bond analysis

1Time Value, Interest Rate

Structures & Bond Valuation

2Outline

Time Value of Money Present Value & Value attime T

Interest Rates Spot Rates, Forward Rates & Discount Factors

Defining Bond

Types of Bond

Bond Pricing I : Simple Bond (non-Callable/ non-Putable Bonds)

Bond Pricing II: Putable/Callable bonds

3Time Value of Money

Suppose an investor has fund amount A today.

Time Value (Value at time T): If she invest thisamount for T years with interest rate r, her fund willgrow to F, which is given by below

In above, r represents yield/interest-rate per annum.

y continuoul compounded If Ae

year per times-m gcompoundin If

m

r1A

F

rT

mTA

T F

4Time Value of Money

So, given the interest rate r, value A at present (i.e.time T=0) can be said equivalent to value F at any time T.

Present Value (PV): In other words, Present Value A ofa sum F at time T can be expressed as

A

T

F

compoundedy continuoul If Fe


)(1

F

A

rT

mT

m

r

5Basic Concepts

- Spot Rates, Forward rates & Discount Factors

Time Point t1 t20 ti tn

Spot Rates at time 0 r1

r2

ri

Forward Rates at time 0

Discount Rates/ Factors

6Spot-rate, Discount Factor and Forward-rate

Spot Rate (also called as Zero-Rate/zero-coupon rate), rt- The n-year spot rate is the rate of interest earned on aninvestment that starts today and lasts for n-years. All theinterest and principal is realised at the end of n years; nointermediate payments.

Discount Factor, (t)- This is simply (t)=(1+rt)

-t or (t) = exp(-trt) as definedearlier, where rts represent spot-rates.

Forward Rate, (denote forward rate from time t-1 to t at time 0 as t)- This is the interest rate implied by current spot-rates, for aspecified future time period. Interesting to see that

(1+rt)t = (1+1) (1+2)(1+ t)

Note: Knowing any one set/sequence of {rt}, {(t)} and {t},we can easily derive other two sets/sequences.

7Alternative Forms of Present Value (PV)- Using Spot Rates

Consider three expressions of PV formulations

The First Expression: Present Value (PV) of all CashFlows (Cks, k=1,2,3,.,n) using Spot Rates.

In above, rt represents Interest-Rate/Yield-to-Maturity(YTM) in zero-coupon bond from time 0 to time t.

gCompoundin Continuous-- FeeC......eCeC

yearper times-m gCompoundin--

)r(1

F

)r1(

C......

)r1(

C

)r(1

C

P

nn21 nrnr

n

r2

2

r

1

n

n

n

n

n

2

2

2

1

1

Note: At time 0, the rt defined above is the Spot-rate formaturity t.

8Alternative Forms of Present Value (PV)

- Using Discount Factors

The Second Expression: Present Value (PV) of allCash Flows (Cks, k=1,2,3,.,n)Using DiscountFactors.

gCompoundin Continuous-- e

1,2,.... tperiods of end gCompoundin--

)r(1(t) where

(n) F(n) C......(2) C(1) CP

trt -

t

t

Note:(i) The (t)s defined above are also called the discount factors.(ii) The PV is linear function of discount factors.

9Alternative Forms of Present Value (PV)

- Using Forward Rates

The Third Expression: Present Value (PV) of all CashFlows (Cks, k=1,2,3,.,n)-using Forward Rates.

Note: The ts defined above represent forward rates at time 0.

gCompoundinly Continuous-- Fe eC

gCompoundin endPeriod--

)1)...(1(

F

)).....(1)(1(1

C

P

).....(-n

1k

).....(-

k

n1

n

1k k21

k

n21k21

Here t represents the interest rate during end of time period (t-1)

to end of time t.

10

Defining Bonds

What is a Bond ?

A debt security, in which the authorized issuer owes the holders a

debt and is obliged to repay the principal a specified later date.

