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
Prof Manish Ramuka Topic – Bond Markets Page 1
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
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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 푋 3.890 + 1000 푋 0.650 = 922.3//
Find price of Bond when yield is 7% Since YTM is same as coupon price =1000
1000 − 922.3 Capital gain = 922.3 = ퟖ. ퟒퟐ%//
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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
1294 − 1134.2 Bond A = = ퟏퟒ. ퟒ% 1134.2
Bond B = ퟏퟒ. ퟕ%
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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 k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2 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.
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Problem #6
Solution
10 Current yield = 110
= 9.09%
If yield goes up by 1% New Yield = 10.09%
10 Price = 10.09%
= 99.1080 //
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
= 3.75*PVIFA (3%, 4) + 10000*PVIF (3%, 4)
= 375*3.7171 + 10000*0.88848
= 10278.78 //
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Problem #7
Solution
YTM = 16% Redemption Price = 5% premium Price of a bond = PV of future cash flows
9 9 9 9 10 10 10 = + + + + + + + 1.16 (1.16)2 (1.16)3 (1.16)4 (1.16)5 (1.16)6 (1.16)7 10 14 14 105 + + + (1.16)8 (1.16)9 (1.16)10 (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
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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
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
10 10 Bond Price = 375 ∗ PVIFA (( )%, 4) + 10000 ∗ PVIF (( )%, 4) 2 2
Bond Price = 375 ∗ 3.546 + 10000 ∗ 0.823
Bond Price = 9559.75
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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
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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
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
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 k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
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 k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
Bond Price = 40 * PVIFA (5%, 30) + 1000* PVIF (5%,30) Bond Price = 846.27
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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
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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
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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) Coupon Intrinsic Value of Perpetual Bond = YTM
14.5 Intrinsic Value of Perpetual Bond = = 90.625 0.16
Since market value>intrinsic value we can conclude that the bond is currently overpriced. Hence the bond should not be purchased.
14.5 YTM = = 15.26% 95
b)
Year Beginning Principle Interest Ending Total PV Factor Present Principal Payment Principle CF @ 18% 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
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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 PV@Lower% − Actual PV desired YTM = Lower % + ∗ (Difference in Yield) (PV@Lower% − PV@Higher%)
1160.59−1147.02 YTC = 6% + x (7% - 6%) 1160.59−1112.9
= 6.28%
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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 PV@Lower% − Actual PV desired YTM = Lower % + ∗ (Difference in Yield) (PV@Lower% − PV@Higher%)
1228.8.59−1165.75 YTC = 7.5% + x (8% - 7.5%) 1228.8−1164
= 7.98%
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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
PV@Lower% − Actual PV desired YTM = Lower % + ∗ (Difference in Yield) (PV@Lower% − PV@Higher%)
91.77 − 90 YTM = 9% + x 9% - 8% 91.77 − 88.14
= 9.4876%
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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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
96.11−95 YTM = 14% + * 1% 96.11−92.43
= 14.30%//
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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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
100 − 97.6 YTM = 11% + ∗ (13% − 11%) 100 − 95.27
= 12%
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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.
