Prof.

Archana Khetan VALUATION OF BONDS

VALUATION OF BONDS

1. IDBI, in its issue of flexi-bonds – 3, offered Growing Interest Bond. The interest will be
paid to the investors every year at the rats given below and the minimum deposits is `
5000/-.
Interest (p.a)
Year 1 10.5%
Year 2 11.0%
Year 3 12.5%
Year 4 15.25%
Year 5 18.0%
Calculate the Yield to Maturity (YTM)?

Sol. At 13%, RHS

525 550 625 762.5 5,900
= + + + + = ` 4,998.43
1.13 (1.13) 2
(1.13) 3
(1.13) 4
(1.13)5
At 12% RHS
525 550 625 762.5 5,900
= + 2
+ 3
+ 4
+ = ` 5,184.47
1.12 (1.12) (1.12) (1.12) (1.12)5
By interpolation,
At 13% ® ` 4,998.43
At r = ? ® ` 5,000
At 12% ® ` 5,184.47
r - 12% 5,000 - 5,184.47
=
13% - 12% 4,998.43 - 5,184.47
r = 12.99%

2. Calculate Market Price of :

i. 10% Government of India security currently quoted at, 110 but interest rate is
expected to go up by 1%.
ii. A bond with 7.5% coupon interest. Face Value, 10,000&term to maturity of 2 years,
presently
Yielding 6% interest payable half yearly.
Sol.
i. CR = 10% Price = `110
`10
C.Y = × 100 = 9.09 %
`110
New Yield = 9.09 % + 1% = 10.09%

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VALUATION OF BONDS KHETAN EDUCATION
New Price = `10 / 10.09% = ` 99.108.

c ` 10, 000 ´ 7.5%

ii. Coupon = = = 375
m 2
Maturity (n) = n × m = 2 years × 2 = 4
n 6%
ROR = = = 3%
m 2
VO = ` 375 × PVA(3%,4) + ` 10,000 × PVIF(3%,4)
= ` 375 × 3.717 + ` 10,000 × 0.8885 = ` 10,278.875

3. X Ispat Ltd. has made an issue of 14 per cent non-convertible debentures on January 1,
2007. These debentures have a face value of ` 100 and is currently traded in the market
at a price of ` 90.
Interest on these NCDs will be paid through post-dated cheques dated June 30 and
December 31. Interest payments for the first 3 years will be paid in advance through post-
dated cheques while for the last 2 years post-dated cheques will be issued at the third year.
The bond is redeemable at par on December 31, 2011 at the end of 5 years.
a. Estimate the current yield at the YTM of the bond.
b. Calculate the duration of the NCD.
c. Assuming that intermediate coupon payments are, not available for reinvestment
calculate the realised yield on the NCD.
Sol. C = 14% FV = 100 Price = 90 n = 5
100 × 14%
Semi-annual coupon = =`7
2
C 7
i. CY = × 100 = × 100 = 7.78%
Price 90
We need to annualize this,
BEY = 7.78% × 2 = 15.56% p.a. (Bond equivalent yield)
BAY = (1.0778)2 – 1 × 100 = 16.17% (Effective annualised yield)

YTM = [ 7+ (100-90) / 2×5 ]× 100

= 8.42%
BEY = 8.42% 2 = 16.84% p.a.
EAY = (1.0842)2 – 1 × 100 = 17.55% p.a.

b. Computation of Duration
Period CF PV@ 8.42% PV WiXi
1 7 0.922 6.454 6.454
2 7 0.851 5.957 11.914
3 7 0.785 5.495 16.485

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Prof. Archana Khetan VALUATION OF BONDS

4 7 0.724 5.068 20.272

5 7 0.668 4.676 23.38
6 7 0.616 4.312 25.872
7 7 0.568 3.976 27.832
8 7 0.524 3.668 29.344
9 7 0.483 3.381 30.429
10 107 0.446 47.722 477.22
` 90.71 ` 669.20
` 669.2
Duration =
= 7.38 semesters
` 90.71
7.38
Duration (in years) = = 3.69 years.
2

c. If coupons are not available for reinvestment

i. Value of coupon = ` 7 × 10 = ` 70
ii. Redemption amount on maturity = ` 100
iii. Total inflow = ` 70 + ` 100 = ` 170
iv. ` 90 (1 + RY)10 = ` 170
170
At 7% , RHS = = ` 86.42
(1.07)10
170
At 6%, RHS = = ` 94.93
(1.06)10
By interpolation
r - 6% 90 - 94.93
=
7% - 6% 86.42 - 94.93
r = 6.58%
RY (pa) = (1.0658)2 – 1 × 100 = 13.59%

4. Consider the following term structure

n
ron = 10+
2
there is a 12% coupon, ` 1000 face value 3 year bond. Compute the market price, duration
and price volatility of this bond. If the yield curve shifts downward parallel by 80 basis
points, find out the new bond price using
a. Price volatility
b. Repricing
n
Sol. r0n = 10 +
2
FV = ` 1,000
n =3