Bonds are usually issued with a par or face value representing

amount of money borrowed and issuer promised to pay a percentage

Bonds Loan

Issuer Borrower

Bond Holder Lender

Coupon Interest

Issue Date Coupon Payment Coupon Payment Coupon Payment

Trading Day

A

Coupon Payment

+ Face Value

11

Types of Bonds

(A) On the Basis of Nature of Coupon

Zero-Coupon Bond

-- It pays no coupon pays only principle at maturity.

Coupon Bond

-- Pays coupon (as interest rate on principle/face value) at certain

pre-defined dates during the life of the bond and pays face-value

with coupon at maturity.

(A) Fixed Coupon Bond coupon amount is fixed

(B) Floating Rate bond/note variable coupon amounts

coupon is usually linked to a reference interest rate and reset

periodically depending upon changes in reference rate.

12

Types of Bonds

(B) On the basis of difference between market

discount rate and coupon rate

Premium Bond Coupon greater than market

discount rate

Par (or Par value) Bond Coupon and market

discount rate same

Discount Bond Coupon lower than discount rate

13

Types of Bonds(C) Other Bond Types/Bond Options

Convertible bond grants the bond holder and/or issuer the

right to convert the bond into a predefined amount of ordinary

stock of the issuing company/entity.

Exchangeable bond - grants the bond holder the right to

convert the bond into a predefined amount of ordinary stock of a

specified company other than the issuing company.

Callable bond - A fixed rate bond where the issuer has the right

but not the obligation to repay the face value of the security at a

pre-agreed value prior to the final original maturity of the

security.

Putable bond Grants the bondholder the right to sell the bond

back to the issuer at its par value on designated dates.

14

Some TerminologiesCurrent Yield

The most basic measure of the yield which is simply the coupon

payment over the current price of the bond.

Example: a bond with current price $92.78 pays $10 annual

coupon. Current yield = 7/92.78 = 0.0754 or 7.54 %

Simple Yield to Maturity (SYM)

- This takes into account capital gains/losses assuming that the

capital gain/loss on the bond occurs evenly over remaining life

of the bond.

Example: Consider the same bond in current yield that is paying

coupon $7 per annum with 5 years until maturity.

Then SYM = 7/92.78 + (100-92.78)/(5 x 92.78) = 0.0910 0r 9.1 %

15

Some Terminologies

Yield to Maturity/Redemption Yield (YTM)

- The problem with SYM is that it does not take into account

the fact that coupon receipts can be reinvested and hence

further interest gained. The YTM or redemption yield takes

into account this aspect.

Thus, YTM is the yield/interest gain made on a bond if it is

held till maturity (assuming that coupons are reinvested).

The YTM, say y of a bond that is trading at price P is

calculated from the relationship (assuming it pays n annual

coupons, and face value F)

nn2 y)(1

F

)1(

C......

)1(

C

y)(1

CP

yy

16

Reinvestment Risk

What is Reinvestment Risk?

- This is the risk arising from uncertainty in the interest rate at

which future cash flows may be invested.

Note:

(1) The YTM is the yield incorporating the fact that coupon

payments can be invested. However, this is subject to

reinvestment risk.

(2) Different coupon payment receipts in future may be

reinvested at different interest rates. YTM is some sort of

overall/average yield earned over the life of the bond if it is

held till maturity.

17

Bond Pricing I

(Non-Callable/Non-Putable Bonds)

18

Bond Pricing

Issue Date Coupon Payment Coupon Payment Coupon Payment

Trading Day

A

Coupon Payment

+ Face Value

Cash Flows from A typical Bond may be represented as follows

Holder of the bond pays price to the issuer/seller at the time of buying

In return, holder of the bond is entitled to get (in future specified dates)coupon payments. If hold till maturity, holder additionally get Face Value.