150 b) Current yield = 1025.86
= 14.62%// c) YTM calculation Calculation Price @ 14% = 1034.3 Price @ 15% = 1000
Using interpolation PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
[ 1034.3−1025.86] YTM = 14% + * (5-14%) [1034.3−1000]
= 14.23%. //
Prof Manish Ramuka Topic – Bond Markets Page 19
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
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
= 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% k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
= 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
PV @ Lower − Actual Desired YTC = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
1263.46 − 1147 YTM = 4% + ∗ (8% − 4%) 1263.46 − 1067.96
YTM = 6.38%//
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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
(F − P) C + YTM = n (F + P) n
110000 − 90000 12000 + = 20 110000 + 90000 2
= ퟏퟑ. ퟎퟎ%//
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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
F − P C + YTM Approximate = n F + P 2
100 − 103.24 10 + YTM = 4 203.24 2
= ퟗ. ퟎퟒ%
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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
13.5 YTC = 12% + ∗ 2% 13.5 + 21.4
퐘퐓퐂 = ퟏퟐ. ퟕퟕퟓ% //
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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
= ퟖ%//
Prof Manish Ramuka Topic – Bond Markets Page 23
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 Face Value Tenure Interest (%) Interest Rs. Rs. (p.a.) 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 DDB* DDB* months IV 10,000 17,800 6 Years 9 DDB* DDB* months 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 − P C + YTM Approximate = n F + P 2
10,000 − 8,500 565 + YTM Approximate = 4 10,000 + 8,500 2 Post Tax YTM = 10.16%
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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 − P C + YTM Approximate = n F + P 2
10,000 − 8,500 700 + YTM Approximate = 6 10,000 + 8,500 2 Post Tax YTM = 10.27%
Bond 3
FV = PV ∗ (1 + Periodic Rate)n∗y 9 (4+ ) 14,750 = 8,500 ∗ (1 + YTM) 12 YTM = 12.31%
Bond 4
FV = PV ∗ (1 + Periodic Rate)n∗y 9 (6+ ) 17,800 = 8,500 ∗ (1 + YTM) 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
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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% k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
= 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% k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
= 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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
1230.55 − 1200 YTM = 3% + ∗ (5% − 3%) 1230.55 − 1131.28
= 3.6155%
Prof Manish Ramuka Topic – Bond Markets Page 26
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
Annual Coupon 140 Current Yield = = = 14.4% Current Price 972
Prof Manish Ramuka Topic – Bond Markets Page 27
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 − P C + YTM Approximate = n F + P 2
1086 − 960 84 + YTM = 5 1086 + 960 2
= ퟏퟎ. ퟔퟕ% 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% Prof Manish Ramuka Topic – Bond Markets Page 28
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 10.5% 11.0% 12.5% 15.25% 18.0% (p.a.)
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
10.5% ∗ 5000 11% ∗ 5000 12.5% ∗ 5000 15.25% ∗ 5000 18% ∗ 5000 5000 = + + + + 1 + YTM 1 1 + YTM 2 1 + YTM 3 1 + YTM 4 1 + YTM 5
By trial and error and using interpolation we get
YTM=13%
Prof Manish Ramuka Topic – Bond Markets Page 29
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
Total of Column 4 Duration = Price
4992.88 = 1000
= 4.99288
= 5 Yrs //
Prof Manish Ramuka Topic – Bond Markets Page 30
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
1818.37 Duration = 964.57
Duration = 1.8852//
Prof Manish Ramuka Topic – Bond Markets Page 31
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
5.3 Modified duration of portfolio = (1+9%)
= 4.8624
10.6 % Change in bond 1 price = ∗ 2 (1 + 9%)
= 19.45%
% change in price of portfolio = 4.8624 x 2 = 9.7248%
Prof Manish Ramuka Topic – Bond Markets Page 32
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
4089.78 Duration = 964.24
= 4.24 //
4.24 Modified Duration = 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 //
Prof Manish Ramuka Topic – Bond Markets Page 33
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
935.77−920.57 % Change = X 100 920.57
= 1.65%//
C) Similar calculation can be performed.
D) Similar calculation can be performed.
Prof Manish Ramuka Topic – Bond Markets Page 34
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
24126 Convexity = 958.42 ∗ (1 + 7%)2
= 21.98//
Prof Manish Ramuka Topic – Bond Markets Page 35
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// Macaulay Duration c) Volatility = (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//
Prof Manish Ramuka Topic – Bond Markets Page 36
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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
968.73 − 950 YTM = 12% + ∗ (12% − 11.5%) 968.73 − 948.61
