CA Final SFM Solve Questions on Bond Valuation by Prof GDTPE140

Prof Manish Ramuka Topic – Bond Markets Page 1

Bond Analysis & Valuation

Solutions

Category of Problems

1. Bond Price…………………………………………………………………………………………………………...2 2. YTM Calculation……………………………………………………………………………………………… 14 3. Duration & Convexity of Bond ………………………………………………………………………… 30 4. Immunization…………………………………………………………………………………………………… 58 5. Forward Rates & Spot Rates Calculation…………………………………………………………... 66 6. Clean Price & Dirty Price…………………………………………………………………………………… 84 7. Bond Refunding Decision………………………………………………………………………………… 88 8. Convertible Bond…………………………………………………………………………….……………… 92 9. Mixed Problems……………………………………………………………………………………………… 102


Category #1: Bond Price Problem #1 Consider three bonds with 8 percent coupon rates, all selling at face value. The short-term bond has a maturity of 4 years, the intermediate-term bond has maturity 8 years and the long-term bond has maturity 30 years. What will happen to the price of each bond if their yields increase to 9 percent? What will happen to the price of each bond if their yields decrease to 7 percent? What do you conclude about the relationship between time to maturity and the sensitivity of bond prices to interest rates? Solution Coupon rate = 8% Bond Maturity 1 – 4 Yrs 2 – 8 Yrs 3 – 30 Yrs If YTM increase to 9% Price of bond will decrease. Bond 1 Price = C X PVIFA (K%, n) + FV X PVIF (K%, n) Price = 80 * PVIFA (4, 9%) + 1000 * PVIF (4, 9%) = (80 x 3.240) + (1000 x 0.708)

= 967.2// Similarly we can calculate other bond prices Bond1 Bond2 Bond3 Yield7% 1033 1059 1124 Yield8% 1000 1000 1000 Yield9% 967 944 897


Problem #2 A bond has a face value of Rs1,000 with maturity of 5 year and a coupon rate of 7% per annum. If interest rates go down from 9% to 7% what will the capital gains from the bond be? Solution Since interest rates are expected to go down from 9% to 7% price will increase as per Meikles theorem Find price of Bond when yield is 9% Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 70 ∗ PVIFA 9%, 5 + 1000 ∗ PVIF 9%, 5 = 70 𝑋 3.890 + 1000 𝑋 0.650 = 922.3// Find price of Bond when yield is 7% Since YTM is same as coupon price =1000

Capital gain = 1000 − 922.3

922.3

= 𝟖. 𝟒𝟐%//


Problem #3 Bonds A and B have Rs1000 face values, 8% YTM and 10 year terms to maturity. Bond a pays coupon of 10% and Bond B trades at par, both making annual coupon payments. If the yields decline to 6% what is the percentage price change in both bonds? Solution Step I: Find Price of 2 bonds today Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond A = 100 ∗ PVIFA 8%, 10 + 1000 ∗ PVIF 8%, 10 = 1134.20 Bond B = 1000 Since Yield is same as coupon Step II: Find price of 2 bonds when rates changes to 6% Bond A = 1294 Bond B = 1147 Step III : % Change in Price

Bond A = 1294 − 1134.2

1134.2= 𝟏𝟒. 𝟒%

Bond B = 𝟏𝟒. 𝟕%


Problem #4 Shyam owns an Rs1000 face value bond with three years maturity. Bond makes an annual coupon of 7.5%. The first coupon is due one year from now. Bond is selling today at Rs975.48. If the YTM is 10%, should shyam sell the bond or hold it? Solution Step I Find intrinsic value (Price) of bond Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) IV = 75 ∗ PVIFA 10%, 3 + 1000 8 PVIF 10%, 3 = 937.8 Step 2 Compare it with actual market value Actual Market price is 975.48 Since Market Price > Intrinsic Value Shyam should sell the bond Problem #5 Consider a two-year Rs. 1000 face value 10% coupon rate bond which pays coupon semi-annually. Find out the intrinsic value of the bond if the required rate of return is 14% p.a. Compounded semi-annually. Should the bond be purchased at the current market price of Rs. 965? Solution

Bond Price = Coupon ∗ PVIFA ((k

2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Bond Price = 50 * PVIFA (7%, 4) + 1000* PVIF (7%,4) Bond Price = 932.25 Since intrinsic Value (932.25)< Market Price (965) implies bond is trading at premium. Hence bond should not be purchased at the current market price.


Problem #6

Solution

Current yield = 10

110

= 9.09% If yield goes up by 1% New Yield = 10.09%

Price = 10

10.09%

= 99.1080 //


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

= 3.75*PVIFA (3%, 4) + 10000*PVIF (3%, 4)

= 375*3.7171 + 10000*0.88848

= 10278.78 //


Problem #7

Solution YTM = 16% Redemption Price = 5% premium Price of a bond = PV of future cash flows

=9

1.16+

9

(1.16)2 + 9

(1.16)3 + 9

(1.16)4 + 10

(1.16)5 + 10

(1.16)6 + 10

(1.16)7 +

10

(1.16)8 + 14

(1.16)9 + 14

(1.16)10 + 105

(1.16)10

Year Cash Flow PV Factor @ 16% Present Value

1 9 0.8621 7.76 2 9 0.7432 6.69 3 9 0.6407 5.77

4 9 0.5523 4.97

5 10 0.4761 4.76

6 10 0.4104 4.10 7 10 0.3538 3.54

8 10 0.3050 3.05 9 14 0.2630 3.68 10 14 0.2267 3.17

10 105 0.2267 23.80 Total 71.29


Problem #8 An Investor is considering the purchase of the following Bond. Find the Price. Face Value 1000 Coupon Rate 8% Maturity 3 years Expected Return 15% Solution Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 80 * PVIFA (15%, 3) + 1000* PVIF (15%,3) Bond Price = 80 * 2.283 + 1000 * 0.658 Bond Price = 840.64// Problem #9 A bond with 7.5% coupon interest – payable half yearly, Face Value 10,000 & Term to maturity of 2 years in traded in the market. Find the Market Price of the Bond if the YTM is 10%. (Nov 2010) Solution


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Bond Price = 375 ∗ PVIFA ((10

2)%, 4) + 10000 ∗ PVIF ((

10

2)%, 4)

Bond Price = 375 ∗ 3.546 + 10000 ∗ 0.823 Bond Price = 9559.75


Problem #10 Calculate the price and analyze the results: Name Coupon Term-Years YTM Price Bond A 10% 5 10% Bond B 10% 5 12% Bond C 10% 5 8% Bond D 10% 10 10% Bond E 10% 10 12% Bond F 5% 5 10% Bond G 5% 5 12% Bond H 10% 15 10% Bond I 10% 15 12% Solution Bond Price is calculated using following formula Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Name Coupon Term-Years YTM Price Bond A 10% 5 10% 1000 Bond B 10% 5 12% 927.5 Bond C 10% 5 8% 1080.3 Bond D 10% 10 10% 1000 Bond E 10% 10 12% 887 Bond F 5% 5 10% 810.55 Bond G 5% 5 12% 747.25 Bond H 10% 15 10% 1000 Bond I 10% 15 12% 864.1 State the results for YTM, Coupon Rate and Maturity


Problem #11 A Rs. 1,000 face value ABC bond has a coupon rate of 6%, with interest paid semi-annually, and matures in 5 years. If the bond is priced to yield 8%, what is the bond’s value today? Answer – Price = Rs. 918.89

Solution


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Bond Price = 30 * PVIFA (4%, 10) + 1000* PVIF (4%,10) Bond Price = 918.89 Problem #12

The KLM bond has a 8% coupon rate, with interest paid semi-annually, a maturity value of

Rs. 1,000 and matures in 5 years. If the bond is priced to yield 6%, what is the bond’s

current price?

Answer – Price = Rs. 1085.2 Solution


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Bond Price = 40 * PVIFA (3%, 10) + 1000* PVIF (3%,10) Bond Price = 1085.2

Problem #13

Consider the following information related to a bond:

Par Value Rs. 1000 Time to Maturity 15 Years Coupon rate (interest payable annually) 8% Current Market Price Rs. 847.88 Yield to Maturity (YTM) 10% Other things remaining the same, if the bond starts paying interest semi-annually, find the change in the market price of the bond. Answer – New Price of Bond = Rs. 846.27 Solution


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Bond Price = 40 * PVIFA (5%, 30) + 1000* PVIF (5%,30) Bond Price = 846.27


Problem #14 ABC Ltd. Has the following outstanding Bonds.

Bond Coupon Maturity Series X 8% 10 Years Series Y Variable changes annually 10 Years

Comparable to 10 years prevailing rate

Initially these bonds were issued at face value of Rs. 10,000 with yield to maturity of 8%. Assuming that: i. After 2 years from the date of issue, interest on comparable bonds is 10%, then what should be the price of each bond? ii. If after two additional years, the interest rate on comparable bond is 7%, then what should be the price of each bond? iii. What conclusions you can draw from the prices of Bonds, computed above. Solution Price of a floating rate bond remains same on every coupon reset date. a)Price after 2 Yrs Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 80 ∗ PVIFA 10%, 8 + 1000 ∗ PVIF 10%, 8 = 893.3 b)Price after 4 Yrs Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 80 ∗ PVIFA 7%, 6 + 1000 ∗ PVIF 7%, 6 = 1047.6


Problem #15 A 7% Bond issued several years ago when the market interest rate was also 7%. Now the bond has a remaining life of 3 years when it would be redeemed at par value of Rs. 1,000. The market rate of interest has increased to 8%. Find out the current market price, price after 1 year and price after 2 years from today. Solution a) Bond Price Today (Remaining Life 3 Yrs) Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 70 * PVIFA (8%, 3) + 1000* PVIF (8%,3) Bond Price = 974.22 b) Bond Price after 1 yr (Remaining Life 2 Yrs) Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 70 * PVIFA (8%, 2) + 1000* PVIF (8%,2) Bond Price = 982.16 c) Bond Price after 2 yrs (Remaining Life 1 Yr) Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 70 * PVIFA (8%, 1) + 1000* PVIF (8%,1) Bond Price = 990.7 Problem #16 A Deep Discount Bond (DDB) was issued by a financial institution for a maturity period of 10 years and having a par value of Rs. 25,000. Find out the value of the Bond given that the required rate of return is 16%. Solution Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Since the bond is a zero coupon bond coupon rate will be zero. Bond Price = 25000* PVIF (16%,10) Bond Price = 5667.1


Problem #17 (a) A Rs. 100 perpetual bond is currently selling for Rs. 95. The coupon rate of interest is 14.5 percent and the appropriate discount rate is 16 percent. Calculate the value of the bond. Should it be bought? What is its yield at maturity? (b) A Company proposes to sell ten-year debentures of Rs. 10,000 each. The company would repay Rs. 1,000 at the end of every year and will pay interest annually at 15 percent on the outstanding amount. Determine the present value of the debenture issue if the capitalization rate is 18 percent. Solution a)

Intrinsic Value of Perpetual Bond =Coupon

YTM

Intrinsic Value of Perpetual Bond =14.5

0.16= 90.625

Since market value>intrinsic value we can conclude that the bond is currently overpriced. Hence the bond should not be purchased.

