Valuation of Bonds &

Bond Duration

Concept of Return

Return: The return is the basic motivating force and the principal reward in the investment process. The return may be defined in terms of

(i) realized return, i.e., the return which has been earned, and

(ii) expected return, i.e., the return which the investor anticipates to earn over some future investment period. The expected return is a predicted or estimated return and may or may not occur.

Concept of Return…

The realized returns in the past allow an investor to estimate cash inflows in terms of dividends, interest, bonus, capital gains, etc. available to the holder of the investment. The return can be measured as the total gain or loss to the holder over a given period of time and may be defined as a percentage return on the initial amount invested. With reference to investment in equity shares, return is consists of the dividends and the capital gain or loss at the time of sale of these shares.

Concept of Value

Book value per share is determined as net worth divided by the number of shares outstanding. Book value reflects historical cost, rather than value.

Replacement value is the amount that a company would be required to spend if it were to replace its existing assets in the current condition.

Concept of Value…

Liquidation value is the amount that a company could realize if it sold its assets, after having terminated its business.

Going Concern Value is the amount that a company could realize if it sold its business as an operating business.

Market value of an asset or security is the current price at which the asset or the security is being sold or bought in the market.

Bonds

A bond is a long-term debt instrument or security. Bonds issued by the government do not have any risk of default. The government always honour its obligations on its bonds.

Bonds of the public sector companies in India are generally secured, but they are not free from the risk of default (credit risk).

The private sector companies also issue bonds, which are called debentures in India.

In case of bonds the rate of interest is generally fixed and known to the investors

The principal of a redeemable bond or a bond with a maturity is payable after a specified period called the maturity period.

Features of a Bond

Face value Interest rate (coupon rate) Maturity Redemption value Market value

Categories of Bonds

Bonds with maturity (redeemable bonds) Deep discount bonds (zero-coupon bonds) Perpetual bonds (irredeemable bonds)

Types of Bonds Bonds can be issued by both the government and the

corporate entities. The corporate bonds can be categorized as:

Straight bonds (plain vanilla redeemable bonds) Zero- coupon bonds Floating rate bonds (linked to a bench mark rate) Bonds with embedded options

Convertible bonds (option to convert into equity shares after a certain period of time on certain terms)

Callable bonds (option of pre-mature redemption) Commodity – linked bonds (linked to the price of certain

commodity)

Bond with Maturity (Redeemable Bonds)Bond value = Present value of interest

+ Present value of maturity value

01

INT

(1 ) (1 )

nt nt n

t d d

BB

k k

Where; B0 = Present value of a Bond or DebentureINT t = the amount of interest in period t (from 1 to n

years)k d = the market interest rate or the bond’s required

rate of returnB n = the bond’s terminal or maturity value at the

end of n years

Bond with Maturity (Redeemable Bonds)…

Therefore,

B 0 = INT t (PVFA n,kd) X B n (PVFn,kd)

Example

A 10 year, 12 % coupon bond has a par value of Rs.1000. Assume that the required yield on the bond is 13 %. What is the value of the bond?

Soln : INTt = 12 % of Rs.1000 = Rs.120

B0 = 120 X PVFA 13%,10yrs + 1000 X PVF 13%,10yrs

B0 = 120 X 5.426 + 1000 X 0.295

Therefore B0 = 651.1 + 295 = Rs. 946.10

Bond Yields

Current yield : Current yield relates the annual coupon rate to the market price. The current yield calculation reflects the coupon interest rate. It does not consider the capital gain (or loss) that an investor will realize if the bond is purchased at a discount (or premium)

Yield to maturity Yield to call

Price Current Yield =

Annual Interest

Yield to Maturity

When you purchase a bond, you are not quoted a promise rate of return. Using the information on bond price, maturity date and coupon payments you calculate the rate of return of the bond.

The yield-to-maturity (YTM) is the measure of a bond’s rate of return that considers both the interest income and any capital gain or loss. YTM is bond’s internal rate of return.

A perpetual bond’s yield-to-maturity:

01

INT INT

(1 )

n

tt d d

Bk k

Yield to Maturity of a Redeemable Bond

01

INT

(1 ) (1 )

nt nt n

t d d

BB

k k

Yield to Call

Some bonds carry a feature that entitles the issuer to call (buy back) the bond prior to the stated maturity date in accordance with the call schedule (which specifies the call price for each call date).

