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2 Bond Valuation

CA. Nagendra Sah Page 2.1

Coverage of Bond Valuation in Past Examinations:

Year May-2011

Nov-2010

May-2010

Nov-2009

June-2009 Nov-2008

May-2008

SFM (New)

--- 5 --- 4 (6+3+3)+6+8

=26 6+8+4

=18 NA

MAFA (Old)

NA 5+5 = 10 --- 4 --- 5 ---

Contents:

1. What is bond and bond Valuation

2. Basic Bond related term

o Par value

o Redemption value

o Coupon rate

3. Value of Bond

o Value of straight coupon bond or equal interest bond

o Value of Zero coupon bond

o Value of Semi-annual interest bond

o Value of Convertible bond

4. Return or yield

o Current yield

o Holding period return

o Yield to maturity/IRR/Available return from bond

o Yield to call

5. Call feature

6. Bond refunding decision

7. Conversion of Bond

8. Bond duration and Bond Volatility

9. Conditional issue of new bond

10. Principal and interest component of increase/decrease in bond value

11. Bond indifference

12. Implied forward rate based on expectation theory (refer to OTC derivative)

13. Bond Convexity

What is bond and bond valuation?

Bond valuation is the process of determining the fair price/theoretical price/intrinsic value of a bond.

Bond (also term as debenture) is long term loan (Instrument of debt) which pay periodical interest and also principal amount upon maturity.

Basic bond related term

a. Face Value/Par Value: It is the value stated on the face of the bond. Unless otherwise stated, bond is assumed to be issued at face value/par value. The face value/par value may be 100 or 1,000 depending upon the question.

Strategic Financial Management

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b. Redemption value: The value which the bond holder will get on maturity is called redemption value. If no information about redeemable value is given, the bond is always assumed to be redeemed at par value. Otherwise it is redeemed at redeemable value.

c. Coupon Rate: A bond carries a specific interest rate known as the coupon rate. The interest may be paid annually, semi-annually or even monthly. Interest payable to bond holder = Par value of bond Coupon rate

Value of bond

Concept for valuation:

Suppose Mr. X want to invest in any of the following companies. All these companies pay back the same amount (Say 110) at end of the year 1. How much will Mr. X invest now in these companies so that he will get exact as he want?

Logic behind Required return:

- For example suppose you have 1,00,000 surplus today and you want to lend it. Two persons, one is student and another is CA, are approached for it @10% interest rate. What will you do?

Obviously you will provide loan to CA @ 10% because you fill low risk. But if you provide loan to Student you will probably charge high interest rate because you fill high risk.

- It means Low risk low return and high risk high return.

Calculation of Value of investment:

1. Government company:

It means, if Mr. X invest 100 today in Government Company then he will get 10% return from this investment.

Verification: interest = 110-100 = 10

Mr. X

Want to invest

Req. Return of Mr. X = 15%



Govt. Co

(Risk Free)

Company X

(Low risk)

Company Y

(High Risk)

@ 10%

110 Value of investment

= 100

110 PVIF (10%, 1)

Bond Valuation


2. Company X:

It means, if Mr. X invest 98.22 today in Company X then he will get 12% return from this investment.

Verification: interest = 110-98.22 = 11.78

3. Company Y:

It means, if Mr. X invest 95.65 today in Company Y then he will get 15% return from this investment.

Verification: interest = 110 - 95.65 = 14.35

Analysis of above discussion:

- The value of investment will depend upon the “required return” and “cash inflow”. If you want more return then you will invest lesser amounts today and if you want lesser return then you will invest more amounts today for the same future inflow.

- Hence, Investment value today = PV of future inflow

Value of bond

It is the amount which the investor is willing to pay today to purchase the bond. It is also known as current value of bond/present value of bond/intrinsic value of bond/theoretical value of bond/equilibrium value of bond.

Value of bond (B0) = PV of future inflow

[Where inflow will be the future interest and maturity value]

@ 12%


= 98.22 110 PVIF (12%, 1)

@ 15%


= 95.65 110 PVIF (15%, 1)

Bond Share

Balance sheet of Company X

Liability Assets

Share ****

10% Bond ****

Pref Sh. ****

Fixed Assets ***

Current Assets ***

Mr. X

Wants to invest in the Bond of Company X



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The value of the bond is depends upon the investor- that how much the Investor want to invest in that bond.

”10% bond” means company provides interest year by year @ 10% on face value of bond for life of the bond. (i.e. coupon rate is 10%). which is cash inflow for Mr. X.

Required return 12% means Mr. X discount above cash inflow @ 12%. If question does not specify the Required return the coupon rate is assumed to be discount rate.