YTM = 11.97%
Prof Manish Ramuka Topic – Bond Markets Page 37
Step2: Calculate Macaulay duration
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4 5 Yrs CF PV Factor @ PV (3x2) 4 x 1 11.97% 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
4447.4 Macaulay Duration = 950
= 4.6815//
Macaulay Duration Modified 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%//
Prof Manish Ramuka Topic – Bond Markets Page 38
Prof Manish Ramuka Topic – Bond Markets Page 39
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
Macualay Duration Modified Duration = k (1 + 2) 6.72 = 12.5% (1 + 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
Prof Manish Ramuka Topic – Bond Markets Page 40
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
Prof Manish Ramuka Topic – Bond Markets Page 41
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
Prof Manish Ramuka Topic – Bond Markets Page 42
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4 5 Yrs CF PV Factor @ PV (3x2) 4 x 1 10.0% 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
Prof Manish Ramuka Topic – Bond Markets Page 43
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
t ∗ c n ∗ B ∑2n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4 5 Yrs CF PV Factor @ PV (3x2) 4 x 1 10.0% 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
8107 Macaulay Duration = 1000 ∗ 2
= 4.05
Prof Manish Ramuka Topic – Bond Markets Page 44
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
5100 Macaulay Duration = 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
Prof Manish Ramuka Topic – Bond Markets Page 45
Problem #49
Solution
FV = 100000 YTM = 16% Macaulay duration = 4.3202
t ∗ c n ∗ B ∑2n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
1퐶 2퐶 3퐶 4퐶 5퐶 6퐶 100000 ∗ 6 1.6 + 2 + 3 + 4 + 5 + 6 + 6 4.3202 = 1.16 1.16 1.16 1.16 1.16 1.16 3.685퐶 + 41040
15.19C + 177392 = 11.36987C + 246265 4.5402C = 68872 C = 15169.37
Substituting in equation for price we get
Price = 96943.14
Prof Manish Ramuka Topic – Bond Markets Page 46
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
1퐶 2퐶 3퐶 4퐶 5퐶 6퐶 100000 ∗ 6 1.08 + 2 + 3 + 4 + 5 + 6 + 6 4.992 = 1.08 1.08 1.08 1.08 1.08 1.08 4.623퐶 + 63000
Solving we get C = 8000
Substituting in equation for price we get
Price = 1,00,000
Prof Manish Ramuka Topic – Bond Markets Page 47
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.
Prof Manish Ramuka Topic – Bond Markets Page 48
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
PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
982.2 − 970 YTM = 12.5% + ∗ (14% − 12.5%) 982.2 − 931.4
= 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.
Prof Manish Ramuka Topic – Bond Markets Page 49
c)
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
3903 Macaulay Duration = = 4.023 970
Prof Manish Ramuka Topic – Bond Markets Page 50
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
4.44C + 202491 2.6 = 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
Prof Manish Ramuka Topic – Bond Markets Page 51
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 − P C + YTM Approximate = n F + P 2
1000 − 950 107.5 + YTM Approximate = 6 1000 + 950 2
YTM Approximate = ퟏퟏ. ퟖퟖ% YTM Actual = 11.96%
Step 2 Calculate Macaulay Duration
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
4450 Macaulay Duration = = 4.684 950
Prof Manish Ramuka Topic – Bond Markets Page 52
Step 3 Calculate Modified Duration
Macaulay Duration 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
Prof Manish Ramuka Topic – Bond Markets Page 53
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 − P C + YTM Approximate = n F + P 2
1000 − 976.95 90 + YTM Approximate = 6 1000 + 976.95 2
YTM Approximate = ퟗ. ퟓ% YTM Actual = 9.52%
Step 2 Calculate Macaulay Duration
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
Prof Manish Ramuka Topic – Bond Markets Page 54
4763 Macaulay Duration = = 4.87 977
Step 3 Calculate Modified Duration
Macaulay Duration Modified Duration = (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
Prof Manish Ramuka Topic – Bond Markets Page 55
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 PV@Lower% − Actual PV desired YTM = Lower % + ∗ (Difference in Yield) (PV@Lower% − PV@Higher%)
966.47−960 YTC = 15% + x (15.5% - 15%) 966.47−950
= 15.2%
Step 2: Calculate YTM F − P C + YTM Approximate = n F + P 2
1000 − 960 140 + YTM Approximate = 20 1000 + 960 2
YTM Approximate = ퟏퟒ. ퟒퟗ% YTM Actual = 14.63%
Prof Manish Ramuka Topic – Bond Markets Page 56
Step 3 Calculate Macaulay duration
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4 5 Year Cash Flow PV Factor @ 9.52% Present Value (3 x 2) 4 x 1 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
7068 Macaulay Duration = = 7.363 960
Step 3 Calculate Modified Duration
Macaulay Duration Modified Duration = (1 + k)
7.363 = (1 + 14.62%)
= 6.42//
Prof Manish Ramuka Topic – Bond Markets Page 57
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = 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
354.27 Macaulay Duration = = 1.84 192.19
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.
