Bond Valuations (1)

Fill in the inputs in yellow coloured boxes.This model only works with Bonds of

value 100 INR/USD

Inputs

Market Price INR 104.10Face value INR 100.00

Rate 8.4000% 0.08400NPER 2 2YTM 7.77 0.077718

Settelment date 25-Mar-2015Maturity date 28-Jun-2024

Coup Numbers 9.2583333333Coup days till now 87

Coup days to next payment 93No of days in coupan period 180

Coupan INR 8.40

OutputsYTM 0.0777

Accrued Interest INR 4.0600Clean Price USD 104.1008Dirty Price INR 108.1608

Quoted Price

To find the YTM , fill the market price in input box's first box

InputsSettlement Date 31-Dec-2015Maturity Date 16-Dec-2033

Rate 8.91%Effective Rate 0.0891

Coupan Amount INR 89.10YTM 7.2644%

Effective YTM 0.072644Face Value INR 1,000.00

Market priceFrequency 1

Periods 17.96Days from last Coupan 15

Days to next coupan 351Total days in Coupan 366

In order to calculate price of a bond on date which is not a coupan date,accrued interest has to be added. Bonds prices are quoted as dirty price which means it includes accrued interest of days from last coupan to

settlement date.Below is the calculator for calculating dirty price by inputting details in yellow boxes

OutputsPV of Coupans INR 878.46

PV of Face Value INR 283.78

Clean Price INR 1,162.24

Accrued Interest INR 3.65

Dirty Price INR 1,165.90

YTM Err:523

In order to calculate price of a bond on date which is not a coupan date,accrued interest has to be added. Bonds prices are quoted as dirty price which means it includes accrued interest of days from last coupan to

settlement date.Below is the calculator for calculating dirty price by inputting details in yellow boxes

This amount is the accrued interest, which is paid to the seller of bond for holding that bond upto settelment date

This price is the sum of pv of coupans, face value and accrued interest

discountInputs

Settelment date 1-Jan-2000

Maturity date 1-Jan-2005

Frequency 1Rate 7.00%YTM 12.00%

Face Value INR 1,000.00Amount of Coupan INR 70.00

coupan numbers 5Pv of coupans INR 252.33

PV of FV INR 567.43 Dirty Price INR 819.76

Term to maturityDuration

Modified Duration

Rupee Duration

PVBP

To find out the price senstivity to yield, Inputs are filled in yellow coloured boxes . Now in Output Boxes (blue coloured boxes) , we can see that duration (wieghted ,Mduration and Rupee duration will be calculated which can be used to forcast the changes in price of bond due to change in yield.We can verify this by actually calculating the effect on price due to change in yield which is shown in table 2.

PVBT is the interest rate senstivity of bond due to change in 1 basis point in yield.This means that change in price due to change in a basis

point(.01) is .3158 rupee.

the value of beta will explain the senstivity of the bondthe bond senstivity will explain the change in the valuation of the bond wrt to change in the int rate in the market

0 it is to be noted that the senstivity of a bond will explain the risk attach to a bond.if a bond is highly sensitive it means the risk attach to the bond is high therfore one will prefer the low sensitive bond if risk appetite of the investors is less.a high duration in the bond is observed when life of the bond is of higher values on the contrary no duration bond will be prefered if in case the life of the bond is less

discount premium parOutputs

1-Jan-2000 1-Jan-2000 1-Jan-2000

1-Jan-2005 1-Jan-2005 1-Jan-2005

1 1 10.07 15.00% 12.00%

0.12000 12.00% 12.00%INR 1,000.00 INR 1,000.00 INR 1,000.00

INR 70.00 INR 150.00 INR 120.005 5 5

INR 540.72 INR 432.57 ($1,108.14)INR 567.43 INR 567.43

INR 1,108.14 INR 1,000.00 Outputs

5.004.31 819.76 819.763.85 0.0000015

0.3158

0.0032


PVBT is the interest rate senstivity of bond due to change in 1 basis point in yield.This means that change in price due to change in a basis

point(.01) is .3158 rupee.