YTM =14.5

95= 15.26%

b) Year Beginning

Principal Principle Payment

Interest Ending Principle

Total CF

PV Factor @ 18%

Present Value

1 10000 1000 1500 9000 2500 0.8475 2119

2 9000 1000 1350 8000 2350 0.7182 1688

3 8000 1000 1200 7000 2200 0.6086 1339

4 7000 1000 1050 6000 2050 0.5158 1057

5 6000 1000 900 5000 1900 0.4371 831

6 5000 1000 750 4000 1750 0.3704 648

7 4000 1000 600 3000 1600 0.3139 502

8 3000 1000 450 2000 1450 0.2660 386

9 2000 1000 300 1000 1300 0.2255 293

10 1000 1000 150 0 1150 0.1911 220

Total 9082


Category #2: YTM Problem #18 ABC Ltd. Recently issued 15-year bonds. The bonds have a coupon rate of 7.5 percent and pays interest semiannually. The bonds are callable in 5 years at a call price equal to 13 percent premium to par value. The par value of the bonds is Rs1,000. If the yield to maturity is 6 percent, what is the price of the bond today and what is yield to call? Solution Coupon Rate = 7.5% Maturity = 15 Yrs Semiannual coupon payment Bond is callable in 5 Yrs YTM = 6% Price = C X PVIFA (K%, n) + FV X PVIF (K%, n) Price = 37.5 * PVIFA (30, 3%) + 1000 * PVIF (3%, 30) = 37.5 x 19.6 + 1000 x 0.412 = 1147.02 YTC is calculated as follows 1147.02 = 37.5 * PVIFA (10, x %) + 1130 * PVIF (x%, 10) First let’s find if equation matches at YTM of 6% Price = 37.5 * PVIFA (3%, 10) + PVIF 1130 (3%, 10) = 37.5 * 8.530 + 1130 * 0.744 = 1160.59 Here since the price is greater we will solve it using higher rate YTM of 7% to get lower price Calculating price @ YTM of 7% Price = 37.5 * PVIFA (3.5%, 10) + PVIF 1130 (3.5%, 10) = 37.5 * 8.3160 + 1130 * 0.7089 = 1112.9 Now we can use interpolation to get exact answer

YTM = Lower % + PV@Lower% − Actual PV desired

(PV@Lower% − PV@Higher%)∗ (Difference in Yield)

YTC = 6% + 1160 .59−1147.02

1160 .59−1112.9 x (7% - 6%)

= 6.28%


Problem #19 It is now January 1,2010, and Mr. X is considering the purchase of an outstanding Municipal Corporation bond that was issued on January 1,2007, the Municipal bond has a 9.5% annual coupon and a 30-year original maturity (it matures on December 31, 2037). Interest rates have declined since the bond was issued, and the bond now is selling at 116.575% of par, or Rs. 1,165.75. Determine the yield to maturity (YTM) of this bond for Mr. X. Solution Coupon Rate = 9.5% Maturity = 27Yrs Price = C X PVIFA (K%, n) + FV X PVIF (K%, n) First let’s find if equation matches at YTM of 8% Price = 95 * PVIFA (8%, 27) + 1000*PVIF (8%, 27) = 1164 Here since the price is less we will solve it using lower rate YTM of 7.5% to get higher price Calculating price @ YTM of 7.5% Price = 95 * PVIFA (7.5%, 27) + 1000* PVIF (7.5%, 27) = 1228.8 Now we can use interpolation to get exact answer



YTC = 7.5% + 1228 .8.59−1165.75

1228 .8−1164 x (8% - 7.5%)

= 7.98%


Problem #20 There is a 9%, 5 year bond issue in the market. The issue price is Rs90 and the redemption price is Rs105. For an investor with marginal income tax rate of 30% and capital gains tax of 10% (assuming no indexation), what is the post tax yield to maturity? Solution Price of bond can be calculated as follows Price = C X PVIFA (K%, n) + FV X PVIF (K%, n) 90 = [9 X (1-30%)] * PVIFA (K%, 5) + [105 – 10% (15)] * PVIF (K%, 5) 90 = 6.3 PVIFA (K%, 5) + 103.5 PVIF (K%, 5) @ YTM of 10% Price = 88.1472 @ YTM of 9% Price = 91.7727 Using Interpolation



YTM = 9% + 91.77 − 90

91.77 − 88.14 x 9% - 8%

= 9.4876%


Problem #21

Solution Maturity = 6 Yrs Price = 95 Coupon = 13% We will use interpolation Calculate price of bond @ YTM of 14% Price = 13*PVIFA (14%,6) + 100*PVIF (14%, 6) = 96.11 Calculate price of bond @ YTM of 15% Price = 92.43% Using interpolation

YTM = Low % + PV @ Lower − Actual Desired

PV @ Lower − PV @ Higher ∗ (High % − Low %)

YTM = 14% + 96.11−95

96.11−92.43 * 1%

= 14.30%//


Problem #22

Solution Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 11*PVIFA (13%, 3) + 100*PVIF (13%, 3)

= 95.27

Calculate price of bond @ YTM of 11% Since coupon rate = YTM Bond Price = 100 Calculate price of bond @ YTM of 13% Bond Price = 95.27 from above Using interpolation



YTM = 11% + 100 − 97.6

100 − 95.27 ∗ (13% − 11%)

= 12%


Problem #23

Solution a) 364 Day T-bill rate = 9% Hence rate for AA rated bond = 9% + 3% + 2% = 14% Price = 150 * PVIFA (14%, 5) + 1000 * PVIF (14%, 5) = 150 * 3.433 + 1000 * 0.519 = 1034.3// Since intrinsic value of 1034.3 > is greater than market price of 1025.86 he should consider investing in bonds.

b) Current yield = 150

1025 .86

= 14.62%//

c) YTM calculation Calculation Price @ 14% = 1034.3 Price @ 15% = 1000 Using interpolation



YTM = 14% + [ 1034.3−1025.86]

[1034.3−1000] * (5-14%)

= 14.23%. //


Problem #24 Arvind Ltd recently issued 15 year bonds. The bonds have a coupon rate of 7.5 percent and pays interest semi-annually. The bonds are callable in 5 years at a call price equal to 13% premium to par value. If the par value of the bonds is Rs1,000, if the yield to maturity is 6 percent, what is yield to call ? Solution Step 1: To calculate current price of bond


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

= 37.5 * PVIFA (3%, 30) + 1000 * PVIF (3%, 30) = 37.5 * 19.6 + 1000 * 0.412 = 1147 Step 2: Calculate YTC Calculating bond price @ YTM of 8%


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

= 37.5 * PVIFA (4%, 10) + 1130 * PVIF (4%, 10) = 37.5 * 8.111 + 1000 * 0.676 = 1067.96 Similarly calculating bond price @ YTM of 4% Bond Price = 1263.46

YTC = Low % + PV @ Lower − Actual Desired


YTM = 4% + 1263.46 − 1147

1263.46 − 1067.96 ∗ (8% − 4%)

YTM = 6.38%//


Problem #25 A bond is issued at 10% discount to its face value of Rs1lakh. Redemption takes place at the end of 20 years. If the coupon is 12% and bonds are redeemed at Rs110000, what is the YTM as per approximate method? Solution Yield to maturity can be calculated using approximate formula as follows

YTM = C +

(F − P)n

(F + P)n

= 12000 +

110000 − 9000020

110000 + 900002

= 𝟏𝟑. 𝟎𝟎%//


Problem #26 Arvind recently purchased a bond with Rs1000 face value, coupon 10% and four years to maturity. The bond makes annual interest payments and the first one is due one year from now. Arvind paid Rs1032.40 for the bond. What is bond’s YTM? If the bond can be called in two years at Rs1100, what is its yield to call? Solution

YTM Approximate = C +

F − Pn

F + P2

YTM = 10 +

100 − 103.244

203.242

= 𝟗. 𝟎𝟒%

For yield to call we do it using the interpolation logic

Bond Price @ 14% = 100 ∗ PVIFA 14%, 2 + 1100 ∗ PVIF 14%, 2

= 1011

Bond Price @ 12% = 100 ∗ PVIFA 12%, 2 + 1100 ∗ PVIF 12%, 2

= 1045.9

Using interpolation we calculate YTC



YTC = 12% + 13.5

13.5 + 21.4∗ 2%

𝐘𝐓𝐂 = 𝟏𝟐. 𝟕𝟕𝟓% //


Problem #27 Shyam recently purchased at par bond with Rs1000 face value, coupon 9% and four years to maturity. Assuming annual interest payment, calculate shyam’s actual YTM if all interest payments are reinvested at 15% per annum. What is Shyam’s actual YTM if all interest payments are immediately spent on receipt ? Solution Bonds Present Value = 1000 Coupon payments = 90 Reinvestment Income = (90 ∗ 1.153) + (90 ∗ 1.152) + (90 ∗ 1.151) + 90 − 360 = 449.4 − 360 = 89.4 YTM is calculated as follows 1000 = PVIF X, 4 ∗ 1000 + 449.4 YTM = 9.72% When all dividends are spent that means no reinvestment income is received. 1000 = PVIF X, 4 ∗ 1000 + 360 = 𝟖%//


Problem #28 Mr. Praveen is working as a Senior Manager in a Public Sector Undertaking. His gross total income is Rs. 5, 00,000 p.a. He would like to avail the benefit of tax rebate (@15%) under section 88 of the Income Tax Act, by investing Rs. 2, 00,000 in the Tax Saving Bonds issued by the ICICI Bank. Options available of Mr. Praveen in respect of Tax Saving Bonds are given below:

Option Issue Price Rs.

Face Value Rs.

Tenure Interest (%) (p.a.)

Interest Payable

I 10,000 10,000 4 Years 5.65 Annually II 10,000 10,000 6 Years 7.00 Annually III 10,000 14,750 4 Years 9

months DDB* DDB*

IV 10,000 17,800 6 Years 9 months

DDB* DDB*

Deep Discount Bond The marginal tax rate applicable to Mr. Praveen is 30% You are required to:

(a) Determine the post-tax YTM for the four options available to Mr. Praveen

Assume that the interest income is tax exempt.

(b) Suggested an option, if

i) The yield curve is upward sloping

ii) The yield curve is downward slopping

iii) The yield curve is flat

Answer – Price YTM = 10.16%, 10.27%, 12.3%, 11.57% Solution a) Calculating Post Tax YTM for 4 bonds Bond 1 Coupon Received (C) = 5.65% * 10,000 = 565 Current Price (P) = 10,000 – (15% * 10,000) = 8,500 Redemption Amount (F) = 10,000 No of Years (n) = 4yrs


F − Pn

F + P2

YTM Approximate = 565 +

10,000 − 8,5004

10,000 + 8,5002

Post Tax YTM = 10.16%


Bond 2 Coupon Received (C) = 7% * 10,000 = 700 Current Price (P) = 10,000 – (15% * 10,000) = 8,500 Redemption Amount (F) = 10,000 No of Years (n) = 6yrs


F − Pn

F + P2


10,000 − 8,5006

10,000 + 8,5002

Post Tax YTM = 10.27% Bond 3 FV = PV ∗ (1 + Periodic Rate)n∗y

14,750 = 8,500 ∗ (1 + YTM)(4+9

12)

YTM = 12.31% Bond 4 FV = PV ∗ (1 + Periodic Rate)n∗y

17,800 = 8,500 ∗ (1 + YTM)(6+9

12)

YTM = 11.57% b) Yield Curve is upward sloping This implies that interest rates are expected to rise. This will imply that bond prices should fall. Hence we should buy the bonds with lowest maturity i.e Bond 1 Yield Curve is Downward sloping This implies that interest rates are expected to fall. This will imply that bond prices should rise. Hence we should buy the bonds with highest maturity i.e Bond 4 Yield Curve is flat This implies that interest rates are not expected to change. Hence the choice of bond should not depend on maturity. We should simply buy the bond with highest YTM i.e Bond 3