Prof Manish Ramuka Topic – Bond Markets Page 58
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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4=3x2 5 Yrs CF Discount Factor PV 4X1 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
260.37 Macaulay Duration = = 2.49 104.68
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
Prof Manish Ramuka Topic – Bond Markets Page 59
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%
Prof Manish Ramuka Topic – Bond Markets Page 60
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 25.50 19.25 18.25 17.50 19.50 17.50 (Rs Lakhs) 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
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4=3x2 5 Yrs CF Discount Factor PV 4X1 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
245.44 Macaulay Duration = = 2.99 82.16
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.
Prof Manish Ramuka Topic – Bond Markets Page 61
Step 2 Calculate the YTM of each bond using approximate formula
F − P C + YTM Approximate = n F + P 2
Bond X 1050 − 966.38 110 + YTM X = 5 = 12.72% 1050 + 966.38 2
Bond Y 950 − 988.66 130 + YTM Y = 3 = 11.99% 950 + 988.66 2
Step 3: Calculate the duration of each bond
Bond X 1 2 3 4=3x2 5 Yrs CF Discount Factor PV 4X1 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
3960.16 Macaulay Duration = = 4.1 966.38
Bond Y 1 2 3 4=3x2 5 Yrs CF Discount Factor PV 4X1 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
2630.18 Macaulay Duration = = 2.66 988.66
Prof Manish Ramuka Topic – Bond Markets Page 62
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%
Prof Manish Ramuka Topic – Bond Markets Page 63
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 t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0 1 2 3 4=3x2 5 Yrs CF Discount Factor PV 4X1 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
264.31 Macaulay Duration = = 2.04 129.45
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%
Prof Manish Ramuka Topic – Bond Markets Page 64
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 Coupon Duration years (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. 푊퐴퐷퐴 + 푊푆퐷퐵 + 푊퐶퐷퐶 = 6
Solving we get 0.5 X 7.35 + 푊퐵 X 6.15 + 푊퐶 X 4.3 = 6
Also 푊퐴 + 푊퐵 + 푊퐶 = 1 i.e. 푊퐵 + 푊퐶=0.5
We have 2 simultaneous equation solving we get
푊퐵 = 9.5% 푊퐶 = 40.5% //
Prof Manish Ramuka Topic – Bond Markets Page 65
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
푆1 = 5% 1푓1 = 6.5% 1푓2 = 8%
푆3 = ?
0 1 2 3 5 6.5 8 |------|------|------| ------X------
(1 + 푋)3 = (1 + 5%) * (1 + 6.5%) * (1 + 8%)
= 6.49%
Prof Manish Ramuka Topic – Bond Markets Page 66
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%
2 (1 + S2) = 1 + 1f0 ∗ 1 + 1f1 2 (1 + S2) = 1 + 12% (1 + 11.25%)
3 1 + S3 = 1 + 1f0 ∗ 1 + 1f1 ∗ 1 + 2f1 3 1 + S3 = 1.12 1.1125 (1.1075)
C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
90 90 1090 Price = + + (1 + 12%) 1 + 12% (1 + 11.25%) 1.12 1.1125 (1.1075)
= 942.48 //
Since β = 1.02
Price = 942.48*102
= 961.33 //
Prof Manish Ramuka Topic – Bond Markets Page 67
Problem #64
Solution a) Forward rate 1 year from today 2 (1 + S2) = 1 + S1 ∗ 1 + 1f1 (1 + 11.25%)2 = (1 + 10.5%) (1 + X%) 1.11252 1 + X% = = 12% 1.105
