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Introduction to Fixed-Income Securities

Financial Risk Management Nattawut Jenwittayaroje, PhD, CFANIDA Business School

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Topics What is a bond?

Time Value of Money and Bond Pricing

Yield Measurement

Risks of Fixed-Income Securities

Interest Rate Risk Measurement: Duration

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What is a bond?

A bond is a form of loans or debt securities (i.e., fixed-income securities), which is a claim on a specified periodicstream of income.

A borrower issues (i.e., sells) a bond to a lender for someamount of cash. The bond obligates the borrower (i.e., issuer)to make specified payments on specified dates.

An investor who purchases a bond is lender or creditor

A (typical) bond generally makes a regular interest payment,called a coupon, at regular intervals, and a terminal payment,called the face value at the maturity date.

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What is a bond? In general, the bond must specify its

Issuer (the borrower)

Face value or par value (principal of the loan)

• Face value is the amount that is to be paid to an investor at the maturitydate of the security.

Maturity date and term-to-maturity

• The final coupon and the face value of a bond are repaid to the investoron its maturity date.

Coupon rate (per annum)

• used to calculate a coupon payment, or interest payment, paid at regularintervals by the issuer to bondholders.

Frequency of coupon payments

• Usually are semi-annually, annually or quarterly.

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Example 1: Illustrate a cash flow stream of a 3-year bond witha $80 coupon (8% coupon rate), and a face value of $1,000.Coupons are paid annually.

End of year

Today year 1 year2 year 3

$80 $80 $80+1,000 Price = (-)$1,000

Investors buy the bond today, and is entitled to a payment of $80 per year for 3 years and par value of $1,000 at the end of year 3

Lending $1,000 for 3 years

Borrower issues the bond to a buyer for $1,000

Borrowing $1,000 for 3 years

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Example 2: illustrate a zero-coupon bond that makes no couponpayments. Investors receive par value (e.g., $1,000) at the maturity(e.g., 4 years) but receive no interest payments. The zero-coupon bond

are issued at prices considerably below par value, e.g., $750.

End of year

Today year 1 year2 year 3 year 4

$0 $0 $0 $1,000 Price = (-)$750

Investors buy the bond today, and is entitled to a par value of $1,000 at the end of year 4

Lending $1,000 for 4 years

Borrower issues the bond to a buyer for $750

Borrowing $1,000 for 4 years

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Time Value of Money and Bond Pricing

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มูลค่าอนาคตเช่น ฝากเงินวนันี้ = 100 บาท ที่อตัราดอกเบีย้ (อตัราผลตอบแทน) 5% ต่อปี จาํนวนหนึ่งและสองปีFV หรือ มูลค่าของเงิน 100 บาท ในหนึ่งและสองปีขา้งหนา้

มูลค่าปัจจุบัน

i นั้นคือ อตัราดอกเบีย้ต่องวด หรือ อตัราคดิลด (discount rate).PV บางครั้งเรียกวา่ มูลค่าคิดลด (discounted value) หรือ มลูค่าของเงินรวมในอนาคตที่คิดกลบัมาที่จุดเริ่มตน้ หรือ มูลค่าปัจจุบนัPV (มูลค่าปัจจุบนั) ของเงิน 100 บาทในสองปีขา้งหนา้ ที่ discount rate ที่ 4%

มูลค่าอนาคต (Future Value) และ มูลค่าปัจจุบัน (Present Value)

หนึ่งปีขา้งหนา้ สองปีขา้งหนา้

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มูลค่าปัจจุบัน (Present Value ) ของ Ordinary Annuity

ผูล้งทุนตอ้งการซื้อเครื่องมือทางการเงิน ที่จ่ายเงินสด $500 ต่อปีเป็นเวลา 20 ปี โดยเริ่มจ่าย 1 ปี ต่อจากวนันี้

ผูล้งทุนตอ้งการอตัราดอกเบี้ย (ผลตอบแทน/อตัราคิดลด) ต่อปีที่ 5.5% จากการลงทุนดงักล่าว

ถา้ราคาเครื่องมือทางการเงินดงักล่าวซื้อขายกนัที่ $5,300.