the bond senstivity will explain the change in the valuation of the bond wrt to change in the int rate in the market it is to be noted that the senstivity of a bond will explain the risk attach to a bond.if a bond is highly sensitive it means the risk attach to the bond is high therfore one will prefer the low sensitive bond if risk appetite of the investors is less.a high duration in the bond is observed when life of the bond is of higher values on the contrary no duration bond will be prefered if in case the life of the bond is less

Table 2

Price senstivity to Yield

When Yield Increases

Yield 12.010%Price INR 819.45

change In Price 0.3157


Here we can see that change in the price of bond ,due to change in yield is appproximately similar to rupee duration or PVBT(in case increase of 1 BPS).And If we calculate the

percentage change in price due to change in yield, it will be similar to Mduration/PVBT

We can change the yield by .01 BPS or 1% and Change in price is not exactly equal to rupee duration/PVBP because of the impact of convexicity. An increase in a bond’s yield to maturity

results in a smaller price decline than the price gain associated with a decrease of equal magnitude in yield.

if a bond is highly sensitive it means the risk attach to the bond is high therfore one will prefer the low sensitive bond if risk appetite of the investors is less.a high duration in the bond is observed when life of the bond is of higher values on the contrary no duration bond will be prefered if in case the life of the bond is less

Table 2

Price senstivity to Yield

When yield decreases

11.000%INR 852.16

32.4029

Here we can see that change in the price of bond ,due to change in yield is appproximately similar to rupee duration or PVBT(in case increase of 1 BPS).And If we calculate the

percentage change in price due to change in yield, it will be similar to Mduration/PVBT

.01 BPS or 1% and Change in price is not exactly equal to rupee because of the impact of convexicity. An increase in a bond’s yield to maturity

results in a smaller price decline than the price gain associated with a decrease of equal magnitude in yield.

Set date 29-Mar-2001

Freq Maturity Yield Rate Coupan

100 1 25-Aug-2001 0.0860 11.75% INR 11.75 214100 1 9-Jan-2002 0.0741 11.15% INR 11.15 80100 1 7-Apr-2003 0.0915 11.10% INR 11.10 352100 1 23-Mar-2004 0.0925 12.50% INR 12.50 6100 1 12-Aug-2005 0.0942 11.19% INR 11.19 227100 1 10-Apr-2006 0.0974 11.68% INR 11.68 349100 1 28-May-2007 0.0984 11.90% INR 11.90 301100 1 31-Aug-2008 0.0992 11.40% INR 11.40 209100 1 7-Apr-2009 0.1028 11.99% INR 11.99 352100 1 28-Jul-2010 0.1018 11.30% INR 11.30 241100 1 29-Jan-2011 0.1050 12.32% INR 12.32 60100 1 20-Aug-2013 0.1074 12.40% INR 12.40 219

Suppose we have a portfolio of 10 bonds with different coupan, expiry and ytm on settelment date 29 mar 2001. We want to see that by how much amount our portfolio will increase/decrease ,due to change in ytm.We have to calculate the wieghted duration of portfolio according to the price of a particular bond's wieght in total portfolio value.And then we will calculate the mduration of portfolio which will tell us that by what percentage our

portfolio value will increase/decrease with changes in YTM.

Bond Face value

Days for accrued interest

When yield of the portfolio bonds is increased by 1 %, portfolio value is decreased by approximately 3.5 % , which is exactly equal to mduration of the

portfolio(shown in right)

Dirty price M Dur Wieght

0.41 6.98 INR 101.21 INR 108.19 0.37 0.0790.78 2.48 INR 102.73 INR 105.21 0.72 0.0772.02 10.85 INR 103.46 INR 114.31 1.59 0.0842.98 0.21 INR 108.15 INR 108.36 2.45 0.0794.37 7.06 INR 106.11 INR 113.17 3.19 0.0835.03 11.32 INR 107.44 INR 118.76 3.39 0.0876.16 9.95 INR 109.20 INR 119.15 3.99 0.0877.42 6.62 INR 107.52 INR 114.14 4.70 0.0838.02 11.72 INR 109.05 INR 120.77 4.65 0.0889.33 7.56 INR 106.55 INR 114.11 5.37 0.0839.83 2.05 INR 110.84 INR 112.89 5.69 0.083

12.39 7.54 INR 111.09 INR 118.63 6.11 0.087

1367.69 1.00

% change


portfolio value will increase/decrease with changes in YTM.