Similarly 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 1 + 12% 3 1 + 2f = 1 (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
982.19 − 1000 % Change in bond price = = −ퟏ. ퟖ% 1000
Prof Manish Ramuka Topic – Bond Markets Page 68
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
3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1)
3 (1 + S3) = 1 + 8.5% ∗ 1 + 9.5% ∗ (1 + 13.56%)
퐒ퟑ = ퟏퟎ. ퟓퟖ%
Prof Manish Ramuka Topic – Bond Markets Page 69
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 n S = 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
13 13 13 13 13 + 113 Bond Price = + + + + 1.0873 1 1.0889 2 1.0906 3 1.0923 4 1.0939 5
Bond Price = 122.5
2) Calculate YTM using approximate formula
F − P C + YTM Approximate = n F + P 2
113 − 122.5 13 + YTM = 5 113 + 122.5 2
= ퟗ. ퟒퟐ%
Prof Manish Ramuka Topic – Bond Markets Page 70
3) Calculate Macaulay Duration
t ∗ c n ∗ B ∑n + n t=1 (1 + k)t (1 + k)t Macaulay Duration = B0
1 2 3 4 5 Year Cash Flow PV Factor @ 9.42% Present Value (3 x 2) 4 x 1 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
501.3 Macaulay Duration = 122.06
= 4.11//
4) Calculate Modified Duration
Macaulay Duration 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
Prof Manish Ramuka Topic – Bond Markets Page 71
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 FV Price = n 1 + Sn 1 year bond 1000 930.23 = (1 + S1) Solving we get S1 = 7.5%
2 year bond 1000 923.79 = 2 (1 + S2) Solving we get S2 = 4.04%
3 year bond 1000 919.54 = 3 (1 + S3) 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 2 (1 + S2) = 1 + S1 ∗ 1 + 1f1 2 1 + 5.5% = 1 + 5% 1 + 1f1 (1.055)2 (1 + 1f ) = 1 1.05 ퟏ퐟ퟏ = ퟔ%
2 year forward rate 3 2 (1 + S3) = 1 + S2 ∗ (1 + 2f1) 3 2 (1 + 6.5%) = (1 + 5.5%) ∗ (1 + 2f1) ퟐ퐟ퟏ = ퟖ. ퟓퟑ%
3 year forward rate ퟑ퐟ퟏ = ퟖ. ퟓퟏ%
Prof Manish Ramuka Topic – Bond Markets Page 72
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 2 (1 + S2) = 1 + S1 ∗ 1 + 1f1
S2 = 1 + 10% 1 + 9.5 − 1 S2 = 9.75%
3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 S3 = 1 + 10% 1 + 9.5% 1 + 9% − 1 S3 = 9.5%
4 (1 + S4) = 1 + S1 ∗ 1 + 1f1 ∗ 1 + 2f1 ∗ (1 + 3f1) 4 S4 = 1.1 ∗ 1.095 ∗ 1.09 ∗ 1.085 S4 = 9.2%
Prof Manish Ramuka Topic – Bond Markets Page 73
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 FV Price = n 1 + Sn 1 year bond 1000 909.09 = (1 + S1) Solving we get S1 = 10%
2 year coupon bond price is given as
C C + FV Price = + 2 (1 + S1) (1 + S2)
100 1100 991.81 = + 2 (1 + 10%) (1 + S2)
Solving we get S2 = 10.5%
3 year coupon bond price is given as C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
100 100 1100 997.18 = + 2 + 3 (1 + 10%) (1 + 10.5%) (1 + S3)
Solving we get S3 = 10.09%
1 years forward Rate calculation 2 (1 + S2) = 1 + S1 ∗ 1 + 1f1 2 (1 + 10.5%) = 1 + 10% ∗ 1 + 1f1 ퟏ퐟ퟏ = ퟏퟏ%
2 years forward Rate calculation 3 2 (1 + S3) = 1 + S2 ∗ (1 + 2f1) 3 2 (1 + 10.09%) = (1 + 10.5%) ∗ (1 + 2f1) ퟐ퐟ퟏ = ퟗ. ퟒ%
Prof Manish Ramuka Topic – Bond Markets Page 74
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 FV Price = n 1 + Sn
1000 S = − 1 = 7.07% 1 934
2 year coupon bond price is given as
C C + FV Price = + 2 (1 + S1) (1 + S2)
100 1100 985 = + 2 (1 + 7.07%) 1 + S2
Solving we get S2 = 11.07