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VALUATION OF BONDS KHETAN EDUCATION
Coupon = 12%
3
r03 = 10 + = 11.5%
2
Year CF PV@11.50% PV WiXi
1 120 0.897 107.64 107.64
2 120 0.804 96.48 192.96
3 1,120 0.721 807.52 2,422.56
1,011.64 2,723.16
2, 723.16
Duration = = 2.69
1, 011.64
a. Price = ` 1,011.64
D
Price volatility (Modified duration) =
1 + YTM
2.69
=
1.115
= 2.414
This means if interest rates change by 1% or 100 bps (basic points), the price of the bond
would change by 2.414% in the opposite direction.
Þ If interest D by 100 bps ® Price D 2.414%
If interest D by 80 bps ® Price 1.9312%
Price as per price volatility method :
Current price = ` 1,011.64
New price = ` 1,011.64 + 1.9312% = ` 1,031.18
b. Price as per Re-pricing method.
Year CF PV@10.7% PV
1 120 0.903 108.36
2 120 0.816 97.92
3 1,120 0.737 825.44
1,031.72

5. The Investment portfolio of a bank is as follows:

Government Bond Coupon Rate (%) Purchase rate Duration (years)
(FV= ` 100/ bond)
G.O.I. 2006 11.68 106.50 3.50
G.O.I. 2010 7.55 105.00 6.50
G.O.I. 2015 7.38 105.00 7.50
G.O.I. 2022 8.35 110.00 8.75
G.O.I. 2032 7.95 101.00 13.00
Face value of total investment is ` 5 crores in each bond.

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Prof. Archana Khetan VALUATION OF BONDS

a. Calculate actual investment in Portfolio.

b. What is a suitable action to churn out investment portfolio in the following
scenario?
i. Case I- interest rates are expected to lower by 25 basis points.
ii. Case II- Interest rates are expected to rise by 75 basis points.
Also calculate the revised duration of investment portfolio in each scenario.
Sol.
Wi (` in lakhs)
Bonds Purchase rate Value Di WiDi
GOI 2,006 106.5 532.5 3.50 1,863.75
GOI 2,010 105.0 525.0 6.50 3,412.50
GOI 2,015 105.0 525.0 7.50 3,937.50
GOI 2,022 110.0 550.0 8.75 4,812.50
GOI 2,032 101.0 505.0 13 6,565.00
` 2,637.5 ` 20,591.25
FV (Total) = ` 50,000
FV Per Bond = ` 100
\ No of bonds = ` 5 Cr ÷ ` 100 = 5,00,000 bonds
a. Actual investment in the portfolio = ` 2,637.5 lakhs
20,591.25
b. Present duration of portfolio = = 7.81 years
2, 637.5
i. If interest rates go down, the price of all bonds would go up and more for longer maturity
bonds in comparison to short term the fund manager should sell off short maturity bond
as bonds and invest the proceeds in long term bond / bonds one of the, combinations
would be to shift GOI 2,010 to GOI 2,032.
20,591.2 - 3, 412.5 + 525 ´ 13
New DP = = 9.10 Years.
2,637.5
ii. If interest rates go up, the price of all bonds would go down and more for longer maturity
bonds in comparison to short term the fund manager should sell off long maturity bonds
and invest the proceeds in short term bonds one of the combinations would be to shift
GOI 2032 to GOI 2010.
` 20,591.2 - ` 6,565 + ` 50,556.5
New DP =
` 2,637.5
= 6.56 Years

7. ABC Ltd. issued 9%, 5 year bonds of ` 1,000/- each having a maturity of 3 years. The
present rate of interest is 12% for one year tenure. It is expected that Forward rate of
interest for one year tenure is going to fall by 75 basis points and further by 50 basis
points for every next year in further for the same tenure. This bond has a beta value of
1.02 and is more popular in the market due to less credit risk. Calculate:

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VALUATION OF BONDS KHETAN EDUCATION
l Intrinsic value of bond
l Expected price of bond in the market
Sol. CR = 9%
FV = 1,000
r01 = 12%
ƒ12 = 12-0.75
f23 = 11.25% -0.5 = 10.75%
n = 3year = 11.25%
The sum is based on the concept of upward rising curve (as the sum mentions spot &
forward rates), we need to discount cash follow of different year with different rates.
® (1 + r02)2 = (1 + r01) (1 + f12)
(1 + r02)2 = (1.12) (1.1125)
(1 + r02)2 = 1.246
1
1 + r02 = (1.246) 2
r02 = 11.62%
® (1 + r03)3 = (1 + r02)2(1 + f23)
(1 + r03)3 = (1 + r01) (1 + f12) (1 + f 23)
= (1.12) (1.1125) (1.1075)
= 1.380
90 90 1,090
a. V0 = + +
2
(1 + r01) 1
(1 + r02) (1 + r03)3
90 90 1, 090
= + +
1.12 1.246 1.380
= ` 942.44
b. Expected market price
= Intrinsic value × Beta (b)
= ` 942.44 × 1.02
= ` 961.30

8. The following is the yield structure of AAA rated debenture:

Period Yield (%)
3 months 8.5
6 months 9.25
1 year 10.50
2 years 11.25
3 years and above 12.00
Based on the expectation theory calculate the implicit one year forward rates in year 2
and 3.