Problem #29 A Rs. 1,000 face value EFG bond has a coupon of 10% (paid semi-annually) matures in 4 years, and has current price of Rs. 1140 what is the EFG bond’s yield to maturity? Answer – BEY = 6.08% Compounded Semi annually Solution Calculating bond price @ YTM of 8%


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

= 50 * PVIFA (4%, 8) + 1000* PVIF (4%, 8) = 1067.32 Similarly calculating bond price @ YTM of 6% Bond Price = 1140.39 YTM = 6% Problem #30 A NOP bond has an 8% coupon rate (semi-annual interest), a maturity value of Rs. 1,000, matures in 5 years, and a current price of Rs. 1,200. What is the NOP’s yield-to-maturity? Answer – 3.6155% Solution Calculating bond price @ YTM of 5%


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

= 40 * PVIFA (2.5%, 10) + 1000* PVIF (2.5%, 10) = 1131.28 Similarly calculating bond price @ YTM of 3% Bond Price = 1230.55 Using interpolation



YTM = 3% + 1230.55 − 1200

1230.55 − 1131.28 ∗ (5% − 3%)

= 3.6155%


Problem #31 Consider a Rs. 1000 face value, 5 years bond presently trading at Rs. 972. The bond has coupon rates of 14% payable semiannually. Compute its current yield? Answer – Current Yield = 7.20% Solution

Current Yield = Annual Coupon

Current Price =

140

972= 14.4%


Problem #32 (a) Consider a 1 year Rs. 1000 face value, 12% coupon bond which pays coupon annually. The bond was issued 5 Years and was trading at Rs. 960. The bond is redeemable at a premium of 10% on maturity. If income tax rate is 30% and capital gains tax is 10%, find out post tax YTM. If the post tax required rate of return is 12.5%, give your investment advice. (b) Suppose in the previous sum there are no taxation issues. Moreover the bond is to be redeemed at a premium of 10% in 2 equal annual installments at the end of 9th year and 10th year, find out the YTM of the bond. Answer – YTM (a) 10.67%, (b) 15% Solution a) Coupon Payment After tax = Coupon * (10Tax Rate)

= 12% * 1000 (1-30%) = 84 Face Value After Tax = Face Value – Capital Gain Tax =1100 – 10%(1100-960) =1086


F − Pn

F + P2

YTM = 84 +

1086 − 9605

1086 + 9602

= 𝟏𝟎. 𝟔𝟕%

b) Year Coupon Cash Flow Principal Total Cash Flow

1 120 120 2 120 120 3 120 120 4 120 550 670 5 60 550 610

YTM is calculated as follows Outflow = PV of Future Cash Inflows Solve Using interpolation such that it satisfies the following equation 960 = 120*PVIFA(YTM,3) + 670*PVIF(YTM,4)+ 610*PVIF(YTM,5) YTM=15%


Problem #33 IDBI, in its issue of Flexi bonds – 3, offered Growing Interest Bond. The interest will be paid to the investors every year at the rates given below and the minimum deposits is Rs. 5000/-,

Year 1 2 3 4 5 Interest

(p.a.) 10.5% 11.0% 12.5% 15.25% 18.0%

Calculate the yield to maturity (YTM) Answer – YTM = 13% Solution YTM should be calculated in such a way that it satisfies the following equation Outflow = PV of Future Cash Inflows

5000 =10.5% ∗ 5000

1 + YTM 1+

11% ∗ 5000

1 + YTM 2+

12.5% ∗ 5000

1 + YTM 3+

15.25% ∗ 5000

1 + YTM 4+

18% ∗ 5000

1 + YTM 5

By trial and error and using interpolation we get YTM=13%


Category #3: Duration & Convexity of Bond Problem #34 Calculate duration of a six year bond whose face value is Rs1000 and which pays a coupon of 8%. Assume the yield to be 8%. Solution Duration of bond =?

Year(1) CF(2) PV Factor(3) 4=1 x 2 x 3 1 80 0.9259 74.07 2 80 0.8573 137.168 3 80 0.7938 190.51 4 80 0.7350 235.2 5 80 0.6806 272.25 6 1080 0.6302 4083.6 Total 4992.88

Duration = Total of Column 4

Price

= 4992.88

1000

= 4.99288

= 5 Yrs //


Problem #35 Calculate duration of a semi annual coupon bond with an 8% coupon on 1000 face value bond with 2 years to maturity and an YTM of 10% Solution YTM = 10%

1 2 3 4 - 3x2 5 Year CF PV Factor PV 4x1 0.5 40 0.9524 38.0960 19.0480 1 40 0.9070 36.2800 36.28

1.5 40 0.8638 34.5520 51.828 2 1040 0.8227 855.6080 1771.22 = 964.57 = 1818.37

Duration =1818.37

964.57

Duration = 1.8852//


Problem #36 An inflow of Rs25lakhs is to be invested in the following bond portfolio in the percentages specified. Bond % of money invested Macaulay Duration of bond 1 10 10.6 2 27 6.9 3 7 12.5 4 50 2.0 5 6 8.3 The face value of all bonds is Rs1000 and the YTM is 9%. Calculate the duration of the portfolio. What would be the percentage change in price of bond 1 if the interest rates fall to 7%? Also ascertain the percentage change in Portfolio value Solution Macaulay duration of bond portfolio

= 10% x 10.6 + 27% x 6.9 + 7% x 12.5 + 50% x 2+ 6% x 8.3 = 5.3

Modified duration of portfolio = 5.3

(1+9%)

= 4.8624

% Change in bond 1 price =10.6

(1 + 9%)∗ 2

= 19.45% % change in price of portfolio = 4.8624 x 2 = 9.7248%


Problem #37 The following data are available for a bond:

a. Face value 1,000 b. Coupon Rate 16% c. Years to maturity 6 d. Redemption value 1,000 e. Yield to maturity 17%

What are the current market price, duration and volatility of this bond? Calculate the expected market price, if we witness an increase in required yield by 75 basis points. Solution Price = 160 x PVIFA (17%, 6) + 1000 X PVIF (17%, 6) = 3.589 X 160 + 1000 X 0.390 = 964.24// Duration Calculation

1 2 3 4 Yrs CF PV 1x2x3 1 160 0.8547 136.75 2 160 0.7305 233.76 3 160 0.6244 299.71 4 160 0.5337 341.56 5 160 0.4561 364.8 6 1160 0.3898 2713.6 = 4089.78

Duration = 4089.78

964.24

= 4.24 //

Modified Duration = 4.24

1+17%

= 3.6261 // % Change is bond price

= 0.75 x 3.6261 = 2.7196%

Bond Price will decrease by 2.7196% New Price = 964 X [1 – 2.7196%] = 937.78 //


Problem #38 The modified duration for a 12 year 6% annual coupon bond yielding 7% is calculated to be 8.245.

a) If the yield falls to 6.8%, what is the percentage price change for this bond using the modified duration value?

b) What is the actual percentage price change for this bond? c) If the yield falls to 6.0%, what is the percentage price change for this bond using the

modified duration value? d) What is the actual percentage price change for this bond?

Solution a) % change in price of bond = 0.2% X 8.245

= 1.6490%// b) Actual % change in price New Price = 60 X PVIFA (6.8%, 12) + 1000 X PVIF (6.8%, 12) = 935.77 Original Price = 60 X PVIFA (7%, 12) + 1000 X PVIF (7%, 12) = 920.57

% Change = 935.77−920.57

920.57 X 100

= 1.65%// C) Similar calculation can be performed. D) Similar calculation can be performed.


Problem #39 Calculate Convexity given the following with respect to a coupon bond. Coupon rate = 6%, Term = 5 years, Yield to maturity = 7% (3.5% semi-annual) and Price = 958.42. Solution

1 2 3 4 5 Yrs Yrs (Yrs + 1) CF PVF 2x3x4 0.5 0.75 30 0.9662 21.73 1.0 2.00 30 0.9335 56.00 1.5 3.75 30 0.9019 101.46 2.0 6.00 30 0.8714 156.85 2.5 8.075 30 0.8420 221.00 3.0 12.00 30 0.8135 292.86 3.5 15.75 30 0.7860 371.38 4.0 20.00 30 0.7594 455.64 4.5 24.75 30 0.7337 544.77 5.0 30.00 1030 0.7089 21905

= 24126

Convexity = 24126

958.42 ∗ (1 + 7%)2

= 21.98//


Problem #40

Solution

a) Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 160 * PVIFA (17%, 6) + 1000 * PVIF (17%, 6) = 964.11 //

b) Duration Yrs CF PV Factor 1x2x3 1 160 0.8547 136.75 2 160 0.7305 233.76 3 160 0.6244 299.71 4 160 0.5337 341.56 5 160 0.4561 364.88 6 1160 0.3898 2713.1 ∑ 4089.5

=4089.58

964.1082

=4.247 Years//

c) Volatility = Macaulay Duration

(1+K)

= 4.247

1.17

= 3.63%//

d) Expected Market Price % change = -3.63% * 0.75 = - 2.7224% New Price = 964.24 (1 – 2.7224%) = 937.98//


Problem #41 Arvind wants to invest in a bond that matures after 6 years from now. The face value of the bond is Rs1000 and carries a coupon rate of 10.75%. If the bond is currently trading at Rs950, Calculate Modified duration of bond Price change if interest rate increases by 0.5% Solution In order to find modified duration we need YTM and Macaulay duration Step1: Calculate YTM using interpolation Since price is lower than face value we select YTM to be higher than coupon rate. Calculating bond price @ YTM of 12% Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 948.6074 Since above price is less than actual price we select next rate to be lower than 12% Calculating bond price @ YTM of 11.5% Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 968.7228



YTM = 12% + 968.73 − 950

968.73 − 948.61 ∗ (12% − 11.5%)

YTM = 11.97%


Step2: Calculate Macaulay duration

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5

Yrs CF PV Factor @ 11.97%

PV (3x2) 4 x 1

1 107.5 0.8931 96.00 96 2 107.5 0.7976 85.70 171.4

3 107.5 0.7123 76.56 229.68 4 107.5 0.6361 68.39 273.56 5 107.5 0.5682 61.08 305.4

6 1107.5 0.5074 561.9 3371.4 ∑ 950 ∑ 4447.4

Macaulay Duration = 4447.4

950

= 4.6815//

Modified Duration = Macaulay Duration

(1 + k)

= 4.6815

(1 + 11.97%)

= 4.1818// % Change in Bond Price = - [Modified Duration] *[% Change in Yield] = -4.1818 * 0.5 = -2.1%//



Problem #42 The duration for a bond paying semi-annual coupon is 6.72 years for a maturity of 10 years. If the YTM of bond is 12.5% with a coupon rate of 11% and the face value is Rs100, what is the modified duration of the bond? Solution Macualay Duration = 6.72

Modified Duration = Macualay Duration

(1 +k2)

= 6.72

(1 +12.5%

2 )

= 6.72

1 + 6.25%

= 6.72

1.0625

= 6.3247

Problem #43 Four bonds are held by Ram (Durations and Proportion given below) Bond Duration Proportion A 4.50 years 0.20 B 3.00 years 0.25 C 3.50 years 0.25 D 2.80 years 0.30 What is the duration of Ram’s Bond Portfolio? Solution

