FINVEST Fundamentals of Bond Valuation Slides

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FUNDAMENTALS OF BOND VALUATION

Investment Analysis & Portfolio Management (FINVEST)

K31 & K32 – 2nd Term – AY 2009-2010

Mr. Clive Manuel O. Wee Sit, CTP

Chapter Outline

• Bond Definition• Typical Issuers of Bonds• Types of Bonds as to Place and Currency of Issue• Traditional Types of Bonds (as to Timing of Cash

Flows, Optionality, and Convertibility)• Bond Payment Structures• Elements of a Bond• Bond Price vs. Yield-to-Maturity• General Bond Pricing Formula• Bond Formula Adjustments• Price Distinctions• Holding Period Return

De La Salle University De La Salle University -- Manila Manila –– College of Business and EconomicsCollege of Business and Economics

Bonds

• Are capital market instruments – have a maturity of more than 1 year

• Are fixed income securities– a contract specifying the timing and amounts

of cash flows over time

• Is a contract of an institution which binds the institution that issued the bond to pay certain amounts of money to the owner of the bond on certain dates.


Typical Issuers of Bonds

• Supranationals

– i.e. government or government-related issuers; in the Philippines, an example would be PLDT

• Multilateral Organizations

– e.g. World Bank, Asian Development Bank, etc.

• Private Sector Corporates

– e.g. British Petroleum

• Banks

• Municipal Governments


Types of Bonds as to Place

and Currency of Issue

• Domestic Bonds

• Foreign Bonds

• Eurobonds


Domestic Bonds

• Issued by a borrower resident in the

country and denominated in the local currency

• Example: Singapore Airlines issues bonds denominated in Singapore Dollars

in Singapore.


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Foreign Bonds

• Issued by a borrower which is non-

resident in the country in which the bond is being issued

• Example: General Electric USA issues a 10-year bond in Singapore Dollars and

sells this in Singapore.


Eurobonds

• A Eurobond is any bond with a payment currency which differs from the currency of the country in which the bond is issued.

• Eurobond markets developed as a means of escaping domestic regulations, especially (withholding) tax, and now rival the size of domestic markets.

• “Euro” is used in the same sense as Eurodollars.

• Example: PLDT issues USD-denominated bond in Europe.


Traditional Types of Bonds

• Level Coupon or Fixed Rate Bonds– The coupon is a fixed amount paid throughout the

bond’s tenor.– The principal is a bullet payment at maturity.

• Variable Coupon or Floating Rate Bonds– The coupon is not a fixed amount (variable).– The coupon is normally based on a benchmark

PLUS a spread.– Floating rate bonds have coupon rates that are reset

periodically according to some pre-determined benchmark.

– The principal is a bullet payment at maturity.


Bond Payment Structures

• Bullet

– Bullet repayment of principal means that 100% of the principal is paid at the maturity

of the bond.

• Amortizing

– Amortizing principal repayment means that some of the principal is paid prior to maturity.



• Zero Coupon Bonds (Zeroes)

– A pure discount bond that does not have periodic coupon payments

– The investor merely receives a bullet payment of its maturity value.

– Since zero coupon bonds repay the holder only on maturity, the price is substantially below the principal value (sold at a deep

discount).



• Income Bonds

– Bonds whose coupons are dependent on the profitability or income generation of the issuer

– The coupons are usually non-cumulative.

• Callable Bonds

– Normally a fixed rate bond with a call option embedded by the issuer

– This call option involves a price higher than the face/maturity value and a call date which is significantly ahead of its maturity date.


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• Convertible Bonds

– Bonds with a convertibility option embedded by the issuer

– These bonds are normally convertible to the issuer’s equity at a stated strike price and number of shares (other issuer assets may

also be used).

– The convertibility date is normally its maturity

date.


Elements of a Bond

• Bonds are a claim to a set or stream of future cash flows.

• They have two primary cash flow components:– Face/Par/Redemption/Maturity Value (Principal)

Component– Periodic Coupon (Interest) Payment Component

• The face value (principal) of a bond is the amount that the issuer agrees to repay the bondholder at the maturity date.

• The periodic amount of interest payment to bondholders during the life of the bond is called the coupon.


Elements of a Bond

• The coupon payments represent the interest on the bond and are made at regular intervals.

• Final interest payment and principal are paid at specific date of maturity (terminal value; final

cash flows).