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Prof. Archana Khetan VALUATION OF BONDS

If the interest rate increases by 50 basis points, what will be the percentage change in the
price of the bond having a maturity of 5 years? Assume bond is fairly priced at the
moment at ` 1000.
Sol.
a. Computation of 1 year fund rate in year 2 i.e. f12
(1 + r02)2 = (1 + r01) (1 + f12)
(1.1125)2 = (1.1050) (1 + f12)
1 + f12 = 1.12
f12 = 12%
Computation of 1 year fund rate in year 3 i.e. year f23
(1 + r03)3 = (1 + r02)2 (1 + f23)
(1.12)3 = (1.1125)2 (1 + f23)
(1.4049) = (1.2377) (1 + f23)
f23 = 13.51%

b. Since the bond is fairly priced at `1,000 it means bond price = FV = `1,000.
It also means coupon rate = YTM = 12%
Year CF PV@12.5% PV
1 120 0.889 106.68
2 120 0.790 94.80
3 120 0.702 84.24
4 120 0.624 74.88
5 1,120 0.555 621.16
` 981.76
P1 - P0
\D in price = × 100 = [(` 981.76 - ` 1000)/ ` 1000] × 100
P0
= -1.824%

9. From the following data for Government securities, calculate the forward rates:
Face value (`) Interest rate Maturity (year) Current price (`.)
1,00,000 0% 1 91,500
1,00,000 10% 2 98,500
1,00,000 10.5% 3 99,000
Sol. With the help of given data, we can compute r01, r02, r03.
For Bond A,
FV
VA =
(1 + r)1
1,00,000
` 91,500 = = r01 = 9.29%
(1 + r)

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VALUATION OF BONDS KHETAN EDUCATION

For Bond B,
C C + R.A.
VB = + ….. (R.A. → Redemption Amount)
(1 + r01)1
(1 + r02)2
C C + R.A.
= +
(1 + r01) (1 + r01) (1 + f12)
` 10,000 ` 10,000 + ` 1,00,000
` 98,500 = +
1.0929 (1.0929)(1 + f12)
` 1,00,649.65
` 89,350 =
(1 + f12)
f12 = 12.65%
For Bond C,
C C C + R.A.
VC = + 2
+
(1 + r01)1
(1 + r02) (1 + r03)3
C C C + R.A
= + +
(1 + r01) (1 + r01) (1 + f12) (1 + r02)2 (1 + f23)
` 10,500 ` 10,500 ` 10,500 + ` 1,00,000
` 99,000 = + +
1.0929 (1.0929) (1.1265) (1.0929) (1.1265) (Hf23)
[ (1.0929)(1.1265)(1+f23)]
` 99,000 – ` 9,607.47 – ` 8,528.6 = ` 89,753.35
(1 + f23)
f23 = 10.99%

10. M/s. Transindia Ltd. is contemplating calling ` 3 crores of 30 years, <1,000 bond issued
5 years ago with a coupon interest rate of 14 percent. The bonds have a call price of
<1,140 and had initially collected proceeds of<2.91 crores due to a discount of<30 per
bond. The initial floating cost was <3,60,000. The Company intends to sell <3 crores of
12 per cent coupon rate, 25 years bonds to raise funds for retiring the old bonds. it
proposes to sell the new bonds at their par value of <1,000. The estimated floatation cost
is <4,00,000. The company is paying 40% tax and its after cost of debt is 8 per cent. As
the new bonds must first be sold and their proceeds, then used to retire old bonds, the
company expects a two months period of overlapping interest during which interest must
be paid on both the old and new bonds.
What is the feasibility of refunding bonds?
` 3 Cr
Sol. No. of Bonds = = 30,000 bonds.
` 1, 000
Computation of outflows
Post tax cost of premium = ` 25,20,000
[(`140 × 30,000) × (1- 40%)]

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Prof. Archana Khetan VALUATION OF BONDS

+ Floatation cost on new bonds = ` 4,00,000

+ Post tax overlapping interest on old bonds = ` 4,20,000
2
[( ` 3cr × 14% × ) (1- 40%)]
12
(-) Tax shield on floatation cost on old bond (unamortized) = (` 1,20,000)
é `3, 60, 000 ù
ê 30 year ´ 25 years ´ 40% ú
ë û
(-) Tax shield on discount on old bonds (unamortized) = (` 3,00,000)
é `30 × 30, 000 ù
ê 30 year ´ 25 years ´ 40% ú = ` 29,20,000
ë û
Computation of present value of inflows.
Particulars Old Bonds New Bonds
i. Post tax interest ` 3 Cr × 14% × (1 – 40%) ` 3Cr × 12% × (1 – 40%)
= 25,20,000 = 21,60,000
ii. Tax benefit on `3, 60, 000 `4, 00, 000 × 40%
= × 40% =
floatation cost 30 year 25 year × 40%

= (`4,800) = (` 6,400)
iii. Tax benefit on -
= `30 × 30, 000 × 40%
discount 30 × 40%
= (`12,000)
` 25,03,200 ` 21,53,600
Saving p.a. = ` 25,03,200 – ` 21,53,600
= ` 3,49,600
Present value = ` 3,49,600 × PVA (8%, 25)
= ` 3,49,600 × 10.675
= 37,31,980.
NPV = ` 37,31,980 – ` 29,20,000
= ` 8,11,980.
Since, the NPV is positive, we advise the firm to go ahead with bond refunding.