Portfolio Duration = WiDi

Portfolio Duration = 4.5 ∗ 0.2 + 3 ∗ 0.25 + 3.5 ∗ 0.25 + 2.8 ∗ 0.3

Portfolio Duration = 3.37


Problem #44 Without calculating rank the following in the descending order of duration. Bond Maturity Coupon % YTM A 30 years 10 10 B 30 years 0 10 C 30 years 10 7 D 5 years 10 10 Solution Face value of a bond forms significant portion of cash flows from bond. Therefore longer maturity bonds will return the cash flows later than a shorter maturity bond. Hence bonds A, B, C will have higher duration than bond D. Bond D would be ranked last. Zero coupon bonds do not give intermediate cash flows and the only cash flow from zero coupon bond is its face value. This implies duration of a zero coupon bond is always higher than the duration of coupon paying bond. Since bond B is a zero coupon bond with same years to maturity as that of A & C, bond B will have higher duration as compared to A & C and hence would be ranked first. With all parameters same in 2 bonds, bond with higher yield to maturity will have lower duration than a corresponding bond with a lower yield to maturity. This is because when YTM is more, the reinvestment income is more and hence the cash flows from the bond is received earlier since the coupons are reinvested at higher rates from the beginning. Hence bond C with YTM equal to 7% is ranked second as compared to bond A which is ranked third. B, C, A, D


Problem #45 Rank the following bonds in the descending order of duration: (Calculate not allowed)

Bond Coupon Rate YTM Maturity A 10% 14% 10 years B 12% 14% 10 years C 0% 14% 10 years D 12% 16% 10 years

Solution Since all the bonds have same maturity we will draw our conclusions from the relationship between coupon rate and YTM Bond C is a zero coupon bond and hence will have maximum duration. Bond B & D have same coupon rate however have different YTM. Bond having higher YTM will have lower duration. Hence Bond B has higher duration in comparison to bond D. B>D Bond A & B have same YTM, however they have different coupon rate. Bond having higher coupon rate has lower duration. Hence Bond B has lower duration in comparison to A Hence we have following relationship for duration of the bonds mentioned above. C>A>B>D


Problem #46 Find the duration of a five year bond with Coupon = 10% and YTM = 10%, With FV = 1000 and coupons payable annually. Solution Duration of a bond is calculated using following formula

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5


PV (3x2) 4 x 1

1 100 0.909 90.9 90.9 2 100 0.826 82.6 165.2

3 100 0.751 75.1 225.3 4 100 0.683 68.3 273.2 5 1100 0.621 683.1 3475.5

∑ 1000 ∑ 4170.1 Duration = 4.17yrs


Problem #47 Find the duration of a five year bond with Coupon = 10% and YTM = 10% With FV = 1000 and coupons payable semi-annually. Is the answer different from the duration of the same bond with annual coupons? Why? Solution Duration of a semiannual bond is calculated using following formula

Macaulay Duration =

∑t ∗ c

(1 + k)t +2nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5


PV (3x2) 4 x 1

1 50 0.952 47.61905 47.61

2 50 0.907 45.35147 90.70

3 50 0.864 43.19188 129.57 4 50 0.823 41.13512 164.54 5 50 0.784 39.17631

195.88

6 50 0.746 37.31077

223.86

7 50 0.711 35.53407

248.73

8 50 0.677 33.84197

270.73

9 50 0.645 32.23045

290.07

10 1050 0.614 644.6089 6446.08

∑ 1000 ∑ 8107

Macaulay Duration = 8107

1000 ∗ 2

= 4.05


Problem #48 Consider a bond selling at its par value of Rs1000 with 6yrs to maturity and 7% annual coupon rate. What is bonds duration? If the YTM of this bond increases to 10%, how it affects the bonds duration? And Why? Why should the duration of a coupon carrying bond always be less than the time to its maturity? Solution Duration of a bond is calculated using following formula

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5

Yrs CF PV Factor @ 7% PV (3*2) 4*1

1 70 0.935 65.42 65.42

2 70 0.873 61.14 122.28

3 70 0.816 57.14 171.42

4 70 0.763 53.40 213.61

5 70 0.713 49.91 249.55

6 1070 0.666 712.99 4277.92

Grand Total 1000 5100.20


1000

= 5.1//

If k increases from 7% to 10%, coupons of Rs 70 would be reinvested at higher rates. This will give us higher reinvestment income ahead of schedule


Problem #49

Solution FV = 100000 YTM = 16% Macaulay duration = 4.3202

Macaulay Duration =

∑t ∗ c

(1 + k)t +2nt=1

n ∗ Bn

(1 + k)t

B0

And price of a bond is given as Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = C*PVIFA (16%, 6) + 100000 PVIF (16%, 6) = 3.685 C + 0.4104*100000 Substituting in above eqn

4.3202 =

1𝐶1.6 +

2𝐶1.162 +

3𝐶1.163 +

4𝐶1.164 +

5𝐶1.165 +

6𝐶1.166 +

100000 ∗ 61.166

3.685𝐶 + 41040

15.19C + 177392 = 11.36987C + 246265 4.5402C = 68872 C = 15169.37 Substituting in equation for price we get Price = 96943.14


Problem #50 Find the current market price of a bond having face value of Rs1L redeemable after 6yrs maturity with YTM at 8% payable annually and duration =4.9927yrs Solution FV = 100000 YTM = 8% Macaulay duration = 4.992

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

And price of a bond is given as Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = C*PVIFA (8%, 6) + 100000 PVIF (8%, 6) = 4.623 C + 0.63*100000 Substituting price in equation for bond

4.992 =

1𝐶1.08 +

2𝐶1.082 +

3𝐶1.083 +

4𝐶1.084 +

5𝐶1.085 +

6𝐶1.086 +

100000 ∗ 61.086

4.623𝐶 + 63000

Solving we get C = 8000 Substituting in equation for price we get Price = 1,00,000


Problem #51 The modified duration for a 5 year 10% annual coupon bond yielding 10% is calculated to be 3.79. Now if the yield falls to 8% what is the percentage price change for this bond using the modified duration value? Is the answer same as that obtained using bond pricing formula? Solution N=5yrs C=10% YTM = 10% Modified Duration = 3.79 Change in yield = -2% % Change in Bond Price = - [Modified Duration] *[% Change in Yield] % Change in Bond Price = - 3.79*-2% = 7.58% Actual percentage price change is calculate by calculating price of a bond at new YTM of 8% Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 1080.3 % Change in Bond Price = (1080.3 – 1000)/1000 % Change in Bond Price = 8.03% The answers are not same because duration is first derivative of the bond pricing formula and assumes a linear relationship between price and yield. Actually the relationship is not linear but convex which is explained by the concept of convexity which is second derivative of bond pricing formulae and gives more accurate answer.


Problem #52 Consider a 12% Rs. 1000 FV, 5 Year bond presently trading at Rs. 970. 1) Compute its YTM 2) State the limitations of YTM. 3 Compute Macaulay’s duration. 4 Prove that Macaulay’s duration is the immunizing period. Answer – YTM = 12.84% Solution a) Calculating bond price @ YTM of 14% Bond Price = Coupon ∗ PVIFA (YTM%, n) + Bn ∗ PVIF (YTM%, n) = 120* PVIFA (14%, 5) + 1000* PVIF (14%, 5) = 931.4 Similarly calculating bond price @ YTM of 12.5% Bond Price = 982.19 Using interpolation



YTM = 12.5% + 982.2 − 970

982.2 − 931.4 ∗ (14% − 12.5%)

= 12.84% b) YTM assumes that the intermediate cash flows are reinvested at the rate of YTM. This is not always true as interest rates keeps on changing in the market, which could distort the reinvestment income and hence change the realized YTM.


c)

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5

Year Cashflow PV Factor @ 12.84% Present Value 4*1

1 120 0.8862 106 106

2 120 0.7854 94 188

3 120 0.6960 84 251

4 120 0.6168 74 296

5 1120 0.5466 612 3061

Total 970 3903


970= 4.023


Problem #53 Consider a 3 year Rs. 100000 face value bond presently yielding 14%. Its duration is 2.6 years. Find its coupon rate and price. Answer – C = 16.93%, Price = Rs. 106802.38 Solution a) 1 2 3 4 5 Year Cashflow PV Factor @ 14% Present Value 4*1 1 C 0.8772 0.8772C 0.8772C 2 C 0.7695 0.7695C 1.5390C 3 C + 100000 0.6750 0.6750C + 67,500 2.025C+2,02,491 Total 2.3216C + 67,500 4.44C + 2,02,491

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

2.6 = 4.44C + 202491

2.3216C + 67500

Solving above equation we get C = 16,930 b) Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 16,930* PVIFA (14%, 3) + 100,000 * PVIF (14%,3) Bond Price = 106802.38


Problem #54 RAMESH wants to invest in a bond that matures after six years from now. The face value of the bond is Rs. 1,000 and it carries a coupon rate of 10.75%. If the bond is currently trading at Rs. 950, You are required to calculate: a) The duration of the bond b) The price of the bond if interest rate increases by 0.50%. Answer – D = 4.68 Yrs, Revised Price = Rs. 930.15 Solution Step 1 Calculate the YTM of the bond


F − Pn

F + P2

YTM Approximate = 107.5 +

1000 − 9506

1000 + 9502

YTM Approximate = 𝟏𝟏. 𝟖𝟖%

YTM Actual = 11.96%

Step 2 Calculate Macaulay Duration

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5

Year Cash Flow PV Factor @ 11.96% Present Value (3 x 2) 4 x 1 1 107.5 0.8932 96.02 96.02 2 107.5 0.7978 85.76 171.52 3 107.5 0.7125 76.60 229.80 4 107.5 0.6364 68.42 273.66 5 107.5 0.5684 61.11 305.54 6 1107.5 0.5077 562.30 3373.79 Total 950.20 4450.32


950= 4.684


Step 3 Calculate Modified Duration


(1 + k)

= 4.684

(1 + 11.96%)

= 4.18// Step 4 Change in bond [price % Change in Bond Price = - [Modified Duration] *[% Change in Yield] = -4.18 * 0.5 = -2.09%// New bond price = 950 – 2.09% = 930.15


Problem #55 The following is the information related to a bond issued by a firm:

Date of Issue Years of Maturity Face Value (Rs) Coupon Rate (%) 01.04.2003 6 1000 9

The bond will be redeemed at its face value and coupon is paid annually. The bond is currently trading at Rs. 976.95. You are required to:

(a) Calculate the duration of the bond

(b) Calculate the percentage change in the price of the bond if the yield increases by

50 basis points.