19

Bond Pricing Price of a Coupon-Bearing Bond

Consider a 5-year bond: Face Value F = 100

Years of maturity = 5 ; Coupon = 7% (annual payment)

Assume annual interest for next five years at 10%.Given frequency of compounding m=1 per year

Year Coupon Principal Cash-Flow Present Value

1 7 0 7 7/1.1 = 6.36

2 7 0 7 7/1.12 = 5.78

3 7 0 7 7/1.13 = 5.26

4 7 0 7 7/1.14 = 4.78

5 7 100 107 107/1.15 = 66.44--------------------------------------------------------------------

Value of the Bond (Total PV) = 88.63

20

Bond Pricing Zero-Coupon Bond

The residual maturity of a bond today is T years.Holder of the bond receives face value F at maturity.There is no intermediate cash flow to the holder. Whatwould be the Bond price P today?

P Would be the Present Value (PV) of F today.

In above, r represents yield/interest-rate per annum.

compoundedy continuoul If Fe


)(1

F

P

rT

mT

m

r

Issue Date Trading DayA Face Value (F)T

21

Bond PricingCoupon -Bearing Bond

One buy the bond today and on maturity (after nperiods) receive a known amount F (Face Value). Inaddition gets Coupon amount C at the end of eachperiod (say, half-year). What would be the Bond price Ptoday?

P Would be the Present Value (PV) of all Cash Flows

In above, r represents Interest-Rate/Yield per annum.

gCompoundin Continuous-- FeCe......CeCe

yearper times-m gCompoundin--

r/m)(1

F

)r/m1(

C......

)r/m1(

C

r/m)(1

C

P

nrnrr2r

nn2

22

Bond PricingMore General Form

One buy the bond today and on maturity (after nperiods) receive a known amount F (Face Value). Inaddition gets Coupon amount C at the end of eachperiod. What would be the Bond price P today?

P Would be the Present Value (PV) of all Cash Flows

In above, rk represents Interest-Rate during k-th period,k=1,2,.,n. These rks represent forward-interest ratesat time 0.

gCompoundinly Continuous-- Fe eC

gCompoundin endPeriod--

)r1)...(r1(

F

)r).....(1r)(1r(1

C

P

)r.....r(r-n

1k

)r.....r(r-

k

n1

n

1k k21

k

n21k21

23

Pricing Floating-Rate Bonds/NotesConsider a floating-rate bond which pays coupon same as a reference rate, say,

LIBOR.

For pricing this bond, note that it is worth the Face Value immediately after acoupon payment. This is because at that time bond is a fair deal where issuer

pays LIBOR for each subsequent accrual period.

Example:Consider a FRN has 1.25 year residual life; Face Value F = $ 100 million

Floating coupon = 6-month LIBOR; Coupon payment = half-yearly

Assume 6-month LIBOR at last coupon date was 10.2 %

Time Cash-flow Cash-flow Cash-flow Present Value0.25 5.1 5.1 105.1 # 102.55

0.75 0.5 x L2 0.5 x L2+100*

1.25 0.5 x L3+100

--------------------------------------------------------------------

* Value of (0.5 x L3+100) at time 0.75 is 100 (discount rate L3)# Value of (0.5 x L2+100) at time 0.25 is 100 (discount rate L2)

Value of the Bond (Total PV) at time 0 = 102.55

24

Bond Pricing

Bond Price is sensitive to changes in one or moreof the following Factors (Given Face Value F)

* Maturity

* Coupon

* Yield/interest-rate

Note:

We primarily focus on bond price sensitivity to changes ininterest rate/yield.

25

Convexity(Price-Yield Relationship)

Increase in price for unit decrease in yield is

greater than decrease in price for the same

increase in yield

Price

YTM

YTM

Price

or

ro r1r2

pop1

p2

po

ro

r1

p2

r2

p1

26

Convexity

The price-yield curve is convex meaning that theslope of the curve is continuously changing

At any point on the curve, say A, slope is the slope of the tangent (the straight line TT) at that point

YTM

Price

A

T

T

27

Convexity

In above Graph, XX and YY curves represent Price-Yieldrelationship for two different portfolios X and Y,respectively.