11. ABC Ltd. has ` 300 million, 12 per cent bonds outstanding with six years remaining to
maturity. Since interest rates are falling, ABC Ltd. is contemplating of refunding these
bonds with a ` 300 million issue of 6-year bonds carrying a coupon rate of 10 per cent.
Issue cost of the new bond will be ` 6 million and the call premium is 4 per cent. ` 9
million being the unamortized portion of issue cost of old bonds can be written off no
sooner the old bonds are called off. Marginal tax rate of ABC Ltd. is 30 per cent.
You are required to analyze the bond refunding decision.

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VALUATION OF BONDS KHETAN EDUCATION
Sol. NPV of bond refunding
NPV of bond refunding = P.V of Savings – Outflow
Computation of outflow
Post tax cost of Premium = ` 84,00,000
[` 300 m × 4% × (1 – 30%)]
+ Floatation cost on new bonds = ` 60,00,000
- Tax shield on unamortized Cost on old bonds = (` 27,00,000)
(` 90,00,000 × 30%)

= ` 1,17,00,000

Computation of savings
Particulars Old New
Post tax Coupon ` 3,000L × 12% × (1 – 30%) ` 3,000L × 10% × (1 – 30%)
cost
= 2,52,00,000 = 2,10,00,000

- Tax benefit on ` 90,00,000 ` 60,00,000

× 30% × 30%
amortization of 6 yrs. 6 yrs.
floatation Cost
= (` 4,50,000) =(` 3,00,000)
` 2,47,50,000 ` 2,07,00,000
Net savings p.a.
= ` 40,50,000
P.V. of savings = ` 40,50,000 × PVA(7%,6)
= ` 1,93,06,350
If kd is not given use
Coupon Rate (1 – T)
= 10% (1 – 30%) = 7%
NPV = 1,93,06,350 – 1,17,00,000
= 76,06,350
® We recommend the firm to go ahead with bond refunding.

12. XYZ company has current earnings of ` 3 per share with 5,00,000 shares outstanding.
The company plans to issue 40,000, 7% convertible preference shares of ` 50 each at par.
The preference shares are convertible into 2 shares for each preference shares held. The
equity share has a current market price of ` 21 per share.
i. What is preference share’s conversion value?
ii. What is conversion premium?

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Prof. Archana Khetan VALUATION OF BONDS

iii. Assuming that total earnings remain the same, calculate the effect of the issue on
the basic earning per share
a. before conversion
b. after conversion
iv. If profits after tax increases by ` 1 million what will be the basic EPS
a. before conversion and
b. on a fully diluted basis?
Sol.
i. Conversion value of preference share
Conversion Ratio × Market Price
2 × ` 21 = ` 42
ii. Conversion Premium
(` 50/ ` 42) - 1 = 19.05%
iii. Effect of the issue on basic EPS
Particulars `
Before Conversion
Total (after tax) earnings ` 3 × 5,00,000 15,00,000
Dividend on Preference shares 1,40,000
Earnings available to equity shares 13,60,000
No. of shares 5,00,000
EPS 2.72
On Diluted Basis
Earnings 15,00,000
No of shares (5,00,000 + 80,000) 5,80,000
EPS 2.59

iv. EPS with increase in profit

Particulars `
Before Conversion
Earnings 25,00,000
Dividend on Preference shares 1,40,000
Earnings available to equity shares 23,60,000
No. of shares 5,00,000
EPS 4.72
On Diluted Basis
Earnings 25,00,000
No of shares 5,80,000

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VALUATION OF BONDS KHETAN EDUCATION
EPS 4.31

13. The data given below relates to a convertible bond:

Face value ` 250
Coupon rate 12%
No. of shares per bond 20
Market price of share ` 12
Straight value of bond ` 235
Market price of convertible bond ` 265
Calculate:
i. Stock value of bond.
ii. The percentage of downside risk.
iii. The conversion premium
iv. The conversion parity price of the stock
Sol. i. Stock value or conversion value of bond
12 × 20 = ` 240
ii. Percentage of the downside risk
`265 - ` 235
= 0.1277 or 12.77%
` 235
OR,
`265 - ` 235
= 0.1132 or 11.32%
` 265
This ratio gives the percentage price decline experienced by the bond if the stock becomes
worthless.
iii. Conversion Premium
Market Price - Conversion Value `265 - ` 240
× 100 = × 100 = 10.42%
Conversion Value ` 240
iv. Conversion Parity Price
Bond Price 265
= = ` 13.25
No. of Shares on Conversion 20
This indicates that if the price of shares rises to ` 13.25 from ` 12 the investor will neither
gain nor lose on buying the bond and exercising it. Observe that ` 1.25 (` 13.25 - ` 12.00)
is 10.42% of ` 12, the Conversion Premium.

14. Suppose Mr. A is offered a 10% Convertible Bond (par value ` 1,000) which either can
be redeemed after 4 years at a premium of 5% of get converted into 25 equity shares
currently trading at ` 33.50 and expected to grow by 5% each year. You are required to
determine the minimum price Mr. A shall be ready to pay for bond if has expected rate
of return is 11%.