Answer – D = 4.87 yrs, Revised Price = Rs. 955.21 Solution Step 1 Calculate the YTM of the bond


F − Pn

F + P2


1000 − 976.956

1000 + 976.952

YTM Approximate = 𝟗. 𝟓%

YTM Actual = 9.52%

Step 2 Calculate Macaulay Duration

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5

Year Cash Flow PV Factor @ 9.52% Present Value (3 x 2) 4 x 1

1 90 0.9131 82.18 82.18

2 90 0.8337 75.03 150.07

3 90 0.7612 68.51 205.53

4 90 0.6951 62.56 250.22

5 90 0.6346 57.12 285.59

6 1090 0.5795 631.63 3789.81

Total 977.03 4763.40



977= 4.87



(1 + k)

= 4.875

(1 + 9.52%)

= 4.45// Step 4 Change in bond price % Change in Bond Price = - [Modified Duration] *[% Change in Yield] = -4.45 * 0.5 = -2.225%// New bond price = 976.95 – 2.225% = 955.21


Problem #56 Consider a 14%, 20 year bond trading at Rs. 960. It is callable at a premium of 10% at the end of 5 years. If not called it is redeemable on maturity at par. Find yield duration and price volatility. Solution Step 1 Calculate Yield to Call Coupon Rate = 14% Maturity = 20Yrs Bond is callable in 5 Yrs Price = C X PVIFA (K%, n) + FV X PVIF (K%, n) 960 = 140 * PVIFA (YTC, 5) + 1000 * PVIF (YTC, 5) First let’s find if equation matches at YTM of 15.5% Price = 140 * PVIFA (15.5%, 5) + 1000 * PVIF(15.5%, 5) = 140 * 3.3128 + 1000 * 0.4865 = 950 Calculating price @ YTM of 15% Price = 140 * PVIFA (15%, 5) + 1000 * PVIF(15%, 5) = 966.47 Now we can use interpolation to get exact answer



YTC = 15% + 966.47−960

966.47−950 x (15.5% - 15%)

= 15.2% Step 2: Calculate YTM


F − Pn

F + P2


1000 − 96020

1000 + 9602

YTM Approximate = 𝟏𝟒. 𝟒𝟗%

YTM Actual = 14.63%


Step 3 Calculate Macaulay duration

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4 5


1 140 0.8724 122.14 122.14

2 140 0.7611 106.55 213.10

3 140 0.6640 92.96 278.87

4 140 0.5793 81.10 324.38

5 140 0.5053 70.75 353.74

6 140 0.4409 61.72 370.33

7 140 0.3846 53.85 376.92

8 140 0.3355 46.98 375.80

9 140 0.2927 40.98 368.83

10 140 0.2554 35.75 357.52

11 140 0.2228 31.19 343.10

12 140 0.1944 27.21 326.53

13 140 0.1696 23.74 308.60

14 140 0.1479 20.71 289.94

15 140 0.1291 18.07 271.01

16 140 0.1126 15.76 252.19

17 140 0.0982 13.75 233.76

18 140 0.0857 12.00 215.93

19 140 0.0748 10.47 198.85

20 1140 0.0652 74.35 1486.92

Total 960.00 7068.46


960= 7.363



(1 + k)

= 7.363

(1 + 14.62%)

= 6.42//


Category #4: Immunization Problem #57 Consider a Pension Fund which has the following Liability Structure:

Years Liability (Amount in Rs.) 1 80 2 110 3 60

Opportunity Cost of Capital = 15% pa. Hence, the pension fund wants to invest funds in such a manner that its liabilities are exactly met despite change in Interest Rate. Solution In order to immunize any liability using bond portfolio in such a way that change in interest rates will have no impact on the value of the portfolio, the duration of the portfolio should be equal to the investment horizon or duration of the liability Calculating duration of our liability

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4=3x2 5

Yrs CF Discount Factor PV 4X1

1 80 0.8696 69.57 69.57

2 110 0.7561 83.18 166.35

3 60 0.6575 39.45 118.35

Total 192.19 354.27


192.19= 1.84

Hence we should create a portfolio of bonds in such a way that its duration is equal to 1.84yrs. Hencce the portfolio will be immunized to changes in interest rate movements.


Problem #58 Consider a pension fund with the following liability structures:

Years Liability amount (Rs. In lakhs) 1 30 2 40 3 20 4 50

Opportunity cost of funds = 12% pa. The fund manager has short listed 2 ZCB’s – bond X and bond Y, with maturities of 2 years and 5 years respectively. Both are presently yielding 12%.

(a) What proportions of funds need to be invested in these bonds for immunization.

Also compute the face value of each bond.

Solution In order to immunize any liability using bond portfolio in such a way that change in interest rates will have no impact on the value of the portfolio, the duration of the portfolio should be equal to the duration of the liability Step 1: Calculate duration of liability

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4=3x2 5


1 30 0.8929 26.79 26.79

2 40 0.7972 31.89 63.78

3 20 0.7118 14.24 42.71

4 50 0.6355 31.78 127.10

Total 104.68 260.37


104.68= 2.49

Hence we should create a portfolio of bonds in such a way that its duration is equal to 2.49yrs. Hencce the portfolio will be immunized to changes in interest rate movements. Step 2 Calculate the duration of each bond Since both the bonds are zero coupon bonds their duration will be equal to their maturity


Step 3: Calculate the proportion of each bond in the portfolio Let X and Y denote the proportion of weights of Bond X and Y respectoively 2X+5Y = 2.49 X+Y = 1 Solving above 2 equations simultaneously we get X=83.67% Y=16.63%


Problem #59 Mr. Rohit Sharma is required to make the following payments at the end of each year for the next 6 years.

Year 1 2 3 4 5 6 Payment

(Rs Lakhs) 25.50 19.25 18.25 17.50 19.50 17.50

He is planning to immunize his liability by investing in the following into bonds. Bond X: 11% Coupon bond of face value Rs. 1,000 maturing after 5 years, redeemable at 5% premium and currently traded at Rs. 966.38. Bond Y: 13% Coupon bond of face value Rs. 1,000 maturing after 3 years, redeemable at 5% discount and currently traded at Rs. 988.66. Required:

a. If the interest rate is 12%, calculate the proportions of funds to be invested in

bonds X and Y, so that Mr. Sharma’s payments are immunized.

Answer – DL = 2.99 Yrs, Wx = .23, Wy = .77 Solution In order to immunize any liability using bond portfolio in such a way that change in interest rates will have no impact on the value of the portfolio, the duration of the portfolio should be equal to the duration of the liability Step 1: Calculate duration of liability

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4=3x2 5


1 25.5 0.8929 22.77 22.76786

2 19.25 0.7972 15.35 30.69196

3 18.25 0.7118 12.99 38.96997

4 17.5 0.6355 11.12 44.48627

5 19.5 0.5674 11.06 55.32412

6 17.5 0.5066 8.87 53.19627

Total 82.16 245.44


82.16= 2.99

Hence we should create a portfolio of bonds in such a way that its duration is equal to 2.99yrs. Hencce the portfolio will be immunized to changes in interest rate movements.


Step 2 Calculate the YTM of each bond using approximate formula


F − Pn

F + P2

Bond X

YTM X = 110 +

1050 − 966.385

1050 + 966.382

= 12.72%

Bond Y

YTM Y = 130 +

950 − 988.663

950 + 988.662

= 11.99%

Step 3: Calculate the duration of each bond Bond X 1 2 3 4=3x2 5


1 110 0.8871 97.58 98

2 110 0.7870 86.57 173

3 110 0.6981 76.79 230

4 110 0.6193 68.13 273

5 1160 0.5494 637.31 3187

Total 966.38 3960.16


966.38= 4.1

Bond Y 1 2 3 4=3x2 5


1 130 0.8929 116.08 116

2 130 0.7973 103.65 207

3 1080 0.7120 768.93 2307

Total 988.66 2630.18


988.66= 2.66


Step 4: Calculate the proportion of each bond in the portfolio Let X and Y denote the proportion of weights of Bond X and Y respectoively 4.1X+2.66Y = 2.99 X+Y = 1 Solving above 2 equations simultaneously we get X=23% Y=77%


Problem #60 Consider a pension with the following liability structure:

Years Liability amount (Rs in lakhs) 1 40 2 70 3 60

Opportunity cost – 14% p.a. Short listed bonds – 2 year and 7 year ZCB, both yielding 14%. Find out the proportion of funds to be invested in each bond for immunization? Solution In order to immunize any liability using bond portfolio in such a way that change in interest rates will have no impact on the value of the portfolio, the duration of the portfolio should be equal to the duration of the liability Step 1: Calculate duration of liability

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0

1 2 3 4=3x2 5


1 40 0.8772 35.09 35.09

2 70 0.7695 53.86 107.73

3 60 0.6750 40.50 121.49

Total 129.45 264.31


129.45= 2.04

Hence we should create a portfolio of bonds in such a way that its duration is equal to 2.04yrs. Hencce the portfolio will be immunized to changes in interest rate movements. Step 2 Calculate the duration of each bond Since both the bonds are zero coupon bonds their duration will be equal to their maturity Step 3: Calculate the proportion of each bond in the portfolio Let X and Y denote the proportion of weights of Bond X and Y respectoively 2X+7Y = 2.04 X+Y = 1 Solving above 2 equations simultaneously we get X=99.2% Y=0.8%


Problem #61 The following corporate bonds are considered for investment by the portfolio manager. His aim is to immunize the liability due in six years. All bonds have face value of Rs1000. Bond Maturity

(Years) Coupon

% Duration years

Arvind Mills 10 8 7.35 BILT 8 9 6.15 Cipla 5 7 4.30 If the portfolio manager wishes to invest 50% in Arvind Mills, What is the percentage of total amount that can be invested in the other two bonds to immunize the portfolio? Solution In order to immunize the portfolio the duration of the portfolio should be equal to the investment horizon This implies Portfolio duration = 6 i.e. 𝑊𝐴𝐷𝐴 + 𝑊𝑆𝐷𝐵 + 𝑊𝐶𝐷𝐶 = 6 Solving we get 0.5 X 7.35 + 𝑊𝐵 X 6.15 + 𝑊𝐶 X 4.3 = 6 Also 𝑊𝐴 + 𝑊𝐵 + 𝑊𝐶 = 1 i.e. 𝑊𝐵 + 𝑊𝐶=0.5 We have 2 simultaneous equation solving we get 𝑊𝐵 = 9.5% 𝑊𝐶 = 40.5% //


Category #5: Forward Rates & Spot Rates Calculation Problem #62 If the 1 year spot is 5%, 1 year forward, starting one year from today is 6.5% and 1 year forward starting two years from today is 8%, what is three year spot rate? Solution 𝑆1 = 5% 1𝑓1 = 6.5% 1𝑓2 = 8% 𝑆3 = ? 0 1 2 3 5 6.5 8 |-----------------------|-----------------------|--------------------------| -------------------------------X---------------------------------- (1 + 𝑋)3 = (1 + 5%) * (1 + 6.5%) * (1 + 8%) = 6.49%


Problem #63

Solution Current 1 year rate = = 12% 1 year forward rate = (12-0.75) = 11.25% 2 year forward rate = (11.25-0.50) = 10.75% (1 + S2)2 = 1 + 1f0 ∗ 1 + 1f1 (1 + S2)2 = 1 + 12% (1 + 11.25%) 1 + S3

3 = 1 + 1f0 ∗ 1 + 1f1 ∗ 1 + 2f1 1 + S3

3 = 1.12 1.1125 (1.1075)

Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

Price = 90

(1 + 12%)+

90

1 + 12% (1 + 11.25%)+

1090

1.12 1.1125 (1.1075)

= 942.48 //

Since β = 1.02 Price = 942.48*102

= 961.33 //


Problem #64

Solution a) Forward rate 1 year from today (1 + S2)2 = 1 + S1 ∗ 1 + 1f1 (1 + 11.25%)2 = (1 + 10.5%) (1 + X%)

1 + X% = 1.11252

1.105= 12%

Similarly (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1)

1 + 2f1 = 1 + 12% 3

(1.105 ∗ 1.12)

2f1 = 13.52% b) If bond is fairly priced then it implies its coupon rate is 12%. This implies if interest rates increase by 50 basis points then YTM will be 12.5% Calculate the price of bond at YTM of 12.5% Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Price = 120 * PVIFA(12.5%, 5) + 1000 * PVIF(12.5%, 5) = 982.19

% Change in bond price = 982.19 − 1000

1000= −𝟏. 𝟖%


Problem #65 Following are the annual interest rates of a security :

Spot rate on one year 8.5% Forward rate after one year for one year 9.50% Forward rate after two years for one year 13.56%

What is the yield of the security for three years? Solution (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + S3)3 = 1 + 8.5% ∗ 1 + 9.5% ∗ (1 + 13.56%) 𝐒𝟑 = 𝟏𝟎. 𝟓𝟖%


Problem #66 A bond issued by ABC Ltd. is selling presently at a face value of Rs100 and pays coupon at the rate of 13% p.a. in arrears, which will be redeemed at Rs113 after five years. The ‘n’ years spot rate of interest is (8.56+ n/6)% where, n=1, 2,3,4 and 5. The term structure of interest rates is flat and pure expectation theory holds good. You are required to calculate:

The value of the bond at time 0 The duration of the above bond Change in bond price for 50 basis point increase in interest rates.

