Valuation of Bonds

Arti Pradhan

Basics of Bonds Types of Bonds Bond Risk Bond Return Valuation of Bonds Bond Value Theorems Term Structure of Interest Rates Bond Portfolio Management

A bond is a long-term promissory note that promises to pay the bondholder a predetermined, fixed amount of interest each year until maturity.

Coupon Rate Par Value Maturity Date Current Yield

Corporate Bonds 1. Straight or plain vanilla Bonds2. Zero Coupon Bonds or Deep Discount

Bonds3. Mortgage Bonds4. Subordinate Bonds5. Bonds with Embedded Options6. Floating Rate Bonds7. Commodity Linked Bonds8. Junk Bonds Government Bonds

Interest Rate Risk Default Risk Marketability Risk Callability Risk

Coupon yield or Nominal yield is coupon payment (C) as a percentage of the face value (F) = C / F

Current yield is coupon payment (C) as a percentage of the bond price (P ) = C / P

Yield to Maturity is the discount rate, which returns the market price of the bond I.e. internal rate of return of an investment in the bond made at the observed price

Market Price = Yield to Call Some bonds carry a call feature that

entitles the issuer to call the bond prior to the stated maturity date in accordance with a call schedule. For such bonds, it is a practice to calculate the yield to call (YTC) as well as YTM.

Calculate current yield and coupon yield of a 10 year, 12 percent coupon bond with a par value of Rs. 1000 and selling for Rs. 950.

Determine the yield to maturity if a 8 percent coupon bond with a face value of Rs 1000 is available for Rs 750. The maturity period is 5 years.

process of determining the fair price of a bond fair value of a bond is the present value of the

stream of cash flows it is expected to generate A company’s bonds have a par value of Rs

100, mature in seven years, and carry a coupon rate of 12 percent annually. If the appropriate discount rate is 16 percent, what price should the bond command in the market? If calculated price > market price, buy the

bond from the market. If calculated price < market price, sell the

bond in the market. If calculated price = market price, neutral.

value of the bonds depends upon three factors namely, the coupon rate, years to maturity and the expected yield to maturity or the required rate of return. Relationship between bond prices and changes in the factors are known as the ‘Bond Value/Pricing Theorems’

(i) Bond prices will move inversely to market interest changes

(ii) Bond Price variability is directly related to term of maturity

If bond yield remains same over its life, discount or premium depends on maturity period

  Bond A Bond B

Face value Rs. 1000 Rs. 1000

Coupon Rate 10% 10%

Maturity Period 5 years 10 years

Yield 15% 15%

Price Rs. 832.39 Rs. 749

Bond’s sensitivity to changes in market interest rate decreases at a diminishing rate as the time to maturity decreases.

For example, a Rs. 1000 bond with maturity period of 5 years with 10% yield to maturity will have following present value or price:Years to Maturity Present Value/Price (in Rs.)

5 620

4 683

3 751

2 826.4

1 909.1

The price changes resulting from equal absolute increases in market interest rates are not symmetrical.

A raise in the bond’s price for a decline in the bond’s yield is greater than the fall in the bond’s price for a raise in the yield. Take a bond of 10% coupon rate, maturity period of 5 years with face value of Rs. 1000. If the yield declines by 2%, that is, to 8%, then the bond price will be Rs. 1079.87.

If the yield increases by 2%, that is, to 12%, then the bond price will be Rs. 927.88

The change in the price will be lesser for a percentage change in bond’s yield if its coupon rate is higher.

Bond A Bond B

Coupon rate 10% 8%

Yield 8% 8%

Maturity period 3 3

Price Rs. 105.15 Rs. 100

Face value Rs. 100.00 Rs. 100

Yield Raise 1% 1%

Price after yield raises Rs. 102.53 Rs. 97.47

Percentage change in price

2.4% 2.53%

Also known as yield curve Relationship of yield to maturity to its

term to maturity for bonds that are similar in all respects, except the maturity

Method of bond valuation The yield curve for the government

bonds is taken as the benchmark and the corporate bonds are priced according the risk associated with them.

COUPON MATURITY PRICE/YIELD

3-Month 0.000 10/14/2010 0.14 / .15

6-Month 0.000 01/13/2011 0.18 / .18

12-Month 0.000 06/30/2011 0.26 / .26

2-Year 0.625 06/30/2012 100-01½ / .60

3-Year 1.000 07/15/2013 99-31 / 1.01

5-Year 1.875 06/30/2015 100-12½ / 1.79

7-Year 2.500 06/30/2017 100-08 / 2.46

10-Year 3.500 05/15/2020 103-31+ / 3.03

30-Year 4.375 05/15/2040 106-09+ / 4.01

Source:http://www.bloomberg.com/markets/rates-bonds/government-bonds/us/ (15 July,2010)

1. Upward Sloping or normal yield curve: The short-term yield is lower than the long-term yield, i.e. it is cheaper to borrow short-term than it is to borrow long-term.

2. Downward Sloping or inverted yield curve: The short-term yield is higher than the long-term yield, i.e. it is more expensive to borrow short-term than it is to borrow long-term.

3. Flat: The short-term yield is the same as the long-term yield, i.e. the short-term cost of borrowing is the same as the long-term cost of borrowing.