For such a bond both the yield to call and yield to maturity is to be calculated (which are mathematically calculated by the same formula).

Bond Value and Amortisation of Principal

A bond (debenture) may be amortised every year, i.e., repayment of principal every year rather at maturity.

The formula for determining the value of a bond or debenture that is amortised every year, can be written as follows:

Note that cash flow, CF, includes both the interest and repayment of the principal.

01 (1 )

ntt

t d

CFB

k

Pure Discount Bonds

Pure discount bond do not carry an explicit rate of interest. It provides for the payment of a lump sum amount at a future date in exchange for the current price of the bond. The difference between the face value of the bond and its purchase price gives the return or YTM to the investor.

01

nn

d

MB

k

Perpetual Bonds

Perpetual bonds, also called consols, have an indefinite life and therefore, they have no maturity value. Perpetual bonds or debentures are rarely found in practice.

01

INT INT

(1 )

n

tt d d

Bk k

Practical ProblemsQ. You are considering investing in one of the following bonds:

Coupon rate Maturity Price/Rs.100par value

Bond A 11% 8 yrs Rs.80

Bond B 9% 9 yrs Rs.70

Your income tax rate is 34 percent and your capital gains tax is effectively 10 percent. Capital gains taxes are paid at the time of maturity on the difference between the purchase price and par value. What is your post-tax yield to maturity from these bonds?

A. 10.91% & 11.06 % (by short –cut method)

Practical Problems…

Q. A company's bonds have a par value of Rs.100, mature in 5 years, and carry a coupon rate of 10 percent payable semi-annually. If the appropriate discount rate is 14 percent, what price should the bond command in the market place?

A. Rs. 85.92

Bond Duration

The term duration has a special meaning in the context of bonds. It is a measurement of how long, in years, it takes for the price of a bond to be repaid by its internal cash flows.

It is an important measure for investors to consider, as bonds with higher durations carry more risk and have higher price volatility than bonds with lower durations.

Bond Duration…

For each of the two basic types of bonds the duration is the following:

1. Zero-Coupon Bond – Duration is equal to its time to maturity.

2. Vanilla Bond - Duration will always be less than its time to maturity.

Duration of a Zero-Coupon Bond

The red lever above represents the four-year time period it takes for a zero-coupon bond to mature. The money bag balancing on the far right represents the future value of the bond, the amount that will be paid to the bondholder at maturity. The fulcrum, or the point holding the lever, represents duration, which must be positioned where the red lever is balanced. The fulcrum balances the red lever at the point on the time line at which the amount paid for the bond and the cash flow received from the bond are equal. The entire cash flow of a zero-coupon bond occurs at maturity, so the fulcrum is located directly below this one payment.

Duration of a Vanilla or Straight Bond

Consider a plain vanilla bond that pays coupons annually and matures in five years. Its cash flows consist of five annual coupon payments and the last payment includes the face value of the bond. The moneybags represent the cash flows you will receive over the five-year period. To balance the red lever at the point where total cash flows equal the amount paid for the bond, the fulcrum must be farther to the left, at a point before maturity. Unlike the zero-coupon bond, the straight bond pays coupon payments throughout its life and therefore repays the full amount paid for the bond sooner.

Factors affecting the Duration of a Bond

Higher the coupon rate, lower the duration of the bond and less volatile the bond price.

Longer the term to maturity, the longer the duration & more volatile the bond.

Higher the yield to maturity, lower the bond duration and bond volatility & vice – versa.

Calculation of Duration of Bond

Macaulay’s Duration

Where D = Duration of the bond

Ct = Cashflows for period t

R = Current yield to maturity (discount rate)

T = Number of years

Pv(Ct)= Present value of the cashflows

P0 = Sum of the present value of cashflows

Tv t

t =1 0

P (C ) D = × t

P

Example

Calculate the duration for bond A & bond B with 7 % & 8 % coupon rates having maturity period of 4 years. The face value is Rs. 1000. Both the bonds are currently yielding 6 %.