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Prof. Archana Khetan VALUATION OF BONDS

Sol: First, we shall find the Conversion Value of Bond

CV = C (1+g)n × R
Where:
C = Current market price
G = Growth rate of price
R = Conversion ratio
n = No. of years

Accordingly, CV shall be
= ` 33.50 × 1.054 × 25 = ` 33.50 × 1.2155 × 25 = ` 1017.98
Value of bond if conversion is opted = ` 100× PVAF (11%,4) + ` 1017.98
PVF (11%,4)
= ` 100 × 3.102 + ` 1017.98 × 0.659
= ` 310.20 + ` 670.85
= ` 981.05

Since above value of Bond is based on the expectation of growth in market price which
may or may not be as per expectations. In such circumstances the redemption at premium
still shall be guaranteed and bond may be purchased at its floor value computed as
follows:
Value of bond if conversion is not opted = `100 × PVAF (11%,4) + ` 1050 PVF
(11%,4)
= ` 100 × 3.102 + ` 1050 × 0.659
= ` 310.20 + ` 691.95
= ` 1002.15

15. A Ltd. has issued convertible bonds, which carries a coupon rate of 14%. Each bond is
convertible into 20 equity shares of the company A Ltd. The prevailing interest rate for
similar credit rating bond is 8%. The convertible bond has 5 years maturity. It is
redeemable at par at ` 100.
The relevant present value table is as follows.
Present T1 T2 T3 T4 T5
Values
PVIF0.14,t 0.877 0.769 0.675 0.592 0.519
PVIF0.08,t 0.926 0.857 0.794 0.735 0.681
You are required to estimate:
(Calculations be made up to 3 decimal places)
i. Current market price of the bond, assuming it being equal to its fundamental value,

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VALUATION OF BONDS KHETAN EDUCATION
ii. Minimum market price of equity share at which bond holder should exercise
conversion option; and
iii. Duration of the bond.
Sol.
i. Current Market price of Bond:
Time CF PVIF 8% PV (CF) PV(CF)
1 14 0.926 12.964
2 14 0.857 11.998
3 14 0.794 11.116
4 14 0.735 10.290
5 14 0.681 77.634
åPV(CF) i.e. P0 = 124.002
Say ` 124.00

ii. Minimum Market Price of Equity Shares at which Bondholder should exercise conversion
option:
124.00
= = 6.20
20.00

iii. Duration of Bond

Year Cash P.V.@ 8% Proportion of Proportion of Bond
Flow Bond Value Value × Time
(Years)
1 14 0.926 12.964 0.105 0.105
2 14 0.857 11.998 0.097 0.194
3 14 0.794 11.116 0.089 0.267
4 14 0.735 10.290 0.083 0.332
5 14 0.681 77.634 0.626 0.130
124.00 1.000 4.028

16. The following is the data related to 9% Fully convertible (into Equity Shares) debentures
issued by Delta Ltd. at ` 1000.
Market Price of 9% Debenture ` 1,000
Conversion Ratio (No. of shares) 25
Straight Value of 9% Debentures ` 800
Market price of equity share on the date of conversion ` 30
Expected Dividend p r share `1
Calculate:
i. Conversion value of Debenture; ii. Market Conversion Price;

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Prof. Archana Khetan VALUATION OF BONDS

iii. Conversion Premium per share; iv. Ratio of Conversion Premium;

v. Premium over straight value of Debenture;
vi Favourable Income Differential per share; and
vii. Premium pay back period
Sol.
i. Conversion Value of Debenture
= Market Price of one Equity Share X Conversion Ratio
= ` 30 × 25
= ` 750

ii. Market Conversion Price

Market Price Convertible Debenture 1000
= =
Conversion Ratio 25
= ` 40

iii. Conversion Premium per share

Market Conversion Price - Market Price of Equity Share
= ` 40 - ` 30
= ` 10

iv. Ratio of Conversion Premium

= Conversion Pr emium Per Share
market Pr ice of Equity Share
10
= × 100
30
= 33.33%

v. Premium over Straight Value of Debenture

= Market Pr ice of Convertible Bond - 1
Straght Value of BOnd
1, 000
= -1
800
= 25%

vi. Favourable income differential per share

Coupon Interst from Debenture - Conversion Ratio × Dividend per share
=
Conversion Ration

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VALUATION OF BONDS KHETAN EDUCATION
90 - 25 × 1
=
25
= ` 2.6
vii. Premium pay back period
= Conversion primium per share
favourable Income Differntial per share
10
= = 3.85 years
2.6