• The coupon amount divided by the face value is the coupon rate.

• A bond’s price and coupon rate are often expressed as a percentage of the face value.


Sample Bond Table

• Issuer: company, state, or country

• Coupon: fixed interest rate that issuer pays to lender (investor)

• Maturity Date: date when borrower will pay the lender the

face value (principal) back

• Bid Price: a certain percentage of

the bond’s face value (100) that someone is willing to pay to receive

the entire face value upon maturity

• Yield: indicates the annual return

on the bond until it matures


Bonds

• Which would you buy?– FXTN with 5 years to go till maturity earning

a semi-annual coupon of 8.50% and trading at 99.05 per 100 face

– FXTN with 9 years to go till maturity earning a semi-annual coupon of 8.25% and trading at 95.44 per 100 face

– Both FXTNs have the same issuer (Philippine government) and are denominated in PHP.


Price and Yield to Maturity

• How do we obtain the price of a bond?

• Price of a bond is the sum of the present value of its cash flows

– It does not tell us the rate of return of the investment and so it is difficult to compare prices across different bonds.

• Comparing prices on bonds may not be so

helpful

– Different coupons

– Different maturities


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• The single discount rate that makes the

present value of the cash flows equal to the price of the bond is called the Yield to

Maturity (YTM)



• A bond’s price may be higher or lower

than its face value

• A bond’s YTM may or may not be equal

to its coupon rate

– YTM is market dictated/determined

– The coupon rate is the periodic rate of

interest (as a percentage of face value) paid during the life of the bond and is set during the bond’s initial offering/issuance



• Price is inversely proportional to Yield.

Price = Face Value if YTM = Coupon Rate

Par

Price < Face Value if YTM > Coupon Rate

Discount

Price > Face Value if YTM < Coupon Rate

Premium


Price of a Fixed Rate Bond

2 Primary Components:

• Present Value of the Face Value +

• Present Value of the Coupons


Present Value of the Face

Value

• Where:

FV = face valueYTM = yield to maturity

N = Nth (final) cash flow

( )NYTM

FVofFVPV

+=

1


Present Value of the

Coupons

• Where:

C = future coupon amount

= FV x coupon rate

YTM = yield to maturity

n = nth cash flow until maturity (N)

( )∑

= +=

N

nn

YTM

CPVofCoupon

1 1


5

Price of a Fixed Rate Bond

P = PV of Coupons + PV of Face Value

• In notation form:

( ) ( )∑

= ++

+=

N

nNn

YTM

FV

YTM

CP

1 11


YTM – Example 1 (Par)

Year Cash Flow Discount Factor Present Value

1 5 1/1.05 = 0.9524 4.7619

2 5 1/(1.05)^2 = 0.9070 4.5351

3 5 1/(1.05)^3 = 0.8638 4.3192

4 105 1/(1.05)^4 = 0.8227 86.3838

Total 100.0000 (par bond)

100

5 5 5

105

4.7619 4.5351 4.3192 86.3838

Future Values

Present Values

4-year 5% bond yielding 5%


YTM – Example 2 (Discount)


1 5 1/1.06 = 0.9434 4.7170

2 5 1/(1.06)^2 = 0.8900 4.4500

3 5 1/(1.06)^3 = 0.8396 4.1981

4 105 1/(1.06)^4 = 0.7921 83.1698

Total

96.5349 (discount

bond)

96.5349

5 5 5

105

4.7170 4.4500 4.1981 83.1698

Future Values

Present Values



YTM – Example 3 (Premium)


1 5 1/1.04 = 0.9615 4.8077

2 5 1/(1.04)^2 = 0.9246 4.6228

3 5 1/(1.04)^3 = 0.8890 4.4450

4 105 1/(1.04)^4 = 0.8548 89.7544

Total

103.6299 (premium

bond)

103.6299

5 5 5

105

4.8077 4.6228 4.4450 89.7544

Future Values

Present Values



Summary of Examples

• From the previous slides:

Price of the Bond Yield to Maturity

100.0000 5%

96.5349 6%

103.6299 4%


General Bond Pricing

Formula

P = PV of Coupons + PV of Face Value

• In notation form:

( ) ( )∑

= ++

+=

N

nNn

YTM

FV

YTM

CP

1 11


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Understanding Yield to

Maturity

• Yield to Maturity can be understood in different ways:

– YTM is the discount rate that equates the present value of the bond’s future cash flows to its price.