Term to maturity

Accrued interest

Clean Price

When yield of the portfolio bonds is increased by 1 %, portfolio value is decreased by approximately 3.5 % , which is exactly equal to mduration of the

portfolio(shown in right)

Change in value

0.030 107.800.056 104.460.133 112.520.194 105.750.264 109.650.295 114.840.347 114.540.392 108.980.411 115.350.448 108.240.470 106.740.530 111.74

3.57 1320.61

47.084

3.443


Weighted M Dur

New after

increase in

YTM+1%

Case Example

Repayment Details PeriodLoan 500000 0.00Rate 0.08 1.00time 5 2.00

Amount due in 5 years 734664.038 3.004.005.00

Here pension fund which has to pay back pension fund of Rs. 500000- to one of its investor, with guaranteed rate of 8% after 5 years. So, it is obligated to pay Rs. 500000 *(1.08) ^ 5=734664 in 5th year. So, suppose, pension fund company chooses to fund its obligation with Rs. 20,000 , of 8% annual coupon bond selling at par value with 6 years maturity(duration of 5 years). So, if interest rate remains at 8% the amount accrued will exactly be equal to the obligation of Rs.734664 in 5 years. Now we consider two scenarios, where interest rate goes

down to 7% and in second case it reaches 9%. In 7% scenario, amount accrued will be equal to Rs. 734702.46 in 5 years and in 9% scenario it will be Rs. 734801.27in 5 years.So if he will match the duration of his liabilities with the wieghted duration of bond, he is said to be immunized against interest rate risk

Case Example

YieldInflows 0.08 0.07 0.09

40000.00 54419.56 52431.84 56463.2640000.00 50388.48 49001.72 51801.1640000.00 46656.00 45796.00 47524.0040000.00 43200.00 42800.00 43600.0040000.00 40000.00 40000.00 40000.00540000.00 500000.00 504672.90 495412.84

734664.04 734702.46 734801.27

Here pension fund which has to pay back pension fund of Rs. 500000- to one of its investor, with guaranteed rate of 8% after 5 years. So, it is obligated to pay Rs. 500000 *(1.08) ^ 5=734664 in 5th year. So, suppose, pension fund company chooses to fund its obligation with Rs. 20,000 , of 8% annual coupon bond selling at par value with 6 years maturity(duration of 5 years). So, if interest rate remains at 8% the amount accrued will exactly be equal to the obligation of Rs.734664 in 5 years. Now we consider two scenarios, where interest rate goes

down to 7% and in second case it reaches 9%. In 7% scenario, amount accrued will be equal to Rs. 734702.46 in 5 years and in 9% scenario it will be Rs. 734801.27in 5 years.So if he will match the duration of his liabilities with the wieghted duration of bond, he is said to be immunized against interest rate risk

Cash flows at the end of period are discounted by yield prevailing at that time because he will be repaying loan in fifth year beginning

Table 1

PAR VAL 100 Period Coupan Price BEY DF0.5 NA 96.15 0.0801 1.0400 0.51 NA 92.19 0.0830 1.0415 1

1.5 8.5 99.45 0.0893 1.0447 1.52 9 99.64 0.0925 1.0462 2

2.5 11 103.49 0.0947 1.0473 2.53 9.5 99.49 0.0979 1.0489 3

3.5 10 100 0.1013 1.0506 3.54 10 98.72 0.1059 1.0530 4

4.5 11.5 103.16 0.1085 1.0542 4.55 8.75 92.24 0.1102 1.0551 5

5.5 10.5 98.38 0.1117 1.0559 5.56 11 99.14 0.1158 1.0579 6

6.5 8.5 86.94 0.1174 1.0587 6.57 8.25 84.24 0.1199 1.0600 7

7.5 11 96.09 0.1240 1.0620 7.58 6.5 72.62 0.1228 1.0614 8

8.5 8.75 82.97 0.1255 1.0627 8.59 13 104.3 0.1315 1.0658 9

9.5 11.5 95.06 0.1338 1.0669 9.510 12.5 100 0.1362 1.0681 10

We need to look up the price of a 10 yr 10% Treasury note. But, there are no Treasuries with a maturity of 10 yrs that have a 10% coupon rate. we know that you can’t price it the same as an 8%