4. Humped: The intermediate yield is higher than both the short-term and long-term yields.

Time to maturity

YTM

Time to maturity

YTM

Time to maturity

YTM

Time to maturity

YTM

Expectation Theory Liquidity Preference Theory Preferred Habitat Theory Market Segmentation Theory

shape of the yield curve can be explained by the interest rate expectations of those who participate in the market

long-term rate is equal to the geometric mean of current and future year-on-year rates expected by the market participants.

(1+tRn) = [(1+ tR1) (1+ t+1R1) (1+ t+2R1)……(1+

t+n-R1)]1/n

Type of Yield Curve Explanation Ascending Short-term rates are expected to rise in future Descending Short-term rates are expected

to fall in future Humped Short-term rates are expected to rise for a while and

then fall Flat Short-term rates are expected to remain unchanged in

future

According to Hicks, risk-averse investors require an inducement to hold long-term bonds

forward rates should incorporate interest expectations as well as the liquidity premium

This can be represented in form of equation as,(1+tRn) = [(1+ tR1) (1+ t+1R1) (1+ t+2R1+L2)….(1+ t+n-

1R1+

Ln)]1/n

Modigilani and Sutch proposed this. risk aversion implies that investors will

prefer to match the maturity with investment objective

the shape of the yield curve is influenced by expectations of future interest rates as well as risk premia, required to move market participants out of the preferred habitats.

extreme form of preferred habitat theory Investors as well as borrowers don’t

change their preferred habitat. shape of the yield curve entirely depends

upon demand and supply

Concept of Duration Two Types of Strategies 1. Passive Strategies- Buy-and-hold - Indexing2. Active Strategies - Interest rate

anticipation - Valuation analysis - Credit analysis - Yield spread analysis - Bond swaps3. Matched-Funding Techniques4. Portfolio Immunization

Duration measures the time structure of a bond and the bond’s interest rate risk

Time structure means that average time one has to wait to recover the interest and the principal amount

Macaulay’s Duration can be defined as weighted average of time periods to maturity, weights being present values of the cash flow in each time period

01

( )tT

t

PV CD xt

P

1 2 3

0 0 0 0

1 2 3 41 2 3 .......(1 ) (1 ) (1 ) (1 )

tC C C CD T

r r r rP P P P

Where P0 = Market Price of the Bond

Calculate the duration for bond A and bond B with 8 percent and 9 percent coupons having maturity period of 3 years. The face value is Rs 1000. Both the bonds are currently yielding 7 percent.

Bond A Bond B

Face Value Rs 1000 Rs 1000

Coupon rate 8% 9%

Years to maturity 3 3

Macaulay’s Duration 2.7861 2.7654

Higher the yields to maturity, lowers the bond duration and bond volatility, and vice versa.

Larger the coupon rate, lowers the duration and less volatile the bond price

Longer the term to maturity, the longer the duration and more volatile the bond

In a zero coupon bond, the bond’s term to maturity and duration are the same. The zero coupon bond makes only one ballon payment to repay the principal and interest on the maturity date.

Bond with Higher Duration gives higher return when the rates decline and also gives comparatively lower returns when rates rise

Bonds with lower duration are less volatile to the changes in the interest rates

It measures the percentage change in the market value to the percentage change in the yield.

Modified Duration = Duration * 1/(1+i)

Buy and Hold Strategy simply involves buying a bond and holding

it until maturity Bond investors would examine such factors

as quality ratings, coupon levels, terms to maturity, call features and sinking funds

buy-and-hold strategy minimizes transaction costs

if interest rates are currently high and are expected to remain so for an extended period of time, the buy-and-hold strategy will do well.

Indexing involves attempting to build a portfolio that will match the performance of a selected bond portfolio index

These strategies require major adjustments to portfolios, trading to take advantage of interest rate fluctuations, etc. There are five major active bond portfolio management strategies:

1. Interest rate anticipation2. Valuation analysis3. Credit analysis4. Yield spread analysis5. Bond swaps

Riskiest strategy Designed in a way that gives maximum

return when interest rates fall and preserves capital when interest rates rise

Interest rates are directly related to duration

When interest rates are expected to fall, the shift to high duration bonds

Timing of the shift is important For this techniques used are- scenario

analysis, relative return value analysis and strategic frontier analysis

If overvaluation, then sell If under valaution, then buy

Credit analysis involves examining bond issuers to determine if any changes in the firm's default risk can be identified

Portfolio manager would monitor the yield relationships between various types of bonds and look for abnormalities

Spread analysis involves changes in the sectoral relationships

Changes in relative yields (or the spread) may occur due to:

- altered perceptions of the creditworthiness of a sector of the market's sensitivity to default risk

- changes in the market's valuation of some attribute or characteristic of the securities in the sector (such as a zero coupon feature); or

- changes in supply/demand conditions.

incorporate the passive buy-and-hold strategy and active management strategies

manager tries to match specific liability obligations due at specific times to a portfolio of bonds in a way that minimizes the portfolio's exposure to interest rate risk

It attempts to enable one to "lock-in" going interest rates and not have to worry about interest rate shifts. This strategy was being developed by Fisher and Weil in 1971.