– YTM is the expected return to the bond investor

– YTM is the assumed reinvestment rate of the bond’s coupons

– YTM is the equivalent rate on a deposit of the bond’s price for the time to maturity of the bond.


Expected Return

• If an investor buys a 5% five-year annual coupon bond yielding 6%, the investor “expects” to earn a 6% return.

• Yield is forward looking.

• Return is backward looking.

• The return is the single interest rate that equates the final investment proceeds to the future value of the bond’s purchase price.


YTM as Assumed

Reinvestment Rate5% coupon 5-year bond yielding 6%; P = 95.787636

Year Cash Flow Future Value

1 5 5 x (1.06)^4 = 6.312384

2 5 5 x (1.06)^3 = 5.955080

3 5 5 x (1.06)^2 = 5.618000

4 5 5 x (1.06)^1 = 5.300000

5 105 105.000000

Total FV in Five Years 128.185464

• Yield to Maturity is like a bank deposit rate – it is the implied

reinvestment rate of all cash-flows.

Discounting the FV: 128.1854648/(1.06)^5 = 95.78763 or

Compounding the PV: 95.78763 x (1.06)^5 = 128.185464


Bank Account Deposit Rate

• Today Deposit 95.79

• Year 1 Total 101.53 Pay 5 Deposit 96.53




• Year 5 Total 105 Pay 105

• Using the bank account, we have replicated the cash flows of the bond:

• YTM is the equivalent rate to depositing the cash price of the bond in a bank account.


Bond Formula Adjustments

• The general bond valuation equation works on the assumption of no-transaction costs, pricing during issue/coupon date only, and coupon payments made only once a year.

• In reality, bonds are not always purchased during the issue/coupon date (secondary market trading exists), nor do they always have coupons that are paid out once a year, nor are they always free from taxes.

• Because of these apparent realities, the general bond pricing formula must be adjusted to take into account the aforementioned.


Bond Formula Adjustments

• For actual application, the general bond

valuation formula must be adjusted for:

– Accrued Interest

– Coupon Payment Frequency

– Pricing on non-repricing dates or non-issue dates

– Withholding Tax


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Coupon Payment Frequency

• Refers to the number of times a bond’s

coupon is to be paid out within a year

• Coupon Frequency

– Annual = 1

– Semi-annual = 2

– Quarterly = 4

– Monthly = 12



• How is coupon payment frequency

reflected in the general bond pricing formula?

– Both the coupon rate and the YTM must be

adjusted accordingly to take into account the multiple coupon payment periods per year.



• Coupon Rate Adjustment

• YTM Adjustment

• Where:

p = coupon payment frequency

n

p

YTM

+1


CRFVC ×=p

C

Withholding Tax

• In the Philippines, all government

securities are subject to a 20% final withholding tax on interest income.

• How is withholding tax reflected in the general bond pricing formula?

– Both the coupon rate and the YTM must be

expressed “net” of final withholding tax of 20%


Withholding Tax

• Coupon Rate Adjustment

• YTM Adjustment

( )( )n

p

YTM

+

8.01


( )8.0

p

C

Pricing on Non-

Repricing/Non-Issue Dates

• The general bond valuation formula

assumes the bond is being priced either during issue date or during

coupon/repricing dates, which is why the

exponents used when discounting the coupons and face value to their

respective present values are expressed

as whole numbers.


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Pricing on Non-

Repricing/Non-Issue Dates • In reality, it is possible for the bond to have

been purchased or sold after issue date or between coupon/repricing dates (secondary market trading of bonds).

• How is pricing during non-issue/non-coupon dates reflected in the bond pricing formula?– The exponents (which represent the time component

during discounting) will have to be changed to “fractional” exponents to account for the discounting of future cash flows wherein a number of days had already run/accrued.


• The Discount Factor would have to be adjusted in the general bond pricing formula as follows:

• Where:

YTM = yield to maturityp = coupon payment frequency

N = nth cash flow until maturity (N)

DSC = number of days from settlement date to next coupon date

E = number of days in coupon period

Pricing on Non-

Repricing/Non-Issue Dates

( )( ) E

DSCN

p

YTM+−

+

1

8.01


Day-Count Conventions

• Valuation of most bonds in the Philippines uses the 30/360 day-count convention.– This simply means that, when counting the DSC,

each month is assumed to have 30 days while each year is assumed to have 360 days.