3 year coupon bond price is given as C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
120 120 1120 1010 = + 2 + 3 (1 + 7.07%) 1 + 11.07 1 + S3
Solving we get S3 = 11.84
Prof Manish Ramuka Topic – Bond Markets Page 75
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 - 퐅ퟏퟐ = 9.3%, 퐅ퟐퟑ = 9.5%, IV = 109.46
Solution
n S = 9.0 + n 10
Year Spot Rate 1 9.1 2 9.2 3 9.3
1) Calculate Bond Price
Bond Price = Present Value of Future Cash Flows
C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
10 10 120 Bond Price = + + 1.091 1 1.092 2 1.093 3
Bond Price = 109.45
2) 1 years forward Rate calculation 2 1 + S2 = 1 + S1 ∗ 1 + 1f1 2 1 + 9.2% = 1 + 9.1% ∗ 1 + 2f1 Solving we get 1f1 = 9.3%
2 years forward Rate calculation 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 (1 + 9.3) = 1 + 9.1% ∗ 1 + 9.3% ∗ (1 + 2f1%) Solving we get 2f1 = 9.5%
Prof Manish Ramuka Topic – Bond Markets Page 76
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
n S = 9.0 + n 10
Year Spot Rate 1 9.1 2 9.2 3 9.3
1) Calculate Bond Price
Bond Price = Present Value of Future Cash Flows
C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
120 120 1120 Bond Price = + + 1.091 1 1.092 2 1.093 3
Bond Price = 1068.3
Prof Manish Ramuka Topic – Bond Markets Page 77
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 - 퐟ퟎퟏ = 퐫ퟎퟏ = 10%, 퐟ퟏퟐ = 4.25%, 퐟ퟐퟑ = 7.54%, 퐟ퟏퟑ = 12.12%, 퐫ퟎퟏ = 10%, 퐫ퟎퟐ = 7.09%, 퐫ퟎퟑ = 7.24%
Solution
1100 S = − 1 = 10% 1 100
2 year coupon bond price is given as C C + FV Price = + 2 (1 + S1) (1 + S2) 10 110 105 = + 2 (1 + 10%) 1 + S2 Solving we get S2 = 7.09%
3 year coupon bond price is given as C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3 10 10 110 107 = + 2 + 3 (1 + 10%) 1 + 7.09% 1 + S3 Solving we get S3 = 7.24%
1 years forward Rate calculation 2 1 + S2 = 1 + S1 ∗ 1 + 1f1 2 1 + 7.09% = 1 + 10% ∗ 1 + 1f1 Solving we get 1f1 = 4.256% 2 years forward Rate calculation 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 (1 + 7.24) = 1 + 10% ∗ 1 + 4.256% ∗ (1 + 2f1%) Solving we get 2f1 = 7.54%
Prof Manish Ramuka Topic – Bond Markets Page 78
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 - 퐟ퟒퟓ = 4.18%
Solution
4 925 ∗ 1 + S4 = 1000 Solving we get S4 = 1.968%
5 900 ∗ 1 + S5 = 1000 Solving we get S5 = 2.13%
5 4 1 + S5 = 1 + S4 ∗ (1 + 4f1) Solving we get 4f1 = 2.78%
Prof Manish Ramuka Topic – Bond Markets Page 79
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 - 퐫ퟎퟏ = 11.11%, 퐫ퟎퟐ = 11.8%, 퐫ퟎퟑ = 12.6%, 퐟ퟎퟏ = 11.11%, 퐟ퟏퟐ = 12.49%, 퐟ퟐퟑ = 14.22%, 퐟ퟏퟑ = 28.49%
Solution For zero coupon bonds bond price is calculated using following formula FV Price = n 1 + Sn
1 year Zero Coupon Bond 100 S = − 1 = 11.11% 1 90
2 year Zero Coupon Bond 100 80 = 2 1 + S2 Solving we get S2 = 11.8%
3 year Zero Coupon Bond 100 70 = 3 1 + S3 Solving we get S3 = 12.6%
1 year forward Rate calculation 2 1 + S2 = 1 + S1 ∗ 1 + 1f1 2 1 + 11.8% = 1 + 11.11% ∗ 1 + 1f1 Solving we get 1f1 = 12.49%
2 year forward Rate calculation 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 (1 + 12.6) = 1 + 11.11% ∗ 1 + 12.49% ∗ (1 + 2f1%) Solving we get 2f1 = 14.22%%
Prof Manish Ramuka Topic – Bond Markets Page 80
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
1,00,000 S = − 1 1 91,500
Solving we get S1 = 9.23%