bond, nor the same as a 9 or 11 yr bond.We have put together a series of bonds maturing at different times every 6 months(Table1).We will discount the first two coupans of 1.5 year bond with yields of first two bonds and by using solver we will find the third discount rate for 1.5 year bond. keep using

solver upto 10 bonds six months rate.Then we can price the 10 year 10% bond mentioned above

Set these boxes as changing cell for every bond ,so solver could find a solution which will make the total of pv of

cashflows same as price

Set these boxes as changing cell for every bond ,so solver could find a solution which will make the total of pv of

cashflows same as price

COUP RATE NA NA 8.50 9.00 11.00 9.50 10.00 10.00

Coupans 4.25 4.50 5.50 4.75 5.00 5.00

Periods 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00

96.15 92.19 4.09 4.33 5.29 4.57 4.81 4.81

0.5 BOND 96.15 92.19 3.92 4.15 5.07 4.38 4.61 4.61

1YEAR BOND 92.19 91.45 3.95 4.82 4.17 4.39 4.39

PRICE OF 1.5 YEAR BOND 99.45 87.22 4.59 3.96 4.17 4.17

PRICE OF 2 YEAR BOND 99.64 83.72 3.77 3.97 3.97

PRICE OF 2.5 YEAR BOND 103.49 78.64 3.75 3.75

PRICE OF 3 YEAR BOND 99.49 74.30 3.54

PRICE OF 3.5 YEAR BOND 100.00 69.48

PRICE OF 4 YEAR BOND 98.72

PRICE OF 4.5 YEAR BONDPRICE OF 5 YEAR BOND






We need to look up the price of a 10 yr 10% Treasury note. But, there are no Treasuries with a maturity of 10 yrs that have a 10% coupon rate. we know that you can’t price it the same as an 8%

bond, nor the same as a 9 or 11 yr bond.We have put together a series of bonds maturing at different .We will discount the first two coupans of 1.5 year bond with yields of

first two bonds and by using solver we will find the third discount rate for 1.5 year bond. keep using solver upto 10 bonds six months rate.Then we can price the 10 year 10% bond mentioned above

To find out the 6 month rate after 1 year,Set this box as target cell and put value of its price from Table 1

in solver

We will discount the first coupan with 0.5 years zero coupan bond's yield and next coupan by 1 year zero coupan bond's

yield.For last coupan and principal's discounting factor at the end of 1.5 years , we will use solver by setting target cell as same to

price of 1.5 year bond given and setting discount factor as changing value for the third and last cash flows

11.50 8.75 10.50 11.00 8.50 8.25 11.00 6.50 8.75 13.00

5.75 4.38 5.25 5.50 4.25 4.13 5.50 3.25 4.38 6.50

4.50 5.00 5.50 6.00 6.50 7.00 7.50 8.00 8.50 9.00

5.53 4.21 5.05 5.29 4.09 3.97 5.29 3.12 4.21 6.25

5.30 4.03 4.84 5.07 3.92 3.80 5.07 3.00 4.03 5.995.04 3.84 4.61 4.82 3.73 3.62 4.82 2.85 3.84 5.704.80 3.65 4.38 4.59 3.55 3.44 4.59 2.71 3.65 5.434.56 3.47 4.17 4.36 3.37 3.27 4.36 2.58 3.47 5.164.32 3.28 3.94 4.13 3.19 3.10 4.13 2.44 3.28 4.884.07 3.10 3.72 3.89 3.01 2.92 3.89 2.30 3.10 4.603.81 2.90 3.47 3.64 2.81 2.73 3.64 2.15 2.90 4.3065.73 2.72 3.26 3.42 2.64 2.56 3.42 2.02 2.72 4.04103.16 61.04 3.07 3.22 2.49 2.41 3.22 1.90 2.56 3.80