ผูล้งทุนควรจะซื้อเครื่องมือทางการเงินดงักล่าวหรือไม่

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ผลรวมทั้งหมดของค่า PV = $5,975.19

$500

ณ ปลายปี0           1     2     3…..…10…….20

$500

$500

$500

$500

ตวัอย่าง: มูลค่าปัจจุบันของ Ordinary Annuity

ซื้อ12

มูลค่าปัจจุบัน (Present Value ) ของOrdinary Annuityมูลค่าในปัจจุบนัของเงินที่เกิดขึ้นทุกสิ้นงวด อธิบายเป็นสูตรไดด้งันี้

A =เงินงวดเป็นจาํนวนเงินคงที่เท่ากนัทุกงวดi = อตัราดอกเบี้ยทบตน้n = จาํนวนงวดของกระแสเงินสด

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Bond Pricing The price paid for a bond today depends on the value of dollars to be received

in the future under the bond indenture. The fair price of a bond is the present value of future cash flows (coupons and

face value) that it makes.

[1]

where PB = price of bond

C = coupon payment ($) =

n = number of years to maturity

i = discount rate (i.e., required yield = ผลตอบแทนที่ตอ้งการ)F = face value

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As can be seen in [1], the price of a bond is comprised of the PV of a series of the coupon payments the PV of the face value.

The coupon payments are equivalent to an ordinary annuity.

If the coupon payments are made more than one time per year. n = number of periods to maturity i = periodic interest rate

C =

Annuity Factor (i,n)

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Bond Pricing (Con’t) Example 3: Mills Company, a large defense contractor, on January

1, 2007, issued a 10% coupon interest rate, 10-year bond with a $1,000 par value that pays interest annually.

Investors who buy this bond receive the contractual right to receive two cash flows: (1) $100 annual interest (10% coupon interest rate $1,000 par value) at the end of each year, and (2) the $1,000 par value at the end of the tenth year.

Let’s assume the required yield is 10%.

C = $100, i = 10%, F = $1,000, and n = 10 years.

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Bond Pricing (Con’t)

P

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Calculate the price of a 10%coupon bond with 10 years tomaturity and a face value of$1,000. The coupon paymentsare made annually, and therequired yield is 12% perannum.

Calculate the price of thebond in previous example,but with the required yieldof 8% per annum.

Bond Pricing (Con’t)

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i P

Bond Values and Required Returns

Negative relationship between price of a bond and its yield.

As the required yield increases, the PV of the cash flows must decrease;therefore, the price decreases.

As the required yield decreases, the PV of the cash flows must increase;therefore, the price increase.

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Bond Values and Required Returns

i

P

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ราคาของตราสารหนี้แปรผกผันกับอัตราผลตอบแทนที่นักลงทุนต้องการ (required yield; y) ในขณะที่ อตัราผลตอบแทนที่นกัลงทุนตอ้งการแปรผนัไปตามอตัราดอกเบี้ยตลาด (interest rate; i)

ดงันั้นราคาของตราสารหนี้แปรผกผนักบัอตัราดอกเบีย้ตลาด (interest rate; i)

Price/yield relationship (ความสัมพนัธ์ของราคากบัอตัราดอกเบีย้)

interest rates Required yield

interest rates Required yield

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Par, premium and discount bonds1. Par bonds (price = face value (i.e. par value)) required yield = coupon rate When yields in the marketplace equal to the coupon rate at a given

time, the price of the bonds will be at the par value.2. Discount bonds (price < face value) required yield > coupon rate When yields in the marketplace rise above the coupon rate at a given

time, the price of the bonds will decrease below the par value becauseof the higher discount rates.

3. Premium bonds (price > face value) required yield < coupon rate When yields in the marketplace drop below the coupon rate at a given

time, the price of the bonds will increase above the par value becauseof the lower discount rates.

When this condition occurs

When this condition occurs

When this condition occurs

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Bond Pricing: Bond Value BehaviorIn practice, the value of a bond in the marketplace is rarely equal to its par value.