(Answer: a. Rs122.79; b. Duration 4.1 years; c. New Price =Rs120.2 (Hint: a. Five different yields to be used for finding the price) Solution

Sn = 8.56 +n

6

Year Spot Rate 1 8.73 2 8.89 3 9.06 4 9.23 5 9.39

1) Calculate Bond Price Bond Price = Present Value of Future Cash Flows

Bond Price = 13

1.0873 1+

13

1.0889 2+

13

1.0906 3+

13

1.0923 4+

13 + 113

1.0939 5

Bond Price = 122.5 2) Calculate YTM using approximate formula


F − Pn

F + P2

YTM = 13 +

113 − 122.55

113 + 122.52

= 𝟗. 𝟒𝟐%


3) Calculate Macaulay Duration

Macaulay Duration =

∑t ∗ c

(1 + k)t +nt=1

n ∗ Bn

(1 + k)t

B0


122.06

= 4.11// 4) Calculate Modified Duration


(1 + k)

= 4.11

(1 + 9.42%)

= 3.75// 5) Change in bond [price % Change in Bond Price = - [Modified Duration] *[% Change in Yield] = -3.75 * 0.5 = -1.875%// New bond price = 122.5 – 1.875% = 120.20

1 2 3 4 5


1 13 0.9139 11.88 11.88

2 13 0.8352 10.86 21.72

3 13 0.7633 9.92 29.77

4 13 0.6976 9.07 36.28

5 126 0.6376 80.33 401.66

Total 122.06 501.30


Problem #67 Consider three pure discount bonds with maturities of one, two and three years and prices of Rs930.23, Rs923.79 and Rs919.54 respectively. Each bond has a face value of Rs1000. What are the 1 year, 2 year and 3 year spot rates? Solution For zero coupon bonds bond price is calculated using following formula

Price = FV

1 + Sn n

1 year bond

930.23 = 1000

(1 + S1)

Solving we get S1 = 7.5% 2 year bond

923.79 = 1000

(1 + S2)2

Solving we get S2 = 4.04% 3 year bond

919.54 = 1000

(1 + S3)3

Solving we get S3 = 2.84% Problem #68 Given the following spot rates for various periods of time from today, calculate forward rates from years one to two, two to three and three to four. S1 = 5%, S2 = 5.5%, S3 = 6.5%, S4 = 7% Solution 1 year forward rate (1 + S2)2 = 1 + S1 ∗ 1 + 1f1 1 + 5.5% 2 = 1 + 5% 1 + 1f1

(1 + 1f1) = (1.055)2

1.05

𝟏𝐟𝟏 = 𝟔% 2 year forward rate (1 + S3)3 = 1 + S2

2 ∗ (1 + 2f1) (1 + 6.5%)3 = (1 + 5.5%)2 ∗ (1 + 2f1) 𝟐𝐟𝟏 = 𝟖. 𝟓𝟑% 3 year forward rate 𝟑𝐟𝟏 = 𝟖. 𝟓𝟏%


Problem #69 Give the following forward rates for respective years; calculate the spot rates for years one, two, three and four. Year Forward Rate

1 10.0% 2 9.5% 3 9.0% 4 8.5%

Solution (1 + S2)2 = 1 + S1 ∗ 1 + 1f1

S2 = 1 + 10% 1 + 9.5 − 1

S2 = 9.75%

(1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1)

S3 = 1 + 10% 1 + 9.5% 1 + 9% − 13

S3 = 9.5%

(1 + S4)4 = 1 + S1 ∗ 1 + 1f1 ∗ 1 + 2f1 ∗ (1 + 3f1)

S4 = 1.1 ∗ 1.095 ∗ 1.09 ∗ 1.0854

S4 = 9.2%


Problem #70 Assume that the government has issued three bonds. The first which pays Rs1000 one year from today is selling at Rs909.09. The second which pays Rs100 one year from today and Rs1100 a year later is selling at Rs991.81. The third which pays Rs100 one year from today, Rs100, one year later and Rs1100 one year after that, is selling for Rs997.18. What are the forward rates for one, two and three years from today? Solution For zero coupon bonds bond price is calculated using following formula

Price = FV

1 + Sn n

1 year bond

909.09 = 1000

(1 + S1)

Solving we get S1 = 10% 2 year coupon bond price is given as

Price = C

(1 + S1)+

C + FV

(1 + S2)2

991.81 = 100

(1 + 10%)+

1100

(1 + S2)2

Solving we get S2 = 10.5% 3 year coupon bond price is given as

Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

997.18 = 100

(1 + 10%)+

100

(1 + 10.5%)2+

1100

(1 + S3)3

Solving we get S3 = 10.09% 1 years forward Rate calculation (1 + S2)2 = 1 + S1 ∗ 1 + 1f1

(1 + 10.5%)2 = 1 + 10% ∗ 1 + 1f1 𝟏𝐟𝟏 = 𝟏𝟏% 2 years forward Rate calculation (1 + S3)3 = 1 + S2

2 ∗ (1 + 2f1) (1 + 10.09%)3 = (1 + 10.5%)2 ∗ (1 + 2f1) 𝟐𝐟𝟏 = 𝟗. 𝟒%


Problem#71 Consider the following data:

Bonds Years (maturity) Face value Coupon rate Market price A 1 1000 0 934 B 2 1000 10% 985 C 3 1000 12% 1010

Derive the term structure. Answer – Spot rates for years 1, 2 & 3 = 7.07%, 11.07%, 11.84% Solution For zero coupon bonds bond price is calculated using following formula

Price = FV

1 + Sn n

S1 = 1000

934− 1 = 7.07%

2 year coupon bond price is given as

Price = C

(1 + S1)+

C + FV

(1 + S2)2

985 = 100

(1 + 7.07%)+

1100

1 + S2 2

Solving we get S2 = 11.07 3 year coupon bond price is given as

Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

1010 = 120

(1 + 7.07%)+

120

1 + 11.07 2+

1120

1 + S3 3

Solving we get S3 = 11.84


Problem#72 A bond issued by ABC Co. is selling presently at the face value of Rs. 100 and pays coupon at the rate of 10% p.a. in arrears and will be redeemed at Rs. 110 after 3 years. The n year spot rate interest, Yn is given by Yn (%) = 9.0 + n/10 for n = 1,2 and 3. Assuming the pure expectations theory holds good, calculate:-

(i) The implied one year forward rates applicable at times t = 1 and t = 2

(ii) The value of the bond at time t = 0

Answer - 𝐅𝟏𝟐 = 9.3%, 𝐅𝟐𝟑 = 9.5%, IV = 109.46 Solution

Sn = 9.0 +n

10

Year Spot Rate 1 9.1 2 9.2 3 9.3


Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

Bond Price = 10

1.091 1+

10

1.092 2+

120

1.093 3

Bond Price = 109.45 2) 1 years forward Rate calculation 1 + S2

2 = 1 + S1 ∗ 1 + 1f1 1 + 9.2% 2 = 1 + 9.1% ∗ 1 + 2f1 Solving we get 1f1 = 9.3% 2 years forward Rate calculation (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + 9.3)3 = 1 + 9.1% ∗ 1 + 9.3% ∗ (1 + 2f1%) Solving we get 2f1 = 9.5%


Problem #73 Consider the sovereign yield curve. Given rn = 9 + n/10 Find out the intrinsic value of a 12% Rs. 1000 face value 3 year government bond. Solution

Sn = 9.0 +n

10

Year Spot Rate 1 9.1 2 9.2 3 9.3


Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

Bond Price = 120

1.091 1+

120

1.092 2+

1120

1.093 3

Bond Price = 1068.3


Problem #74 Assume you observe the following three coupon bond prices and remaining cash flows (coupons are paid annually and this year’s coupon has already been paid Bond A is currently trading at a price of 107, has a face value of 100 and 10% coupon

and three years to maturity.

Bond B is currently trading at a 105, has a face value of 100 and 10% coupon and two

years to maturity.

Bond C is currently trading at a price of 100, has a face value of 100 and 10% coupon and

1 year to maturity.

Find out the term structure of interest rates by the method of bootstrapping. Also, compute the 1 Yr forward rates. Answer - 𝐟𝟎𝟏 = 𝐫𝟎𝟏 = 10%, 𝐟𝟏𝟐 = 4.25%, 𝐟𝟐𝟑 = 7.54%, 𝐟𝟏𝟑 = 12.12%, 𝐫𝟎𝟏 = 10%, 𝐫𝟎𝟐 = 7.09%, 𝐫𝟎𝟑 = 7.24% Solution

S1 = 1100

100− 1 = 10%

2 year coupon bond price is given as

Price = C

(1 + S1)+

C + FV

(1 + S2)2

105 = 10

(1 + 10%)+

110

1 + S2 2


Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

107 = 10

(1 + 10%)+

10

1 + 7.09% 2+

110

1 + S3 3

Solving we get S3 = 7.24% 1 years forward Rate calculation 1 + S2

2 = 1 + S1 ∗ 1 + 1f1 1 + 7.09% 2 = 1 + 10% ∗ 1 + 1f1 Solving we get 1f1 = 4.256% 2 years forward Rate calculation (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + 7.24)3 = 1 + 10% ∗ 1 + 4.256% ∗ (1 + 2f1%) Solving we get 2f1 = 7.54%


Problem #75 ABC Ltd. is coming out with an issue of two series of zero coupon bonds maturing in 4 and 5 years. Face value of both the bonds is Rs. 1000. Market price of similar traded bonds is Rs. 925 and Rs. 900 respectively. Mr. Tiwari is considering investing in these bonds. You are required to calculate one year interest rates after 4 years. Answer - 𝐟𝟒𝟓 = 4.18% Solution 925 ∗ 1 + S4

4 = 1000 Solving we get S4 = 1.968% 900 ∗ 1 + S5

5 = 1000 Solving we get S5 = 2.13% 1 + S5

5 = 1 + S4 4 ∗ (1 + 4f1)

Solving we get 4f1 = 2.78%


Problem #76 Suppose a zero-coupon bond maturing one year from now costs Rs. 90, a zero-coupon bond maturing two years from now costs Rs. 80, and a zero-coupon bond maturing three years from now costs Rs. 70. Calculate:

1. The zero-coupon yields for one-year, two-year and three-year zero-coupon bonds;

2. The implied 1 year forward interest rates.

Answer - 𝐫𝟎𝟏 = 11.11%, 𝐫𝟎𝟐 = 11.8%, 𝐫𝟎𝟑 = 12.6%, 𝐟𝟎𝟏 = 11.11%, 𝐟𝟏𝟐 = 12.49%, 𝐟𝟐𝟑 = 14.22%, 𝐟𝟏𝟑 = 28.49% Solution For zero coupon bonds bond price is calculated using following formula

Price = FV

1 + Sn n

1 year Zero Coupon Bond

S1 = 100

90− 1 = 11.11%

2 year Zero Coupon Bond

80 = 100

1 + S2 2

Solving we get S2 = 11.8% 3 year Zero Coupon Bond

70 = 100

1 + S3 3

Solving we get S3 = 12.6% 1 year forward Rate calculation 1 + S2

2 = 1 + S1 ∗ 1 + 1f1 1 + 11.8% 2 = 1 + 11.11% ∗ 1 + 1f1 Solving we get 1f1 = 12.49% 2 year forward Rate calculation (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + 12.6)3 = 1 + 11.11% ∗ 1 + 12.49% ∗ (1 + 2f1%) Solving we get 2f1 = 14.22%%