17. A hypothetical company ABC Ltd. issued a 10% Debenture (Face Value of ` 1000) of
the duration of 10 years, currently trading at ` 850 per debenture. The bond is convertible
into 50 equity shares being currently quoted at ` 17 per share.
If yield on equivalent comparable bond is 11.80%, then calculate the spread of yield of
the above bond from this comparable bond.
The relevant present value table is as follows.
Present t1 t2 t3 t4 t5 t6 t7 t8 t9 t10
Values
PVIF0.11,t 0.901 0.812 0.731 0.659 0.593 0.535 0.482 0.434 0.391 0.352
PVIF0.13,t 0.885 0.783 0.693 0.613 0.543 0.480 0.425 0.376 0.333 0.295
Sol. Conversion Price = ` 50 × 17 = ` 850
Intrinsic Value = ` 850
Accordingly the yield (r) on the bond shall be :
` 850 = ` 100 PVAF (r, 10) + ` 1000 PVF (r, 10)
Let us discount the cash flows by 11%
850 = 100 PVAF (11%, 10) + 1000 PVF (11%, 10)
850 = 100 × 5.890 + 1000 × 0.352 = 91
Now let us discount the cash flows by 13%
850 = 100 PVAF (13%, 10) + 1000 PVF (13%, 10)
850 = 100 × 5.426 + 1000 × 0.295 = -12.40
Accordingly, IRR
11% + 90.90 (13% - 11%)
90.90 - (-12.40)
90.90
11% + × (13% - 11% = 12.76%
103.30
The spread from comparable bond = 12.76% - 11.80% = 0.96%

18. XYZ Ltd.’s bond (Face Value of ` 1,000) with 4 years maturity is currently trading at `
900 carrying a coupon rate of 15%. Assuming that the reinvestment rate is 16%, you are
required to calculate Realized Yield to Maturity of the bond.

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Prof. Archana Khetan VALUATION OF BONDS

Sol. We shall computer* (Realized Yield to Maturity) by using following equation:

PV (1+r*) 4 = Future Value of benefits and Future Value of Benefits shall be
computed as follows:
0 1 2 3 4
Investment (`) 900
Annual Interest(`) 150 150 150 150
Compound Factor @ 1.56 1.35 1.16 1.00
16%
Future Value of 234.00 202.50 174.00 150
Intermediate Cash
Flows(`)
Maturity Value (`) 1,000
900 234.00 202.50 174.00 1,150.00
Total Future Benefits 1,760.50
Accordingly,
900(1 + r*)4 = 1,760.50
(1 + r*)4 = 1,760.50/900
(1 + r*)4 = 1.956
(1 + r*) = (1.183)1/4
r* = 0.183 say 18.30%
This portion can also be alternatively done as follows
= ` 964.40 × .75(3.63/100) = ` 26.26
Then the market price will be
= ` 964.40 - ` 26.26 = ` 938.14
19. Mr. A will need ` 1,00,000 after two years for which he wants to make one time necessary
investment now. He has a choice of two types of bonds. Their details are as below:

BOND X BOND Y
Face value ` 1,000 ` 1,000
Coupon 7% payable annually 8%payable annually
Years to maturity 1 4
Current price ` 972.73 ` 936.52
Current yield 10% 10%
Advice Mr. A whether he should invest all his money in one type of bond or he should
buy both the bonds and, if so, in which quantity? Assume that there will not be any call
risk or default risk.
Sol. Since, the liability is one shot,
DL = 2 years.
For immunization, DA = DL

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VALUATION OF BONDS KHETAN EDUCATION
The bonds are coupon paying and hence, we need to compute duration.

For Bond X,
Year CF PV@10% PV Wixi
1 70 0.909 972.63 972.63
+ 1,000
` 972.63
Dx = = 1year.
` 972.63

For Bond Y,
Year CF PV@10% PV Wixi
1 80 0.909 72.72 72.72
2 80 0.826 66.08 132.16
3 80 0.751 60.08 180.24
4 1,080 0.683 737.64 2,950.56
` 936.52 ` 3335.68
` 3,335.68
Dy = = 3.56 years.
` 936.52
In order to immunize, we would invest in a combination of Bond X & Bond Y, such that
the weighted average duration = 2 years.
WxDx + WyDy = 2
Wx + (1-Wx) 3.56 = 2
Wx + 3.56 – 3.56Wx = 2
3.56 – 2 = 3.56Wx - Wx
1.56 = 2.56Wx
Wx = 61%
Wy = 39%
FV of liability at the end of 2nd year = `1,00,000
`1,00,000
PV =
(1.10)2
= ` 82,644.63
Amount to be invested in X = ` 82,644.63 × 61%
= ` 50,413.22
50, 413.22
No. of Bonds =
972.73
= 51.83 » 52
Amount to be invested in Y = ` 82,644.63 – ` 50,413.22

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Prof. Archana Khetan VALUATION OF BONDS

= ` 32,231.41
32, 231.41
No. of Bonds = = 34.42 » 35
936.52
20. The following data are available for three bonds A, B and C. These bonds are used by a
bond portfolio manager to fund an outflow scheduled in 6 years. Current yield is 9%. All
bonds have face value of `100 each and will be redeemed at par. Interest is payable
annually.

