedure used in previous example is equally applicable for this problem by evaluating (using MatLab) the lowest possible PV of all call schemes.