Problem #77 From the following data for Government securities, calculate the forward rates:

Face Value (Rs.) Interest rate Maturity (Year) Current price (Rs.) 1,00,000 0% 1 91,500 1,00,000 10% 2 98,500 1,00,000 10.5% 3 99,000

Solution

S1 = 1,00,000

91,500− 1


Price = C

(1 + S1)+

C + FV

(1 + S2)2

98,500 = 10,000

(1 + 9.23%)+

1,10,000

1 + S2 2


Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

99,000 = 10,500

(1 + 9.23%)+

10,500

1 + 10.96% 2+

1,10,500

1 + S3 3

Solving we get S3 = 10.97% 1 year forward Rate calculation 1 + S2

2 = 1 + S1 ∗ 1 + 1f1 1 + 10.96% 2 = 1 + 9.23% ∗ 1 + 1f1 Solving we get 1f1 = 12.72% 2 year forward Rate calculation (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + 10.97)3 = 1 + 9.23% ∗ 1 + 12.72% ∗ (1 + 2f1%) Solving we get 2f1 = 10.99%


Problem #78 Consider the following date for Government securities:

Face value Interest (Rate %) Maturity (Years) Current Price (Rs.) 1,00,000 0 1 91,000 1,00,000 10.5 2 99,000 1,00,000 11.0 3 99,500 1,00,000 11.5 4 99,900

Calculate the forward interest rates. Solution

S1 = 1,00,000

91,000− 1


Price = C

(1 + S1)+

C + FV

(1 + S2)2

99,000 = 10,500

(1 + 9.89%)+

1,10,500

1 + S2 2


Price = C

(1 + S1)+

C

(1 + S2)2+

C + FV

1 + S3 3

99,500 = 11,000

(1 + 9.89%)+

11,000

1 + 11.15% 2+

1,11,000

1 + S3 3


Price = C

(1 + S1)+

C

(1 + S2)2+

C

1 + S3 3+

C FV

(1 + S4)4

99,900 = 11,500

(1 + 9.89%)+

11,500

1 + 11.15% 2+

11,500

1 + 11.26% 3+

1,11,500

1 + S4 4

Solving we get S4 = 11.64%


1 year forward Rate calculation 1 + S2

2 = 1 + S1 ∗ 1 + 1f1 1 + 11.15% 2 = 1 + 9.89% ∗ 1 + 1f1 Solving we get 1f1 = 12.42% 2 year forward Rate calculation (1 + S3)3 = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) (1 + 11.26)3 = 1 + 9.89% ∗ 1 + 12.42% ∗ (1 + 2f1%) Solving we get 2f1 = 11.48% 3 year forward Rate calculation (1 + S4)4 = (1 + S3)3 ∗ 1 + 3f1 1 + 11.64% 4 = 1 + 11.26% 3 ∗ 1 + 3f1% 3f1 = 12.78%


Category #6: Clean Price & Dirty Price Problem #79

Solution a) YTM as of January 1, 2000 Since the bonds were sold @ Par YTM = CR = 10% b) Step1: Calculate clean price on next coupon date i.e on 30/June/2008


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Clean Price = 50 ∗ PVIFA ((12

2)%, 2 ∗ 7.5) + 1000 ∗ PVIF (

12

2)%, 2 ∗ 7.5)

Clean Price = 902.87 Step2: Calculate Dirty Price on i.e on 30/June/2008 Dirty Price = Clean Price + Coupon = 902.87+50 = 952.87 Step 3: Calculate Dirty Price on 1/March/2008

Dirty Price =952.87

1 + 6% 4/6= 916.22


Step 4: Calculate Clean Price on 1/March/2008 Dirty Price = Clean Price + Accrued Interest Clean Price = Dirty Price – Accrued Interest Clean Price = 916.22 – 50*(2/6) Clean Price = 899.55


Problem #80 Consider a bond with the following features: Face value – Rs. 1, 00,000 Coupon rate – 12% payable at the end of December each year Required return – 15% Valuation date – 1st April 2009. Redemption, i.e. Maturity date – 31.12.2015 Current market price – 92.55%. Redemption at par on maturity. Find out the intrinsic value, that is full price of the bond and split it into the accrued interest and clean price components. Give your investment advice. Answer – Clean Price & Dirty price today = Rs. 90628, 87628 Solution Step1: Calculate clean price on next coupon date i.e on 31/Dec/2009 Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Clean Price = 12,000*PVIFA(15%,6) + 1,00,000*PVIF(15%,6) Clean Price = 88,646.55 Step2: Calculate Dirty Price on i.e on 31/Dec/2009 Dirty Price = Clean Price + Coupon = 88,646.55+12,000 = 1,00,646.55 Step 3: Calculate Dirty Price on 1/Apr/2009

Dirty Price =1,00,646.55

1 + 15% 9/12= 90,630.74

Step 4: Calculate Clean Price on 31/March/2008 Dirty Price = Clean Price + Accrued Interest Clean Price = Dirty Price – Accrued Interest Clean Price = 90,630.74 – 12,000*(3/12) Clean Price = 87,630.74 However the actual price quoted in the market is 92,550 which is greater than intrinsic value. So the bond is trading rich and investor should go short.


Problem #81 Consider a bond with the following features: Face value – Rs. 1000 Coupon rate – 14% payable semi-annually on end June and end December. Required rate – 12% BEY. Maturity date – 31st December 2022. Valuation date – 1st October 2010. Market quoted price = 103%. Give your investment advice by computing the clean price of the bond. Answer – Clean price & Dirty price today = Rs. 1161.79, 1126.79 Solution Step1: Calculate clean price on next coupon date i.e on 31/Dec/2010


2)%, 2n) + Bn ∗ PVIF ((

k

2)%, 2n)

Clean Price = 70 ∗ PVIFA (6%, 24) + 1000 ∗ PVIF (6%, 24) Clean Price = 1125.49 Step2: Calculate Dirty Price on i.e on 31/Dec/2010 Dirty Price = Clean Price + Coupon = 1125.49+70 = 1195.5 Step 3: Calculate Dirty Price on 01/Oct/2010

Dirty Price =1195.5

1 + 6% 3/6= 1161.16

Step 4: Calculate Clean Price on 01/Oct/2010 Dirty Price = Clean Price + Accrued Interest Clean Price = Dirty Price – Accrued Interest Clean Price = 1161.16 – 70*(3/6) Clean Price = 1126.17 Since the market price 1030 is less than intrinsic value the bond is trading cheap and investor should go long


Category #7: Bond Refunding Decision Problem #82

Solution Details of old bond Coupon Rate = 12% FV = 300mn Unamortized cost = 9mn New bond details CR = 10% FV = 300mn Issuance cost = 6mn Call premium of 4% on old bond Tax rate = 30% Discount Rate = 7% Cash outflow for calling old bonds = 300 + 4% of 300

= 312mn

Cash outflow for Issuance cost of new bond = 6mn Cash inflow from new bond = 300mn

Now lets calculate savings & taxes

Premium cost & unamortized cost of old bonds will be deducted now in income statement which will lead to tax savings.

Tax savings = (9 + 12) * 0.3

= 6.3mn There will be savings on coupon also as new coupon is leaser compared to old Difference in coupon = 300 (12% - 10%) = 6mn


However because of savings on coupon tax payment will also go up as a result of which net savings will be net of tax loss = 6mn * (1 – 0.3) = 4.2mn // 4.2mn of saving every year for next 6 years PV of 4.2mn @ 7% for 6 yrs = 4.2 * PVIFA (7%, 6) = 20.02mn // Now here is tricky part Because of new bonds issuance cost of 6mn there will be tax benefits. New bond will be amortized (i.e. its issuance cost will be amortized over next 6 years Amortized cost = 6mn = 6 yrs = 1mn Savings on tax due to amortization cost = 0.3 * 1mn * PVIFA (7%, 6) = 1.42mn However the unamortized cost of 9m of old bond is not there now Hence loss in taxes because of that

= 0.3 * 9

6 (PVIFA) (7%, 6)

= 2.14mn Net savings = –312 – 6 + 300 + 6.3 + 20.02 + 1.42 – 2.14 = 7.6mn // Here there is net savings we should consider refunding of bonds.


Problem #83

Solution Time to maturity = 10 Years Outstanding Value = 2 Cr Coupon Rate = 11% New Coupon Rate = 9% Unamortized issue cost = 3L Insurance cost of New bonds = 2.5L Call Premium = 5%

a) Proceeds from issuance of new bonds = + 2 Cr b) Issuance Cost = - 25 Lacs c) Refunding of old bonds = - 2 Cr d) Premium on old bond = 5% of 2 Cr

= - 10L e) Tax savings due to unamortized portion & Premium

= 30% [10L + 3L] = + 3.9L

f) Savings due to lower coupon rate = 2 Cr * [11% - 9%] * (1 – 30%) = 2.8 Lacs per Year

PV of total savings = 2.8 * PVIFA (7%, 10) = 19.66602

g) Savings on tax due to amortization of issuance cost

= 2.5−3

10 * 0.3 x PVIFA (7%, 10)

= -0.1054L Total savings = 2 Cr – 2.5L – 2 Cr – 10L + 3.9L +19.66L – 0.1054L = 10.9546 Lacs Hence refunding should be considered.


Problem #84


Category #8: Convertible Bond Problem #85

Solution FV = 1000 Price = 1350 CR = 10.5% Conversion rate = 14 Shares CMP = 1475 Share Price = 80 Conversion Premium is % increase in price required from CMP to reach to conversion price

Conversion price = Market Price of Bond

Conversion Rate

Conversion Price = 1475

14

= 105.3571

Conversion Premium = (Conversion Price – Current Share Price)

Current Share Price

Conversion premium = 105.3571−80

80

= 31.7% //


Problem #86

Solution Coupon Rate = 12 Conversion ratio = 20 FV = 100 Maturity = 5 yrs Current Price of bond @ 8% YTM Price = 12 * PVIFA (8%, 5) + 100 * PVIF (8%, 5) = 115.97 We should convert whenever we get more value than 115.97 When share price = 4 Net worth of shares = 20*4

= 80 When share price = 5 Net worth of shares = 100 When share price = 6 Net worth of shares = 120 Hence we should convert only when share price is 6 //


Problem #87

Solution Stock value of bond = Current Market Price * Conversion ratio = 20*12 = 240

Downside Risk = (Market Price – Straight Value)

Straight Value

= 265−235

235

= 12.77%


Current Share Price

= 13.25−12

12

= 10.42%

Conversion Parity = Current Market Price of Bond

Conversion Ratio

= 265

20

= 13.25


Problem #88

Solution Conversion ratio = 10


Current Share Price

OR

Conversion Premium = (Current Market Price of Bond – Conversion Value)

Conversion Value

= 5400−(430∗10)

(430∗10)

= 25.58% // Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 430*10 = 4300 //


Problem #89 A convertible bond with a face value of Rs1,000 has been issued at Rs1, 300 with a coupon rate of 12%. The conversions rate is 20 shares per bond. The current market price of the bond is Rs1,500 and that of stock is Rs60. What is the conversion value premium? Solution


Conversion Rate

Conversion price = 1500

20= 75


Current Share Price

Conversion Premium = (75 − 60)