The curve XX has more curvature than YY. Convexitymeasures this curvature.

At point A, both the portfolios have same price change forvery-small change in yield. But for larger change in yield,portfolio Y experiences higher price change. This is the impact ofconvexity.

YTM

Price

Y

Y

X

X

A

28

Price Sensitivity to Interest Rate Changes

Maturity Effect

The longer the term to maturity, the greater

the sensitivity to interest rate changes.

Example:-Let zero coupon yield curve is flat at 12%.-Bond A pays 176.234 in 5 years-Bond B pays 310.584 in 10 years

Note: Both Bonds are currently priced at 100.

29


Maturity Effect

Example continued...

Bond A: P = 100 = 176.234/(1.12)5

Bond B: P = 100 = 310.584/(1.12)10

Now suppose the interest rate increases by 1%.

Bond A: P = 176.234/(1.13)5 = 95.653

Bond B: P = 310.584/(1.13)10 = 91.494

The longer maturity bond has the greater drop in price.

30


Coupon Effect

Bonds with identical maturities will responddifferently to interest rate changes when thecoupons differ.

It is readily understood by recognizing thatcoupon bonds consist of a bundle of zero-coupon bonds. With higher coupons, more of thebonds value is generated by cash flows whichtake place sooner in time.

31

Price Sensitivity to Interest Rate Changes- Coupon Effect

Sensitivity of 6% Coupon BondMaturity

(n)Yield (r)

7% 6% 5% Range40 86.7 100 117.2 30.5

20 89.4 100 112.5 23.1

10 93.0 100 107.7 14.7

2 98.2 100 101.9 3.7

Sensitivity of 8% Coupon BondMaturity

(n)Yield (r)

9% 8% 7% Range40 89.2 100 113.3 24.1

20 90.9 100 110.6 19.7

10 93.6 100 107.0 13.4

2 98.2 100 101.8 3.6

32


The longer maturity bonds experiencegreater price changes in response to anychange in the discount rate.

The range of prices is greater when thecoupon is lower.

The 6% bond shows greater changes in price in response to a 1% change than the 8% bond. The first bond is riskier.

33

Bond Value Theorems

Price and Yield move in opposite directions.

Price-yield curve is convex in shape.

Discount/Premium decreases with decrease inmaturity period - decreases at an increasingrate as time to maturity decreases (longermaturity bonds have greater change in pricedue to unit change in interest rate/yield).

Sensitivity of bond price to changes in yield islower if coupon is higher.

34

Bond Pricing II

(Callable/Putable Bonds)

35

Callable Bonds Callable bonds are issued to borrow money for whatever

reason.

Being callable, such bonds give the issuerthe right to call home the bonds repay their borrowingswhen seems good/fit, which usually means when interestrates are low.

To pay off the bonds, the issuers usually have to pay theholder the face value of the bonds.

For many callable bonds, however, the issuers need topay some premium on top of the face value. This premiumacts as some compensation for the lenders who upon beingprepaid, have to find new borrowers at generally lowerinterest rates. The price that the issuers have to pay isthe call price.

36

Example: Typical Callable Bond Structure

Take an example of a typical 10 NC 2 bond (10 yearsstated maturity, only callable after 2 years) may havefollowing features

Face Value : $ 100

Lockout period : 2 years (i.e. no call privileges in first 2years)

After the lockout period, issuer might have the right tobuy the bond back at following prices

$ 110 in years 3 & 4

$ 107.5 in years 5 & 6

$ 106 in years 7 &8

$ 103 in years 9 & 10

38

Yields for Callable Bonds

Consider 2-year bond that can only be called at end of year 1 for a callprice $100, has a face value $100 and currently selling at $99. Assumesemi-annual coupon rate of 8% p.a.