Value of bond:

a. Annual equal interest bond:

It is the bond on which investors are entitled to get annual interest at fixed rate and also get maturity value at end of the bond life (i.e. on maturity date)

Value of annual interest bond (B0)

= +

Where: PVIFA (YTM, Life) = Sum of PV factor at YTM rate for specific period (i.e. life)

PVIF (YTM, Life) = PV factor at YTM rate for specific period (i.e. life)

b. Value of Zero coupon bond: Zero coupon bonds are those bonds on which investors are not allowed to any interest but are entitled only to repayment of principal amount on the maturity period.

Value of Zero coupon bond (B0) =

c. Value of Perpetual bond:

Perpetual bond are those bonds on which interest is paid forever i.e. up to infinity period. In other word we can say that the perpetual bonds are irredeemable bond.

Value of perpetual bond (B0) =

d. Value of Semi-annual interest bond: Semi-annual interest bond are those bond which pay interest semi-annually (i.e. on every 6 month)

Value of Semi-annual interest bond (B0)

=

+

e. Value of Convertible bond/Market value of convertible bond: It is a bond that can be converted into a predetermined amount of the company’s equity share at certain times during its life. As this bond is convertible on option of bond holder its value will be affected by conversion value or maturity value whichever is higher.

Value of Convertible bond/Mkt. value of CB (B0)

= +

f. Floor value of convertible bond: The value of convertible bond calculated by ignoring the conversion value of convertible bond is called floor value.

Floor Value of Convertible bond

= +

Bond Valuation


Return or yield

(a) Current yield: The rate of return over next one year on the amount invested is called current yield.

Current yield =

(b) Holding period return:

- The holding period is the investment period and return over this period is known as holding period return.

- It is the % return of total holding period and not for one year period.

Holding period return =

(c) Yield to maturity(YTM)/Annual Redemption Yield:

- It is the available rate of return from bond if it is purchased and hold till maturity period. - It is similar to IRR. Also known as available rate of return from bond - Yield to maturity would be calculated by using linear interpolation between two different rates. However, before interpolation we calculate approximate rate to find the range of two different rates, so that we can save our time which we will waste in trial and error method to find one higher rate & another lower rate.

Approximate YTM =

–

After that, calculate exact YTM by interpolating two rates: Exact YTM =

Low rate +

OR,

Exact YTM =

Low rate +

Note: (1) In examination you may calculate approximate YTM instead of calculating exact YTM, if

question carry small mark or you have shortage of time.

(2) Suppose, a bond pays 80 per year as interest and 1000 at maturity. A person invested 930 in such a bond. What is YTM and what does it mean?

0 Y 1Y 2Y 3Y 4Y 5Y

(930) 80 80 80 80 80 +1000 - YTM is such discount rate at which PV of future inflow of bond will equal to the amount

invested in such bond (i.e. PV of outflow)

930



- In other word, we can say at this discount rate (YTM) PV of future inflow is equal to PV of outflow (i.e. NPV = 0)

- If we can solve the equation written below, then we will get YTM directly,

930 =

+

+

+

+

- But we cannot solve above equation easily. Hence, to calculate YTM, we have to assume any rate in place of YTM on trial and error method. But it is not necessary that the calculated value will arrive to 930 at the rate assumed.

- Hence we have to make interpolation of two different rates to reach at 930 values. Concept of interpolation: Assume rate of YTM as 7%, then PV of inflows = 1041 (from above expression)

Assume rate of YTM as 8%, then PV of inflows = 1000 (from above expression)

Assume rate of YTM as 9%, then PV of inflows = 961 (from above expression)

Assume rate of YTM as 10%, then PV of inflows = 924 (from above expression) But we need such YTM rate at which PV of inflows = 930. We know that the required rate is between 9% and 10%. How it will be calculated?

- It will be calculated by interpolating 9% and 10%

7% 8% 9% Required rate (i.e. YTM) 10%

1041 1000 961 930 924

Difference in price = 31 Difference in price = 6

Price difference = 37

Alternative-1:

If difference in Price is 37 then difference in Rate is 1%

If difference in price is 31 then difference in rate must be

= 0.84%

Hence Required rate is 9% + 0.84% = 9.84%

Alternative-2:

If difference is price is 37 then difference in rate is 1%

If difference is price is 6 then difference in rate must be

= 0.16%

Hence required rate is 10% - 0.16% = 9.84%

Note:

PV of cash inflow has inverse relationship with discount rate. If discount rate increases PV of inflow will be decreases If discount rate decreases PV of inflow will be increases It can be understand by following relationship

PV of inflow =

If we divide anything by higher figure, the resultant figure will be lower and

If we divide anything by lower figure the resultant figure will be higher.