Whenever the required return on a bond differs from the bond’s coupon interest rate, the bond’s value will differ from its par value.

The required return is likely to differ from the coupon interest rate because either

• (1) economic conditions have changed, causing a change in the (long-term) interest rates, or

• (2) the firm’s risk has changed. Increases in the (long-term) interest rates or in risk will raise the

required return; decreases in the (long-term) interest rates or in risk will lower the required return.

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Example 4: Calculate the price of a 8% coupon bond with 20 yearsto maturity and a face value of $1,000. The coupon payments aremade semi-annually, and the required yield is 12% per annum.

Bond Pricing: Semiannual Interest payment

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Bond Pricing: zero coupon bonds No periodic coupon payments.

Single payment of par value at maturity.

The price of bond (PB) is simply the present value (PV) of the face value (F)(i.e., the face value discounted at market rate).

where n = number of years to maturity and i = discount rate (i.e., requiredyield)

The zero bonds are therefore issued at prices considerably below par value,and the return comes solely from the difference between issue price and thepar value received at maturity.

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Bond Pricing: zero coupon bonds

Example 5: Assume that the discount rate in the market is 8.6% p.a., compute the price of a zero-coupon that matures in 10 years and has the face value of $1,000?

Issued at discount from par.

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Yield Measurement

(การคาํนวณผลตอบแทนของตราสารหนี้)

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Today, Bond A has the following feature; a government bond

a coupon rate of 5% per year (coupon paid annually),

6 year to maturity

1,000 face value.

If today I buy Bond A at $940.5 and hold it until maturity, whatwould be the rate of return of my bond investment?

Yield to Maturity (YTM)

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The Yield to maturity (or Promised yield or Internal rate of return(IRR)) can be computed from bond pricing formulas [1] when abond’s price is known. (Solve for i)

YTM will be realised when:

bond is held to maturity.

Coupon payments are reinvested at the same rate as the YTM.

Yield to Maturity (YTM)

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Example 6: Compute the YTM for a 6 year, 5% coupon (annualpayments) bond selling for $940.50.

=IRR(B5:B11)30

YTM for a zero-coupon bond

From

Rearrange the above equation, we get:

Example 7: The YTM for a zero-coupon bond selling for $274.8with a face value of $1,000, maturing in 15 years can becomputed as:

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Risks of Fixed-Income Securities

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Risks of Fixed-Income Securities ความเสีย่งของตราสารหนีม้หีลายประเภทแต ่ความเสีย่ง 2 ประเภทหลกั คอื

ความเสีย่งในการผดินัดชาํระดอกเบีย้หรอืเงนิตน้ (Credit / Default Risk)

Credit Rating

ความเสีย่งจากการเปลีย่นแปลงของอตัราดอกเบีย้ (Interest Rate Risk)

ความเสีย่งดา้นราคา (Market / Price Risk)

วดัไดโ้ดย

• Price volatility

• Duration

ความเสีย่งในการลงทนุตอ่ (Reinvestment Risk)

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Credit ratings ของตราสารหนี้ เป็นเครื่องมือที่ช่วยบ่งบอกถึงความเสี่ยงทางดา้นเครดิต (credit and defaults risk) ของตราสารหนี้บริษทัจดัอนัดบัเครดิต

Moody’s Investor Service (Moody’s), Standard & Poor’s (S&P), FitchTRIS (Thai Rating and Information Service)

ความน่าเชื่อถอืสูงสุด, ความเสี่ยงตํา่สุด (Investment grade bonds)Aaa ถึง Baa สาํหรับ Moody’sAAA ถึง BBB สาํหรับ S&P

ความน่าเชื่อถอืตํา่สุด, ความเสี่ยงสูงสุด (Speculative-grade bonds or junk bonds)ตํ่ากวา่ Baa สาํหรับ Moody’sตํ่ากวา่ BBB สาํหรับ S&P

การจดัอนัดบัเครดติของตราสารหนี ้(Bonds’ Credit Rating)

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กลุ่ม Investment grade: อนัดบัความน่าเชื่อถอื : สูง - ปานกลาง

กลุ่ม Speculative, Junk Bondsอนัดบัความน่าเชื่อถอื : ตํา่ - มโีอกาสทีจ่ะเกดิการผดิชําระคนืเงนิต้นและดอกเบีย้สูง

การจดัอนัดบัของตราสารหนี้

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Source: ThaiBMA

Credit Rating & Agency

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Credit Rating and Spread

spread

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Interest rate risk Risk related to changes in interest rates that cause a bond’s price

to change.