PRICE OF 5 YEAR BOND 92.24 57.87 3.02 2.34 2.27 3.02 1.79 2.41 3.57

PRICE OF 5.5 YEAR BOND 98.38 53.68 2.16 2.10 2.80 1.65 2.23 3.31

PRICE OF 6 YEAR BOND 99.14 49.65 1.96 2.62 1.55 2.08 3.10

PRICE OF 6.5 YEAR BOND 86.94 46.08 2.43 1.44 1.94 2.88

PRICE OF 7 YEAR BOND 84.24 42.78 1.32 1.77 2.64

PRICE OF 7.5 YEAR BOND 96.09 39.80 1.69 2.51

PRICE OF 8 YEAR BOND 72.62 37.10 2.31

PRICE OF 8.5 YEAR BOND 82.97 33.84

PRICE OF 9 YEAR BOND 104.30PRICE OF 9.5 YEAR BOND

PRICE OF 10 YEAR BOND

We will discount the first coupan with 0.5 years zero coupan bond's yield and next coupan by 1 year zero coupan bond's

yield.For last coupan and principal's discounting factor at the end of 1.5 years , we will use solver by setting target cell as same to

price of 1.5 year bond given and setting discount factor as changing value for the third and last cash flows

11.50 12.50

5.75 6.25

9.50 10.00

5.53 6.01

5.30 5.765.04 5.484.80 5.224.56 4.964.32 4.694.07 4.423.81 4.143.57 3.883.36 3.663.16 3.442.93 3.182.74 2.982.54 2.772.33 2.532.22 2.412.04 2.221.83 1.9930.91 1.8395.06 28.44

PRICE OF 10 YEAR BOND 100.00

Period Forward rate Disc factor Coupan PV0.5 0.0801 1.0400 5 4.80751 0.0830 1.0415 5 4.6095

1.5 0.0893 1.0447 5 4.38592 0.0925 1.0462 5 4.1731

2.5 0.0947 1.0473 5 3.96763 0.0979 1.0489 5 3.7539

3.5 0.1013 1.0506 5 3.53824 0.1059 1.0530 5 3.3088

4.5 0.1085 1.0542 5 3.10805 0.1102 1.0551 5 2.9243

5.5 0.1117 1.0559 5 2.74946 0.1158 1.0579 5 2.5441

6.5 0.1174 1.0587 5 2.38137 0.1199 1.0600 5 2.2128

7.5 0.1240 1.0620 5 2.02758 0.1228 1.0614 5 1.9274

8.5 0.1255 1.0627 5 1.77749 0.1315 1.0658 5 1.5889

9.5 0.1338 1.0669 5 1.461310 0.1362 1.0681 105 28.1068

Price 85.3537

From the bootstrapped forward rates, we can valuate 10 year 10% treasury bond

Instrument Day count conv Excel key

Treasury bills Actual/365 3

Yield 7.35

Yield 0.09428Time 182

Par 100

Price 95.51

The implicit yield inthe T-bill is the rate at which the issue price (which is the cut-off price in theauction) has to be compounded, for the number of days to maturity, to equal

the maturity value.Example the price of a 91 day Treasury bill at issue is Rs.98.20, the yield on the same would be

A 182-day T-bill, auctioned on January 18, at a price of Rs.95.510 would have an implicit yield of 9.4280% .Its value is computed as follows

Auction day(91) Auction day(182)

Every WednesdayWednesday preceeding non

reporting Friday

The implicit yield inthe T-bill is the rate at which the issue price (which is the cut-off price in theauction) has to be compounded, for the number of days to maturity, to equal

the maturity value.Example the price of a 91 day Treasury bill at issue is Rs.98.20, the yield on the same would be

A 182-day T-bill, auctioned on January 18, at a price of Rs.95.510 would have an implicit yield of 9.4280% .Its value is computed as follows

Auction day(364) Payment day

Every alternate Wednesday Following Friday

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Bond Valuations (1)