2 year coupon bond price is given as
C C + FV Price = + 2 (1 + S1) (1 + S2)
10,000 1,10,000 98,500 = + 2 (1 + 9.23%) 1 + S2
Solving we get S2 = 10.96%
3 year coupon bond price is given as C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
10,500 10,500 1,10,500 99,000 = + 2 + 3 (1 + 9.23%) 1 + 10.96% 1 + S3
Solving we get S3 = 10.97%
1 year forward Rate calculation 2 1 + S2 = 1 + S1 ∗ 1 + 1f1 2 1 + 10.96% = 1 + 9.23% ∗ 1 + 1f1 Solving we get 1f1 = 12.72%
2 year forward Rate calculation 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 (1 + 10.97) = 1 + 9.23% ∗ 1 + 12.72% ∗ (1 + 2f1%) Solving we get 2f1 = 10.99%
Prof Manish Ramuka Topic – Bond Markets Page 81
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
1,00,000 S = − 1 1 91,000
Solving we get S1 = 9.89%
2 year coupon bond price is given as
C C + FV Price = + 2 (1 + S1) (1 + S2)
10,500 1,10,500 99,000 = + 2 (1 + 9.89%) 1 + S2
Solving we get S2 = 11.15%
3 year coupon bond price is given as C C C + FV Price = + 2 + 3 (1 + S1) (1 + S2) 1 + S3
11,000 11,000 1,11,000 99,500 = + 2 + 3 (1 + 9.89%) 1 + 11.15% 1 + S3
Solving we get S3 = 11.26%
4 year coupon bond price is given as C C C C FV Price = + 2 + 3 + 4 (1 + S1) (1 + S2) 1 + S3 (1 + S4)
11,500 11,500 11,500 1,11,500 99,900 = + 2 + 3 + 4 (1 + 9.89%) 1 + 11.15% 1 + 11.26% 1 + S4
Solving we get S4 = 11.64%
Prof Manish Ramuka Topic – Bond Markets Page 82
1 year forward Rate calculation 2 1 + S2 = 1 + S1 ∗ 1 + 1f1 2 1 + 11.15% = 1 + 9.89% ∗ 1 + 1f1 Solving we get 1f1 = 12.42%
2 year forward Rate calculation 3 (1 + S3) = 1 + S1 ∗ 1 + 1f1 ∗ (1 + 2f1) 3 (1 + 11.26) = 1 + 9.89% ∗ 1 + 12.42% ∗ (1 + 2f1%) Solving we get 2f1 = 11.48%
3 year forward Rate calculation 4 3 (1 + S4) = (1 + S3) ∗ 1 + 3f1 4 3 1 + 11.64% = 1 + 11.26% ∗ 1 + 3f1% 3f1 = 12.78%
Prof Manish Ramuka Topic – Bond Markets Page 83
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
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
12 12 Clean Price = 50 ∗ PVIFA (( )%, 2 ∗ 7.5) + 1000 ∗ PVIF ( )%, 2 ∗ 7.5) 2 2
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
952.87 Dirty Price = = 916.22 1 + 6% 4/6
Prof Manish Ramuka Topic – Bond Markets Page 84
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
Prof Manish Ramuka Topic – Bond Markets Page 85
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
1,00,646.55 Dirty Price = = 90,630.74 1 + 15% 9/12
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.
Prof Manish Ramuka Topic – Bond Markets Page 86
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
k k Bond Price = Coupon ∗ PVIFA (( )%, 2n) + Bn ∗ PVIF (( )%, 2n) 2 2
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
1195.5 Dirty Price = = 1161.16 1 + 6% 3/6
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
Prof Manish Ramuka Topic – Bond Markets Page 87
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
Prof Manish Ramuka Topic – Bond Markets Page 88
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
9 = 0.3 * (PVIFA) (7%, 6) 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.
Prof Manish Ramuka Topic – Bond Markets Page 89
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 = * 0.3 x PVIFA (7%, 10) 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.