Bond Maturity (Years) Coupon rate

A 10 10%
B 8 11%
C 5 9%

i. Calculate the duration of each bond.

ii. The bond portfolio manager has been asked to keep 45% of the portfolio money in
Bond A. Calculate the percentage amount to be invested in bonds B and C that need
to be purchased to immunize the portfolio.
iii. After the portfolio has been formulated, an interest rate change occurs, increasing
the yield to 11%. The new duration of these bonds are: Bond A = 7.15 Years, Bond
B = 6.03 Years and Bond C = 4.27 years.
Is the portfolio still immunized? Why or why not?
iv. Determine the new percentage of B and C bonds that are needed to immunize the
portfolio. Bond A remaining at 45% of the portfolio.
Present values be used follows :
Present Values t1 t2 t3 t4 t5
PVIF0.09,t 0.917 0.842 0.772 0.708 0.650

Present Values t6 t7 t8 t9 t10

PVIF0.09,t 0.596 0.547 0.502 0.460 0.4224
Sol.
i. Calculation of Bond Duration Bond A
Proportion Proportion
Year Cash flow P.V. @ 9% of bond of
value bond value
× time
(years)
1 10 0.917 9.17 0.086 0.086
2 10 0.842 8.42 0.079 0.158
3 10 0.772 7.72 0.073 0.219

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VALUATION OF BONDS KHETAN EDUCATION
4 10 0.708 7.08 0.067 0.268
5 10 0.650 6.50 0.061 0.305
6 10 0.596 5.96 0.056 0.336
7 10 0.547 5.47 0.051 0.357
8 10 0.502 5.02 0.047 0.376
9 10 0.460 4.60 0.043 0.387
10 110 0.4224 46.46 0.437 4.370
106.40 1.000 6.862

Duration of the bond is 6.862 years or 6.86 year Bond B

Year Cash flow P.V. @ 9% Proportion of Proportion of
bond value bond
value × time
(years)
1 11 0.917 10.087 0.091 0.091
2 11 0.842 9.262 0.083 0.166
3 11 0.772 8.492 0.076 0.228
4 11 0.708 7.788 0.070 0.280
5 11 0.650 7.150 0.064 0.320
6 11 0.596 6.556 0.059 0.354
7 11 0.547 6.017 0.054 0.378
8 111 0.502 55.772 0.502 4.016
111.224 1.000 5.833

Duration of the bond B is 5.833 years or 5.84 years Bond C

Year Cash P.V. @ 9% Proportion of proportio
flow bond value n of bond
value ×
time
(years)
1 9 0.917 8.253 0.082 0.082
2 9 0.842 7.578 0.076 0.152
3 9 0.772 6.948 0.069 0.207
4 9 0.708 6.372 0.064 0.256

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Prof. Archana Khetan VALUATION OF BONDS

5 109 0.650 70.850 0.709 3.545

100.00 1.000 4.242
Duration of the bond C is 4.242 years or 4.24 years
ii. Amount of Investment required in Bond B and C
Period required to be immunized 6.000 Year
Less: Period covered from Bond A 3.087 Year
To be immunized from B and C 2.913 Year
Let proportion of investment in Bond B and C is b and c respectively then
b + c = 0.55 (1)
5.883b + 4.242c = 2.913 (2)
On solving these equations, the value of b and c comes 0.3534 or 0.3621 and 0.1966 or
0.1879 respectively and accordingly, the % of investment of B and C is 35.34% or 36.21%
and 19.66 % or 18.79% respectively.

iii. With revised yield the Revised Duration of Bond stands

0.45 × 7.15 + 0.36 × 6.03 + 0.19 × 4.27 = 6.20 year
No portfolio is not immunized as the duration of the portfolio has been increased from 6
years to 6.20 years.

iv. New percentage of B and C bonds that are needed to immunize the portfolio.
Period required to be immunized 6.0000 Year
Less: Period covered from Bond A 3.2175 Year
To be immunized from B and C 2.7825 Year

21. Consider a bond with a face value of ` 100. And coupon = 10% p.a. the bond has a
maturity of 5 years and is trading at par.
Compute:
a. YTM of the bond.
b. Duration and modified duration.
c. New price using PV method if
• interest increases by 1%
• interest decreases by 1%
d. Show that the relationship between interest rate and bond price is inverse & not
linear.
e. Compute Convexity.
f. Compute Convexity effect
g. Compute the new bond price using convexity if
• interest increases by 1%

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VALUATION OF BONDS KHETAN EDUCATION
• interest decreases by 1%
FV = ` 100 C = 10% n = 5 years Price = ` 100 [Since, it is trading at par]
Sol.
a. Since, Price = Face value; Coupon = YTM = 10%
b. Computation of duration.
Year CF PV@10% PV WiXi
1 10 0.909 9.09 9.09
2 10 0.826 8.26 16.52
3 10 0.751 7.51 22.53
4 10 0.683 6.83 27.32
5 110 0.621 68.31 341.55
` 100 ` 417.01
417.01
Duration = = 4.17 years
100
D 4.17
M.D = = = 3.79 years
1 + YTM 1.10
If interest changes by 100 bps or 1%, then price of the bond will change by 3.79% in the
opposite direction.

c. If interest rate interest rate increase by 1%

New YTM = 10 + 1 = 11%
VO = `10 × PVA(11%, 5) + ` 100 × PVIF(11%, 5)
= ` 10 × 3.696 + ` 100 × 0.593
= ` 96.26
If interest rate decreases by 1%
New YTM = 10 % – 1 = 9%
VO = ` 10 × PVA(9%, 5) + ` 100 × PVIF(9%, 5)
= ` 10 × 3.89 + ` 100 × 0.65
= ` 103.9
When interest goes up, P0 = ` 100, New Price = ` 96.26
100 - 96.26
D% = × 100 = 3.79%
100
When interest goes down, P0 = ` 100, New price = ` 103.9
103.9 - 100
%D = × 100 = 3.9%
100
3.74 + 3.90
Average % D = = 3.82 %
2
This means modified duration provides an average of the percentage change in price for
1% D in interest rates on either side. It also means it ignores the convexity effect.