60∗ 100

Conversion Premium = 𝟐𝟓%


Problem #90 Consider the data regarding convertible bonds by M.K. Enterprise:- Par Value = Rs. 1000 Coupon rate = 9% Market price of the Convertible bond = Rs. 925 Conversion ratio = 25 Estimated Straight value of the bond = Rs. 730 Price of common stock = 30 Calculate each of the following:-

a. Conversion Value

b. Market Conversion price

c. Conversion premium per share

d. Conversion premium ratio

e. Premium over straight value

Solution a) Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 30*25 = 750// b)


Conversion Rate


25= 37

c)


Current Share Price


30∗ 100 = 23.33%

d)

Premium over straight value = (Market Price of Bond – Straight Value of Bond )

Straight Value of Bond

Premium over straight value = (925 − 730)

730∗ 100 = 26.71%


Problem #91 The following data is related to 8.5% Fully Convertible (into Equity shares) Debentures issued by JAC Ltd. At Rs. 1000 Market Price of Debenture Rs. 900 Conversion ratio 30 Straight value of Debenture Rs. 700 Market Price of equity share on the date of Conversion Rs. 25 You are required to calculate: a. Conversion Value of Debenture b. market Conversion Price c. Conversion premium per share d. Ratio of Conversion premium e. Premium over straight value of debenture Solution a) Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 25*30 = 750// b)


Conversion Rate


30= 30

c)


Current Share Price


25∗ 100 = 20%

d)

Premium over straight value = (Market Price of Bond – Straight Value of Bond )

Straight Value of Bond

Premium over straight value = (900 − 700)

700∗ 100 = 28.57%


Problem #91 Newchem Corporation has issued a fully convertible 10% debenture of Rs. 10,000 face value, convertible into 20 equity shares. The current market price of the debentures is Rs. 10,800, whereas, the current market price of equity share price is Rs. 480. You are required to calculate (i) the conversion premium and 9ii) the conversion value. Solution a)


Conversion Rate


20= 540

b)


Current Share Price


480∗ 100 = 12.50%

c) Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 480*20 = 9600//


Problem #92 Consider a Rs1000 FV, 5 year 10% Coupon OCD which is convertible into 4 shares of share price Rs. 260. Yield on similar Non Convertible Debenture is 12% Option Value = Rs. 50 Find the IV of the OCD. Solution Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio Conversion Value Stock Value = 260 ∗ 4 = 1040 Investment Value = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Investment Value = 100* PVIFA (12%, 5) + 1000* PVIF (12%,5) Investment Value = 927.88 Floor Value of Bond = Higher of ( Conversion Value, Investment Value) Floor Value of Bond = Higher of (1040 , 927.88) Floor Value of Bond = 1040 Intrinsic Value = Floor Value + Option Premium Intrinsic Value = 1040 + 50

= 1090


Problem #93 Consider the following OCD:- FV = Rs. 100000 Coupon Rate = 12% Conversion Rate = 20.1 (1Bond = 20 Shares) Share Price = Rs. 5210 Maturity of the OCD = 5 Years YTM on similar Bonds = 13% If option value is 5% of the floor Value, Calculate the IV of the OCD. Solution Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio Conversion Value Stock Value = 5210 ∗ 20 = 1,04,200 Investment Value = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Investment Value = 12000* PVIFA (13%, 5) + 100000* PVIF (13%,5) Investment Value = 96,436.4 Floor Value of Bond = Higher of ( Conversion Value, Investment Value) Floor Value of Bond = Higher of (1,04,200, 96,436) Floor Value of Bond = 1,04,200 Intrinsic Value = Floor Value + Option Premium Intrinsic Value = 104,200 + 4%

= 1,09,410


Category #9: Mixed Problem #94

Solution

a)Current Yield = 14

90

= 15.5%

For YTM we need to find X in following equation 90 = 14 * PVIFA (X, 5) + 100 * PVIF (X, 5) We solve it by trial & error and then use interpolation to get to correct answer. At 15% At 18% Price = 96.64 Price = 87.49 Interpolation is used as follow



YTM = 15% + 96.64−90

96.64−87.49 * 3%

= 15% + 2.177% = 17.17%


b)Duration

Yrs CF PV Factor 1x2x3 1 14 0.8535 11.94 2 14 0.728 20.38 3 14 0.622 26.12 4 14 0.531 29.73 5 114 0.453 258.2 346.38

Duration = 3.847 Years iii) Realized Yield 14∗ 5 + 100

90 = (1 + X)5

1.8889 = (1 + X)5 Solving we get X = 13.56%


Problem #95

Solution a) 5 Year Bond Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) = 80 * PVIFA (6%, 5) + 1000 * PVIF (6%, 5) = 1083.96 % change in 5 Yrs bond = 8.3% Price increase due to change in PV of Principal = 1000 * [PVIFA (6%, 5) – PVIF (8%, 5)] = 1000 * [0.747 – 0.681] = 66 So out of total change of Rs. 83.96, 66 comes due to principal Hence % change in bond price due to principal

=66

83.96

= 78.6% % change in bond price due to coupon = 21.4%


20 Year bond Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) = 80 * PVIFA (6%, 20) + 1000 * PVIF (6%, 20) = 1229.6 % change in 20 Yrs bond = 22.9% % change in 20 yrs bond price due to principal = 42.68% % change in bond price due to coupon = 57.32% b)

Yrs CF PV Factor @ 7% 1x2x3 1 70 0.935 65.45 2 70 0.873 122.22 3 70 0.816 171.36 4 70 0.763 213.64 5 70 0.713 249.55 6 1070 0.666 4275.72 ∑ 5097.94

Duration = 5097.74

1000

= 5.097 Years

C) If YTM increase to 10%

New Price = 70 * PVIFA (10%, 6) + 1000 * PVIF (10%, 6) = 868.85 New duration can be calculated as follows as only discounting factor will change

New Duration 4366 .45

868.85 = 5.2025 Years

Since duration is inversely proportion to YTM.


Problem #96 Consider a 10% bond of face value of Rs1,000 and redeemable after 5 years at a premium of 5% What is the total interest on interest earned by the investor at the end of the second year, if the reinvestment rate is 12% ? Solution Coupon earned at the end of year 1 = 10% of 1000 = 100 If reinvestment rates increase to 12% then interest earned on interest = 12% of 100 = 12


Problem #97 Consider a bond with Rs1000 face value, 10 years to maturity and 8% coupon. Bond is selling at an YTM of 10% now. If the yield is expected to decline to 9% at the end of 4 years and if we sell the bond then, what is the total absolute and percentage return earned, if coupons were reinvested at 9.5% Segregate the absolute return into four components: a gain because of passage of time, b. gain because of decrease in yield, c. coupons and d. reinvestment income. Solution Step I Fund PV of bond today n = 10 FV = 1000 I/Y = 10% PMT = 80 Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%, n) PV of Bond = 80 ∗ PVIFA 10%, 10 + 1000 ∗ PVIF 10%, 10 = 877.1087 Step II Value of bond at the end of year 4 n = 6 FV = 1000 I /Y = 9 PMT = 80 Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%, n) Bond value = 955.14 Step III Coupon payments @ end of Year 1 = 80 ∗ 1.0953 = 105.03 2 = 80 ∗ 1.0952 = 95.9 3 = 80 ∗ 1.0951 = 87.6 4 = 80 ∗ 10.950 = 80 Total = 368.53 Total absolute returns = 368.53 + 955 − 877 = 446.56//

% 𝑔𝑎𝑖𝑛 = 446.56

877.1087

= 50.91%


Step IV Gain because of investment income = 368.56 − 320 = 48.56 Step V Gain because of coupons = 80 ∗ 4 = 320 Step VI Gain because of decrease in yield Price of bond w/o change in yield at the end of 4 years = 80 ∗ PVIFA 10%, 6 + 1000 ∗ PVIF 10%, 6 = 912.89 Hence gain due to change in yield = 955.14 − 912.89 = 42.25 Gain due to passage of time

= Absolute Returns − Coupon Income − Gain due to change in yield− Reinvestment income

Gain due to passage of time = 446.56 − 48.56 − 320 − 42.25

= 𝟑𝟓. 𝟕𝟗//


Problem #98 Fill in the table below for the following zero-coupon bonds. The face value of each bond is 1,000.

Price Maturity (Years) Yield to Maturity A 5 10%

312.00 20 b 315.00 c 8%

Solution Solving for a Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) Bond Price = 0 + 1000 * PVIF(10%,5) a = 1000 * 0.621 a = 621 Solving for b

312 = 1000

(1 + b)20

312 = 1000 * PVIF(b%,20) We can either solve the above equation using interpolation or we can look at the PVIF table for value of 0.312 for 20 yrs Solving we get b=6% Solving for c

315 = 1000

(1 + 8%)c

315 = 1000 * PVIF(8%,c) We can either solve the above equation using interpolation or we can look at the PVIF table for value of 0.315 for 8%. Solving we get c=15yrs


Problem #99 ABC Ltd. Recently issued 5-year bonds. The bonds pay an annual coupon rate of 10 percent. The bonds are callable in 3 years at a call price equal to 5 percent premium to par value. The par value of the bonds is 1,000. If the yield to maturity is 10 percent what is the price of the bond today and what is yield to call? Solution Bond Price = C * PVIFA (k%, n) + Bn * PVIF (k%,n) However since the YTM and coupon rate is same the Bond Price today is same as face value which is equal to 1000 In order to calculate YTC we can use approximate formula or we can use interpolation Using approximate formula

YTC Approximate = C +

F − Pn

F + P2

YTC Approximate = 100 +

1050 − 10003

1050 + 10002

YTC Approximate = 𝟏𝟏. 𝟑𝟖%


Problem #100 Jagat Industries Ltd. (JIL) has raised 50 crore though an issue of 9% bond. Each bond has a face value of 500 and 10 years term to maturity. As per the terms of the issue each bond is redeemable in four equal installment starting from the end of 7th year. You are required to find out price of the bond if YTM is 13%. Solution FV = 500; Coupon Rate = 9% ; n = 10yrs ; YTM = 13% ; Price=? Amount Redeemend in 7th Yr =125; Pending Amount =375 Amount Redeemend in 8th Yr =125; Pending Amount =250 Amount Redeemend in 9th Yr =125; Pending Amount =125 Amount Redeemend in 10th Yr =125; Pending Amount =0

Year (1) Coupon Received (2)

Redemption Amount (3)

PV Factor @ 13% (4)

Present Value (5)=

4*(3+2) 1 (9% * 500) = 45 .8849 39.82 2 (9% * 500) = 45 .7831 35.23 3 (9% * 500) = 45 .6931 31.18 4 (9% * 500) = 45 .6133 27.95 5 (9% * 500) = 45 .5427 24.42

6 (9% * 500) = 45 .4803 21.61 7 (9% * 500) = 45 125 .4250 72.25 8 (9% * 375) = 33.75 125 .3762 59.72 9 (9% * 250) = 22.5 125 .3329 49.10

10 (9% * 125) = 11.25 125 .2945 40.12 Total 401


Problem #101 The price of a bond just before a year of maturity is $ 5,000. Its redemption value is $ 5,250 at the end of the said period. Interest is $ 350 p.a. The Dollar appreciates by 2% during the said period. Calculate the rate of return. Solution

Absolute Returns = Ending Value − Beginning Value + Dividend/Interest

Beginning Value

Absolute Returns on $ = 5250 − 5000 + 350

5000∗ 100

Absolute Returns on $ = 12% Total Returns on Rs = 12%(1+2%) + 2% Total Returns on Rs = 14.24%

Documents

CA Final SFM Solve Questions on Bond Valuation by Prof GDTPE140