Example Bond A Bond B

Face Value Rs.1000 Rs.1000

Coupon Rate 7% 8%

Years to maturity 4 4

Macaulay’s Duration 3.631 years 3.592 years

Modified Duration

Modified Duration = D

Where D = Macaulay’s Duration

y = bond’s yield to maturity

k = number of compounding done (annual interest k = 1, semi-annual interest k = 2, & so on)

1 + y / k

Problems

1. The following information has been given:

Face Value : Rs.100

Coupon Rate : 12% payable annually

Years to maturity :6

Current Market price of the bond : RS. 110

What is the duration of the bond? Use the approximate formula for calculating the YTM.

A. YTM = 9.75 %

Duration = 4.665 years

Rules for Duration

1. The duration of a zero – coupon bond is same as its maturity.

2. Higher the coupon rate, lower the duration of the bond and less volatile the bond price.

3. Higher the yield to maturity, lower the bond duration and bond volatility & vice – versa.

4. Longer the term to maturity, the longer the duration & more volatile the bond.

Rules for Duration…

5. Duration of a level perpetuity1 + Yield

6. Duration of a level annuity1 + Yield

7. Duration of a coupon bond1 + y

Yield

Yield

Number of payments

(1 + Yield) Number of Payments – 1 –

y–

(1+y) + T (c – y)

c [ (1 + y)T – 1] + y

Where; y = bond’s yield per payment periodT = the number of payment periodsc = Coupon rate per payment period

Problems…

1. A 10 year annual annuity has a yield of nine percent. What is its duration?

A. 4.8 years

2. A 10 percent coupon bond has a maturity of 12 years. It pays interest semi – annually. Its yield to maturity is four percent per half – year period. What is its duration?

A. 7.58 years

3. Find the duration of a 6% coupon bond making annual coupon payments if it has 3 years until maturity and has a YTM of 6%. What is the duration if the YTM is 10%?

A. 2.836 years, 2.824 years

Problems…

4. A 10 percent coupon bond has a maturity of six years. It pays interest semi-annually. Its yield to maturity is 4 percent per half year period. What is its duration?

A.9.44 half periods (4.72 years) 5. Find the duration of a 7% coupon bond

making annual coupon payments if it has 5 years to maturity and has a YTM of 7%. What is the duration if the YTM is 13%?

Bond Immunization

An investor is interested in bond immunization if they would like to limit their interest rate risk.

In order to reduce the interest rate risk of a portfolio, an investor may buy bonds with a duration equal to the their time horizon (known as asset/liability matching). This strategy is known as bond immunization.

A simplified example is if an investor plans to buy a car in five years for their child, then they may buy a bond that matures in five years. In this case, the investor is indifferent to the bond's price fluctuation because the bond matures at the original par value at the stated maturity and when the investor needs the money.

Need for Immunization

Bonds exposed to two types of risk: The price risk The reinvestment risk

The price risk & the reinvestment risk move in the opposite direction.

When interest rate rises there are two consequences: The price of the bond falls, & The return on reinvestment of the interest income

improves

& vice - versa

Need for Immunization…

These two opposite effects can exactly offset each other if a portfolio of bonds is built whose duration is exactly equal to the investors time horizon & the investor will become immune to the interest rate risk.

If an investor does so whenever there is a change in the interest rate, the losses (or gains) in capital value will be exactly offset by gains (or losses) on reinvestments.

Bond Immunization Process

The bond portfolio manager or investor calculates the duration of the promised outflow of the funds and invests in a portfolio of bonds which has an identical duration.

The bond portfolio duration is the weighted average of the durations of the individual bonds in the portfolio.

For example if an investor has invested equal amounts of money in three bonds A, B & C having durations of 2, 3 & 4 years respectively then the duration of the bond portfolio is

D = (1/3 X 2) + (1/3 X 3) + (1/3 X 4) = 2.99 or 3 years

Example

1. Abhishek has Rs.50000 to make one time investment. He needs his money back after two years for his son’s education. He has a choice of two types of bonds:

i. Bond A has a coupon rate of 7 % and maturity period of four years with a current yield of 10 %. Current price is Rs.904.90

ii. Bond B has a coupon rate of 6 %, a maturity of one year and a current yield of 10 %. The current price is Rs.963.64.

Advice Abhishek on the proportion of funds he needs to put into each bond to build his portfolio.

A. 61.58 % & 38.42 % (Rs. 25,446 & 15,876 = Rs.41,322)