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Prof. Archana Khetan VALUATION OF BONDS

If by MD method, of interest ¯ 1% Present price = 100, New price = 100 + 3.79% =

103.79.
If interest ­ 1%, Po = ` 100 New Price = ` 100 – 3.79% = ` 96.21
P1 + P2 - 2 × Po
e. Convexity =
2 × Po × (ΔYTM)2
P0 = ` 100
P1 = ` 96.26 [Price if interest goes up]
P2 = ` 103.9 [Price if interest goes down]
96.26 + 103.9 - 2 × 100
=
2 × 100 × (0.01)2
=`8
f. Convexity effect
= Convexity × (DYTM)2 × 100 = ` 8 × (0.01)2 × 100 = 0.08%
g. If interest ­ 1%, the price should ¯ = 3.79 - 0.08= 3.71%
New price = 100 - 3.71% = ` 96.29
If interest ¯ 1%, the price should ­ = ` 3.79 + 0.08 = 3.87%
New price = ` 100 + 3.87% = ` 103.87 ! ` 103.9

22. MP Ltd. issued a new series of bonds on January 1, 2000. The bonds we sold at par (`
1,000), having a coupon rate 10% p.a. and mature on 31st December, 2015. Coupon
payments are made semi-annually on June 30th and December 31st each year. Assume
that you purchased an outstanding MP Ltd. Bond on 1st March, 2008 when the going
interest rate was 12%.
Required:
i. What was the YTM of MP Ltd. Bonds as on January 1, 2000 ?
ii. What amount you should pay to complete the transaction? Of that amount how
much should be accrued interest and how much would represent bonds basic value.
Sol.
i. On the date of issue, the price of the bond = face value.
This means, coupon = YTM = 10%
ii. Price on 30th June, 2008 assuming us a ex coupon date
= ` 50 × PVA (6%, 15) + ` 1000 × PVIF (6%, 15)
= ` 50 × 9.712 + ` 1000 × 0.417
= ` 902.6
Price on 30th June with coupon= ` 902.6 + ` 50 = ` 952.6
Price of the bond on 1st March, 2008
952.6
=
4
1 + 0.12 ×
12

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VALUATION OF BONDS KHETAN EDUCATION
= ` 915.96
So on 1st March,
952.6 50
+
4 4
1 + 0.12 × 1 + 0.12 ×
12 12
= ` 867.8 + ` 48.08 = ` 915.96
*Alternate Solution:
Price of the bond on 31st Dec, 2007
= ` 50 × PVA (6%, 16) + ` 1,000 × PVIF (6%, 16)
= ` 50 × 10.106 + ` 1,000 × 0.3936
= ` 898.57
Price on 1st March, 2008
= ` 898.90 + ` 50 × 2/6
= ` 915.57

23. Bank A enter into a Repo for 14 days with Bank B in 10% Government of India Bonds
2028 @ 5.65% for ` 8 crore. Assuming that clean price (the price that does not have
accrued interest) be ` 99.42 and initial Margin be 2% and days of accrued interest be 262
days. You are required to determine.
i. Dirty Price
ii. Repayment at maturity. (consider 360 days in a year)
Sol. i. Dirty Price
= Clean Price + Interest Accrued
10 262
= 99.42 + 100 × × = 106.70
100 360

ii. First Leg (Start Proceed)

Dirty Pr ice 100 - Initial Margin
= Nominal Value × ×
100 100
106.70 100 - 2
= ` 8,00,00,000 × × = ` 8,36,52,800
100 100
No. of days
Second Leg (Repayment at Maturity) = Start Proceed × (1 + Repo rate × )
360
14
= ` 8,36,52,800 × (1 + 0.0565 × ) = ` 8,38,36,604
360

24. Wonderland Limited has excess cash of ` 20 lakhs, which it wants to invest in short term
marketable securities. Expenses relating to investment will be ` 50,000.
The securities invested will have an annual yield of 9%.
The company seeks your advice

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Prof. Archana Khetan VALUATION OF BONDS

i. as to the period of investment so as to earn a pre-tax income of 5%. (discuss)

ii. the minimum period for the company to breakeven its investment expenditure
overtime value of money.
Sol. i. Pre – tax Income required on investment of ` 20,00,000
Let the period of Investment by ‘P’ and return required on investment ` 1,00,000 (` 20,00,000
× 5%)
Accordingly,
9 P
(` 20,00,000 × × ) - ` 50,000 = ` 1,00,000
100 12
P = 10 months
ii. Break – Even its investment expenditure
9 P
(` 20,00,000 × × ) - ` 50,000 = 0
100 12
P = 3.33 months

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