• Coupon Frequency E

Annual (p=1) 360

Semi-annual (p=2) 180

Quarterly (p=4) 90

Monthly (p=12) 30


Accrued Interest

• Accrued interest, by definition, is the

amount of interest that has accrued from the issue/last coupon date up to the

settlement date.


Accrued Interest

• Where:

C = future coupon amount

= FV x coupon rate

p = coupon payment frequency

A = accrued number of days

t = number of days in a coupon period (same as “E”)


( )( )

t

Ap

C

erestAccruedInt

8.0

=

Accrued Interest

• The price of a fixed rate bond after

adjusting for accrued interest may be obtained through the following:

• Present Value of the Coupons +

• Present Value of the Face Value –

• Accrued Interest


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Adjusted Bond Pricing

Formula

• Allowing for the presence of accrued

interest, withholding tax, coupon payment frequency, and pricing during non-

issue/non-coupon dates gives us the

adjusted bond valuation equation:


( )( )

( )

( )( )

( )( )

t

Ap

C

p

YTM

p

C

p

YTM

FVP

N

n E

DSCN

E

DSCN

8.0

8.01

8.0

8.01

111

−

+

+

+

= ∑=

+−+−

Price Distinctions

• Clean Price

– Also known as the Flat Price or Acquisition Cost of the bond

– Is equal to the Net Present Value (NPV) of all the bond’s expected future cash flows lessAccrued Interest

– Is the basis for the marking-to-market of bonds


Price Per Hundred (PPH)

• Is the Clean Price of the bond taken as a

percentage of its Face Value and expressed per 100 Face Value

– Is the basis for secondary market quotations (bid and offer) of bonds

100Pr

×=FV

iceCleanPPH


Price Distinctions

• Dirty Price

– Is the total amount that is to be paid (received) by a buyer (seller) of a bond to

(from) the bond’s seller (buyer)

– Is equal to the Net Present Value (NPV) of all the bond’s expected future cash flows

– May be arrived at by adding the Clean Price of the bond to Accrued Interest


Sources of Return for Bonds

• Coupons

– Current Income

– Interest payments made by issuer

• Capital Gains (Losses)

– Capital Appreciation (Depreciation)

– When security matures, is called, or sold

• Reinvestment Income

– Interest earned from reinvesting interim cash flows (interest and/or principal)


Holding Period Return (HPR)

• Is the rate of return on an investment

based on the actual cash flows received by an investor for the time

he/she holds on to a bond

( )

purchase

sale

purchasen

npurchasesale

DP

CDPDP

HPR

∑=

+−

=


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Problem 1

• What is the price of a 3-year, 11% semi-

annual bond with face value of PHP500,000.00 if the YTM is 10.75%?

Assume that the bond was issued free

from transaction costs. Was the bond sold at par, at a premium, or at a

discount?


Problem 2

• A bank has given you its bid and ask for a

2-year, 11.00% semi-annual bond with a par value of PHP1,000,000.00. Its bid for

the bond is 11.25% while its offer is

10.75%. Assuming that the bond is subject to 20% final withholding tax on

interest income, what is the price per

hundred (PPH) equivalent of its bid and offer quotation for the bond?


Problem 3

• What is the price of a 2-year, 10% semi-annual bond with par value of PHP750,000.00 if the YTM is 10.50%? If it is sold 1 year later at a YTM of 10.25%, what is the price at sale? Did the investor make money from the sale? What is the investor’s holding period return (HPR)? For this problem, assume that the bond is subject to 20% final withholding tax on interest income.


Problem 4

• On May 30, 2006, an investor purchases

a PHP10,000,000.00 par value worth of a Fixed Rate Treasury Note (FXTN) with a

coupon rate of 8.50% p.a. and a maturity

date of March 3, 2011 at a YTM of 8.75%. Provide the following information: a)

Dirty Price of the FXTN; b) Accrued

Interest; c) Clean Price of the FXTN.


References

• Treasury Certification Program (TCP) Money Market Module – Ateneo-BAP Institute of Banking

• Fixed Income Seminar Notes – Fund Managers’ Association of the Philippines (FMAP) and UBS Investment Bank

• Lecture Notes – Treasury Operations –(FIN538M)

• Lecture Notes – Fixed Income Securities –Special Topics in Financial Engineering (FIN570M)


Documents

FINVEST Fundamentals of Bond Valuation Slides