Yield to Maturity

The yield to maturity of this callable bond is calculated assumingthat the bond will be held till maturity regardless.

Therefore, the cash flows from the bond will simply be:

At time 0.5: $4 At time 1.0: $4

At time 1.5: $4 At time 2.0: $104

The yield to maturity of the bond will then be y such that:

Solve this for y, we have y=8.55 %

432

21

104

21

4

21

4

2

y1

499

yyy

39

Yields for Callable Bonds

Yield to Call (YTC)

In our example, Yield to Call is calculated assuming that thebond will be called with certainty (at end of first year).

Therefore, the cash flows from the bond will simply be:

At time 0.5: $4 & At time 1.0: $104

The yield to call of the bond y will then be such that:

Solve this for y, we have y=9.07 %

Yield to Worst

Yield to Worst = Minimum (YTM, YTC)

In our example, Yield to Worst = Min(8.55%, 9.07%) = 8.55 %

2

21

104

2

y1

499

y

40

Pricing Callable Bonds

Consider the Cash Flow of the Callable bond in our example

----------------------------------------------------------------------------------

Cash Flow at time

--------------------------------------------------------

Bond 0.5 1.0----------------------------------------------------------------------------------1-year Non-Callable $ 4 $1042-year Non-Callable $ 4 $ 4 + Price of 2-year Non-Callable at time 1Callable $ 4 $ 4 + Min (100, Price of 2-year Non-Callable----------------------------------------------------------------------------------------

Note that the Cash Flow of Callable bonds identical at time 0.5.But at time 1, Cash Flow is smallest in the case of Callable Bond

So, the Callable Bond here will be cheaper than

1-year Non-callable as well as 2-year Non-callable

bonds.

41


To value the Callable Bond in our example, assume following

tree of semi-annual interest rates

42


For Similar Non-Callable Bond, Cash Flow would be

Time 0 0.5 1.0 1.5 2.0CF 4 4 4 104-------------------------------------------------------------------------------------------------------Price of the Non-Callable Bond using Interest rate Tree would be(assume probability of moving up or down in the tree at any time is 0.5)

99.1005

96.7878

95.7549

98.7160

101.0012100.4394

96.5451

98.3727

99.7719

100.8344

96.5451 = 104/(1+15.54%/2)

95.7549 = [0.5*(96.5451+98.3727)+4] / (1+11.91%/2)

43


Now turn to Price the Callable Bond in our Example At time 1, issuer may call the bond. However, issuer will call only if value of bond at

time 1 is higher than call price; $100.

Time 0 0.5 1.0 1.5 2.0----------------------------------------------------------------------------------------------------------------------------- --Checking 3-scenarios in time 1, it will make sense to buy back the bond if its value is 101.0012.

Paying $100, he gains $1.0012Price of the Callable Bond using Interest rate Tree would be (assume probability of moving up or

down in the tree at any time is 0.5)

99.1005

98.8667

96.7878

95.7549

98.7160

101.0012

100.00

100.4394

99.9552

96.5451

98.3727

99.7719

100.8344

96.5451 = 104/(1+15.54%/2)

99.9552 = [0.5*(98.7160+100.00)+4] / (1+11.91%/2)

44

Pricing Callable/Putable Bonds

Issues and further detail on pricing of callable/putable bondsare discussed seperately in the context of callable bondsfollowing the write-up/chapter by Professor Anh Le, Thewrite-up/Chapter on Callable Bonds.

Also we discussed relevant issues using hands-on in MS-Excelplatform.

45

Select References

Hull, John C. (2004), Options, Futures, andOther Derivatives, Fifth Edition, Prentice-Hall of India Pvt. Ltd.

or

Hull, John C. (2005), Options, Futures andOther Derivatives, Sixth Edition (Chapters4 & 6), Prentice-Hall of India Pvt. Ltd.

Professor Anh Le, The write-up/Chapter onCallable Bonds.

46

Thank You

bond analysis

Documents