Interest rate risk comprises:

Price risk is the variability in bond prices caused by theirinverse relationship with interest rates.

Reinvestment risk is caused by changing market rates atwhich coupons can be reinvested.

• Only face the price risk if to sell the bond before maturity.• Since the bond price can change due to the change in the interest (yield) during the holding period.

• the reinvestment rate risk occurs even though a bond is held until maturity.

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Bond price volatility

Bond prices are inversely related to bond yields.

Measure sensitivity of a change in a bond’s price to a changein yield.

Percentage change in price for given change in interest rates:

where %∆PB = percentage change in price

Pt = new price in period t

Pt – 1 = bond’s price one period earlier

A measure of PRICE RISK

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Bond price volatility and maturity

Considering 1-year bond,when i rises from 5% to 6%

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Bond price volatility and coupon rate

100.56

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Bond price volatility, coupon rate, and maturity

Consider three loan plans, each of which have different maturities in years. The loan amount is $1,000 and the current interest rate is 3%.

Loan #1 (3 years), is a three-payment loan with three equal payments of $353.53 each.

Loan #2 (2 years) is structured as a 3% annual coupon bond.

Loan # 3 (1.5 years) is a discount loan, which has a single payment of $1,045.33.

Which loan(s) are more “risky”????42

Interest Rate Risk Measurement:

Duration

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The duration measures the approximate sensitivity of a change in a bond’s price to a change in yield.

We take the first derivative of the price of bond equation withrespect to the required yield to determine the approximatedollar price change for a change in yield,

Duration - a measure of interest rate risk (price risk)

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The dPB/di is the dollar change of the price when the yieldchanges. The percentage change of the price can be computedby dividing both side of the equation above by the price (PB).

Macaulay duration (D) is :

Duration - a measure of interest rate risk (price risk)

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Modify duration (MD) is :

Duration - a measure of interest rate risk (price risk)

percentage changein price

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Example 9: Suppose we have a bond with a 3-year term tomaturity, an 8% coupon paid annually, and a market yield of10%. Durations are:

PB

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From example 9, MD = 2.5249. If the yield rises by 100 basispoints (1.00%), the % price change can be estimated by:

Using duration to estimate % price change

% price change

In other words, MD = 2.5249 means that if the yield rises(drops) by 1.00%, the bond price will decrease (increase)2.5249%

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Interpretation and Significance of Duration

Duration is used to measure the price sensitivity of a fixed income security due to a change in interest rates.

The higher the duration number, the greater interest rate risk.

Combines the effects of differences in coupon rates and differences in maturity.

Duration is the weighted average time to maturity of a bond.

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Properties of duration: Duration Versus Maturity

12% Coupon

1.00

1.89

2.70

3.42

4.07

Premium bond

Discount bonds

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Properties of duration:Duration Versus Maturity

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Duration as Index of Interest Rate Risk

Consider three loan plans, each of which have different maturities in years. The loan amount is $1,000 and the current interest rate is 3%.

Loan #1 (3 years), is a three-payment loan with three equal payments of $353.53 each.

Loan #2 (2 years) is structured as a 3% annual coupon bond.

Loan # 3 (1.5 years) is a discount loan, which has a single payment of $1,045.33.

D1 = 1.920, D2 = 1.971, D3 = 1.552

Rearrange from lowest to highest duration

a) 6-year maturity with 6% coupon paid semiannually.

b) 6-year maturity with 6% coupon paid annually.

c) 6-year maturity with zero coupon rate.

d) 3-year maturity with 7% coupon paid semiannually.

e) 4-year maturity with 6.5% coupon paid semiannually.

f) 5-year maturity with 6.5% coupon paid semiannually

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