Bond Valuation


(d) Yield to call (YTC) It is available rate of return from bond if it purchased and hold till call period. It is also similar to IRR or YTM

Approximate YTC =

–

After that, calculate exact YTC by interpolating two rates: Exact YTC =

Low rate +

OR,

Exact YTC =

Low rate +

Call feature

Call feature allow the bond issuer to call in the bond and repay them at a pre-determined price before maturity.

Bond issuers use this features to protect themselves from paying more interest.

Example:

- Suppose IDBI had issued 16 Years 1000 bonds 10 year ago @14% p.a. But now interest rate in market is around 9% to 10%.

- If the issuer wants to take the benefit of the call features, it will call back the earlier issued bonds and re-issue it @9% p.a.

- The new proceeds from new bonds may be used to re-pay the existing bonds. In this way IDBI now enjoy lower cost for its borrowed money.

- Some bonds offer “call protection” i.e. it would guarantied not to be called before maturity period and it will affect the “bond price”.

Bond refunding decision When bonds redeems before maturity the issuer does not have sufficient cash in hand to repay bond holders. The issuer can issue new bonds and use the proceeds to redeem the older bonds. This process is called bond defunding.

Implications on bond refunding:

(i) Unamortized floatation cost on old bond is being debited to P/L Account on which tax saving is available.

However, floatation cost incurred earlier (i.e. at the time of issue of old bond) is not relevant for bond refunding decision as it was sunk cost.

(ii) Unamortized discount on issue on Old bond is being debited to P/L Account on which tax saving is available. However, cost incurred earlier is not relevant.

(iii) Premium on redemption of old bond is outflow and tax saving is also available on this premium.

(iv) Repayment of old bond is also outflow

(v) Proceeds from new issue of bond is an inflow

(vi) Floatation cost on new issue is also outflow on which tax saving will available over the bond life.

(vii) Coupon rate of old bond is always greater than new bond. Decrease in interest cost on new bond is inflow.



Note: (i) The face value of new bond is always equal to face value of old bond.

(ii) Life of new bond is equal to remaining life of old bond. If question informs otherwise, then you have to take decision on the basis of “Annual Equal inflow” (refer capital budgeting)

Steps for evaluation of refunding decision:

(1) Calculation of Net initial Outflow:

(a) Repayment of old bond with premium………………. Outflow

(b) Net proceeds from New issue…………………………….. Inflow

(i.e. Value of bond – issue cost –Discount)

(c) Tax saving………………………………………………………….. Inflow

(on Unamortized discount of old bond

+on unamortized floatation cost of old bond

+on premium)

(d) Overlapping Interest Net of tax…………………………… Outflow

Net initial out flow Outflow

(2) Calculation of Annual Saving:

Amount

Annual saving in Interest (Existing – New) ***** Inflow

Less: Decrease in Tax saving

(on interest cost)

*****

Outflow

Less: Decrease in Tax saving OR

Add: Increase in Tax saving

(on Amortize Discount and floatation cost)

*****

*****

Outflow

Inflow

Annual saving *****

(3) Alternatively we may calculate Annual Saving as below:

Existing Bond New Bond

Annual Interest ***** *****

Add: Amortized

- Discount cost - Floatation cost

*****

*****

*****

*****

***** *****

Less: Tax saving @given rate ***** *****

***** *****

Less: Amortized(Discount + floatation cost) ***** *****

***** “A” ***** “B”

(4) Calculate net present Value using given discount rate. If discount rate is not given use new interest rate net of tax as discount rate. NPV = Annual saving PVIFA (Periodic Rate, period) – Net initial outflow Decision: If NPV is + ve then bond refunding should be carried. If NPV is –ve then bond refunding should not be carried.

Annual Saving: “A” – “B”

Bond Valuation


Concept for assumption of IRR

Calculation of IRR (Available rate of return) assumes that the intermediate period return will be re-invested at a rate of IRR itself.

Example-1:

0 Y 1 Y 2 Y

Today Value = ? 10 (Int.)

Today’s value of future inflow = (10 PVIF(10%,1 ) + (110 PVIF (10%,2)

(Assuming 10% as discount rate) = (10 .909) + (110 .826)

= 100 (appx.)

- It means, if a person invests 100 today and he will get 10 at 1 year end and (10 + 110 at end of 2 year end then he will get return of 10%.