Prof Manish Ramuka Topic – Bond Markets Page 90
Problem #84
Prof Manish Ramuka Topic – Bond Markets Page 91
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
Market Price of Bond Conversion price = Conversion Rate
1475 Conversion Price = 14
= 105.3571
(Conversion Price – Current Share Price) Conversion Premium = Current Share Price
105.3571−80 Conversion premium = 80
= 31.7% //
Prof Manish Ramuka Topic – Bond Markets Page 92
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 //
Prof Manish Ramuka Topic – Bond Markets Page 93
Problem #87
Solution
Stock value of bond = Current Market Price * Conversion ratio = 20*12 = 240
(Market Price – Straight Value) Downside Risk = Straight Value
265−235 = 235
= 12.77%
(Conversion Price – Current Share Price) Conversion Premium = Current Share Price
13.25−12 = 12
= 10.42%
Current Market Price of Bond Conversion Parity = Conversion Ratio
265 = 20
= 13.25 Prof Manish Ramuka Topic – Bond Markets Page 94
Problem #88
Solution
Conversion ratio = 10
(Conversion Price – Current Share Price) Conversion Premium = Current Share Price
OR
(Current Market Price of Bond – Conversion Value) Conversion Premium = Conversion Value
5400−(430∗10) = (430∗10)
= 25.58% //
Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 430*10 = 4300 //
Prof Manish Ramuka Topic – Bond Markets Page 95
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
Market Price of Bond Conversion price = Conversion Rate
1500 Conversion price = = 75 20
(Conversion Price – Current Share Price) Conversion Premium = Current Share Price
(75 − 60) Conversion Premium = ∗ 100 60
Conversion Premium = ퟐퟓ%
Prof Manish Ramuka Topic – Bond Markets Page 96
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) Market Price of Bond Conversion price = Conversion Rate
925 Conversion price = = 37 25 c) (Conversion Price – Current Share Price) Conversion Premium = Current Share Price
(37 − 30) Conversion Premium = ∗ 100 = 23.33% 30 d) (Market Price of Bond – Straight Value of Bond ) Premium over straight value = Straight Value of Bond
(925 − 730) Premium over straight value = ∗ 100 = 26.71% 730
Prof Manish Ramuka Topic – Bond Markets Page 97
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) Market Price of Bond Conversion price = Conversion Rate
900 Conversion price = = 30 30 c) (Conversion Price – Current Share Price) Conversion Premium = Current Share Price
(30 − 25) Conversion Premium = ∗ 100 = 20% 25 d) (Market Price of Bond – Straight Value of Bond ) Premium over straight value = Straight Value of Bond
(900 − 700) Premium over straight value = ∗ 100 = 28.57% 700
Prof Manish Ramuka Topic – Bond Markets Page 98
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)
Market Price of Bond Conversion price = Conversion Rate
10800 Conversion price = = 540 20
b) (Conversion Price – Current Share Price) Conversion Premium = Current Share Price
(540 − 480) Conversion Premium = ∗ 100 = 12.50% 480 c) Conversion Value Stock Value = Current Market Price ∗ Conversion Ratio = 480*20 = 9600//
Prof Manish Ramuka Topic – Bond Markets Page 99
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
Prof Manish Ramuka Topic – Bond Markets Page 100
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
Prof Manish Ramuka Topic – Bond Markets Page 101
Category #9: Mixed
Problem #94
Solution 14 a)Current Yield = 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 PV @ Lower − Actual Desired YTM = Low % + ∗ (High % − Low %) PV @ Lower − PV @ Higher
96.64−90 YTM = 15% + * 3% 96.64−87.49
= 15% + 2.177%
= 17.17%
Prof Manish Ramuka Topic – Bond Markets Page 102
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 = (1 + X)5 90
1.8889 = (1 + X)5
Solving we get
X = 13.56%
Prof Manish Ramuka Topic – Bond Markets Page 103
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%
Prof Manish Ramuka Topic – Bond Markets Page 104
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
5097.74 Duration = 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
4366.45 New Duration = 5.2025 Years 868.85
Since duration is inversely proportion to YTM.
Prof Manish Ramuka Topic – Bond Markets Page 105
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
Prof Manish Ramuka Topic – Bond Markets Page 106
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//
446.56 % 푔푎푖푛 = 877.1087
= 50.91% Prof Manish Ramuka Topic – Bond Markets Page 107
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 = ퟑퟓ. ퟕퟗ//
Prof Manish Ramuka Topic – Bond Markets Page 108
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
1000 312 = (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
1000 315 = (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
Prof Manish Ramuka Topic – Bond Markets Page 109
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
F − P C + YTC Approximate = n F + P 2
1050 − 1000 100 + YTC Approximate = 3 1050 + 1000 2
YTC Approximate = ퟏퟏ. ퟑퟖ%
Prof Manish Ramuka Topic – Bond Markets Page 110
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 Redemption PV Factor @ Present (2) Amount (3) 13% (4) 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
Prof Manish Ramuka Topic – Bond Markets Page 111
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
Ending Value − Beginning Value + Dividend/Interest Absolute Returns = Beginning Value
5250 − 5000 + 350 Absolute Returns on $ = ∗ 100 5000
Absolute Returns on $ = 12%
Total Returns on Rs = 12%(1+2%) + 2% Total Returns on Rs = 14.24%
Prof Manish Ramuka Topic – Bond Markets Page 112