- Hence, we can say available rate of return from above investment is 10%. OR We can say IRR = 10%

Example – 2:

0 Y 1 Y 2 Y

Today Value = ?

Today’s value of future inflow = (121 PVIF (10%,2) (Assuming 10% as discount rate) = (121 .826) = 100 (appx.)

- It means, if a person invests 100 today and he will get 121 at 2 year end then he will get return of 10%.

- Hence, we can say available rate of return from above investment is 10%. OR We can say IRR = 10%

Compare example-1 and example –2

In both example today’s investment = 100

Available return = 10%

It means, the value of future inflow as on 2 year end of example-1 must be 121.

0 Y 1 Y 2 Y

Today Value = ? 10 (Int.)

Conclusion:

We can say, calculation of IRR assumes that the intermediate period return is re-invested at IRR rate itself.

10 (Int.)

+100 (Maturity)

?

21 (Int.)

+100 (Maturity)

?

10 (Int.)

+100 (Maturity)

+11 100

=121 It means intermediate period receipt is being re-invested at the rate of 10% (i.e. IRR rate)



Bond duration

- Bond duration is the Average time taken to re-collect back the present value of bond. - Average is being calculated by taking weight. The weights are the present values of future

payments. (Duration is also called as Macaulay Duration). Example: Suppose a person lends Rs.3,20,000 to his friend on interest free basis. The friend returns him Rs.10,000 at the end of 1st year, Rs.10,000 at the end of 2nd year and Rs.3,00,000 at the end of 3rd year. What is average period of the loan? Answer Year Weight Year Weight 1 10,000 10,000 2 10,000 20,000 3 3.00.000 9,00,000

ΣW= 320000 ΣYW = 9,30,000 Weighted average period = 9,30,000 / 320000 = 2.90625 In case interest is considered, weights being present value of cash flows arising from the investment.

- If bond carry a coupon rate means there would be receipt of a part of his investment over the

time interest before maturity. Hence the duration of bond would be lesser than the maturity of bond

- If coupon rate is high, the duration should be low. Similarly, if YTM is high, he duration should be low.

- Duration of a non-zero coupon bond is always shorter than its maturity. Duration of a Zero - coupon bond is exactly equal to its maturity.

Mathematically,

Bond Duration =

OR

Bond Duration =

(

Bond Volatility/Modified Duration

Volatility refers to the sensitivity of the bond price to change in current interest rate. In other word Volatility is a measure of rate of change in the bond price on change in yield to maturity. Mathematically,

Bond Volatility =

(

The market price and YTM has opposite relation i.e. when YTM goes up the price goes down or vice-versa. Example Suppose, Modified duration is 3, it means if the Current interest rate changes by 100 basis points (i. e., 1%) the price of bond will change by 1% = 3% in opposite direction. Note: Always see the % change in YTM in absolute term. If YTM change from 15% to 16% then change is 1% (or we can say change is 100 basis point).

Bond Valuation


Conversion of bond

- Convertible bonds are the bonds which have to be converted into specified number of equity shares.

- In India most of the convertible bonds have been issued on the basis of compulsory conversion i.e. the bonds are compulsorily convertible into number of specified number of shares, there is no discretion of the bond holder.

- In USA and European countries, convertible bonds are option convertible bonds i.e. conversion takes place if the bond holder so desire.

Premium over conversion value/Conversion premium:

Conversion premium =

Premium over investment value:

Premium over investment value =

Conversion parity price of share:

Conversion parity price is the price where the market price of bond is equal to the conversion value.

Conversion parity price of share =

Derivation:

At conversion parity price: Market price of bond = Conversion value

Or, market price of bond= Conversion ratio conversion parity price

Or, conversion parity price = arket price of bond

Percentage of downside risk:

Percentage of downside risk =

Where, Straight value of bond (i.e. floor value of bond) = Market value of convertible bond when stock value of bond (i.e. Conversion value) become worthless.

Break even period for convertible bond:

It is the time period, when bond would be converted into equity share so that the loss on conversion would be set off by income from interest income.

Break even period =

=

Bond

Equity

1 Bond @ 100

= 1,000/-

50 Equity share

@ 22 each

= 1,100/-

Convertible into

Conversion ratio

Conversion Value

Or,

Stock value of Bond

Conversion price Mkt Price of bond



Bond Convexity

Convexity is a better technique to find the change in the value of the bond on change in YTM. When we calculate change in bond value due to change in YTM using the convexity, values are nearer to the values obtained by direct calculations (By direct calculation we mean finding the value of bond as present value of all cash flows from

that bond).

Mathematically, Bond Convexity = ear of in lows ( ear

on in lows

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