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1.
Money markets and fixed income securities
Report Submitted
By
Student Name
University Name
Document: JOB # MAY 221
Author: Stuart Thomas
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Page 1 of 22
2.
Data from the RBA.GOV.AU
June 2011
Closing
Gross
Accrued
Capital
Series Number
Coupon
Maturity
Yield
Price
Interest
Price
TB123
5.75%
15 Apr 12
4.700
101.993
1.194
100.799
TB127
4.75%
15 Nov 12
4.715
100.635
0.594
100.041
TB118
6.50%
15 May 13
4.735
103.937
0.813
103.125
TB129
5.50%
15 Dec 13
4.745
101.956
0.225
101.731
Closing
Gross
Accrued
Capital
Yield
Price
Interest
Price
3.930
101.692
1.194
100.498
3.480
101.670
0.587
101.083
3.285
105.091
0.804
104.287
3.160
104.635
0.225
104.410
Dec 2011
Series Number
Coupon
Maturity
15
TB123
5.75%
12
15
TB127
4.75%
6.50%
5.50%
May
13
15
TB129
Nov
12
15
TB118
Apr
13
Dec
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3.
Part 1:
A. Calculate the dirty price, clean price, modified duration and modified convexity of the
Government bonds as at as at the end of June 2011 and the end of December 2011.
The value is calculates based on the fact that it pays 4 times a year and it uses the 30/360 US
day count convention. These calculations are at the end of June 2011.
Term
Value
Value
Value
Value
Series Number
TB123
TB127
TB118
TB129
Coupon Rate
5.75%
4.75%
6.50%
5.50%
Par Value
$100
$100
$100
$100
Full Market Value
$101.993
$100.635
$103.937
$101.956
Dirty Price
10.20
10.06
10.40
10.20
Accrued Interest
1.194
0.594
0.813
0.225
Flat Market Value
$100.799
$100.041
$103.125
$101.731
Clear Price
10.08
10.00
10.31
10.18
Table - 1: Calculation of dirt price and clear price at the end of June 2011
These calculations are at the end of December 2011
Term
Value
Value
Value
Value
Series Number
TB123
TB127
TB118
TB129
Coupon Rate
5.75%
4.75%
6.50%
5.50%
Par Value
$100
$100
$100
$100
Full Market Value
$101.692
$101.670
$105.091
$101.956
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4.
Dirty Price
10.17
10.16
10.51
10.20
Accrued Interest
1.194
0.594
0.804
0.225
Flat Market Value
$100.498
$101.083
$104.287
$104.410
Clear Price
10.05
10.10
10.31
10.44
Table - 1: Calculation of dirt price and clear price at the end of December, 2011
For TB123: Calculation of the modified duration and convexity
Bond Price
$96.05
$95.15
If Yield Changes By
1.00%
Bond Price Will Change
Face Value
100
100
Coupon Rate
5.75%
By
6%
Modified
-0.90
-0.93%
Duration
Life in Years
1
1
Predicts
-0.90
-0.94%
Yield
10.00%
11%
Convexity Adjustment
0.01
0.01%
Frequency
2
2
Total Predicted Change
-0.90
-0.93%
0.99
Actual New Price
$95.15
Duration
0.94
Predicted New Price
$95.15
Convexity
1.33
Difference
$0.00
Macaulay
Duration
Modified
PV
Period
Cash Flow
0
Flow
Cash Duration
Calc
+0.0000%
Convexity
Calc
($96.05)
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5.
1
2.88
2.74
2.74
4.97
2
102.88
93.31
186.62
507.81
Total
189.36
512.78
Total
For TB127- Calculation of the modified duration and convexity
Bond Price
$95.12
$94.23
If Yield Changes By
1.00%
Bond Price Will Change
Face Value
100
100
Coupon Rate
4.75%
By
5%
Modified
-0.89
-0.93%
Duration
Life in Years
1
1
Predicts
-0.90
-0.94%
Yield
10.00%
11%
Convexity Adjustment
0.01
0.01%
Frequency
2
2
Total Predicted Change
-0.89
-0.93%
0.99
Actual New Price
$94.23
Duration
0.94
Predicted New Price
$94.23
Convexity
1.34
Difference
$0.00
Macaulay
Duration
Modified
+0.0000%
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6.
PV
Period
Cash Flow
0
2
Convexity
($95.12)
1
Cash Duration
Flow
Calc
Calc
2.38
2.26
2.26
4.10
102.38
92.86
185.71
505.34
Total
187.98
509.45
Total
For TB 118- Calculation of the modified duration and convexity
Bond Price
$96.75
$95.85
If Yield Changes By
1.00%
Bond Price Will Change
Face Value
100
100
Coupon Rate
6.50%
By
7%
Modified
-0.90
-0.93%
Duration
Life in Years
1
1
Predicts
-0.91
-0.94%
Yield
10.00%
11%
Convexity Adjustment
0.01
0.01%
Frequency
2
2
Total Predicted Change
-0.90
-0.93%
0.98
Actual New Price
$95.85
Duration
0.94
Predicted New Price
$95.85
Convexity
1.33
Difference
$0.00
Macaulay
Duration
Modified
+0.0000%
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7.
PV
Period
Cash Flow
0
2
Convexity
($96.75)
1
Cash Duration
Flow
Calc
Calc
3.25
3.10
3.10
5.61
103.25
93.65
187.30
509.66
Total
190.40
515.28
Total
For TB129 - Calculation of the modified duration and convexity
Bond Price
$95.82
$94.92
If Yield Changes By
1.00%
Bond Price Will Change
Face Value
100
100
Coupon Rate
5.50%
By
6%
Modified
-0.89
-0.93%
Duration
Life in Years
1
1
Predicts
-0.90
-0.94%
Yield
10.00%
11%
Convexity Adjustment
0.01
0.01%
Frequency
2
2
Total Predicted Change
-0.89
-0.93%
0.99
Actual New Price
$94.92
Duration
0.94
Predicted New Price
$94.92
Convexity
1.34
Difference
$0.00
Macaulay
Duration
Modified
+0.0000%
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8.
PV
Period
Cash Flow
0
2
Convexity
($95.82)
1
Cash Duration
Flow
Calc
Calc
2.75
2.62
2.62
4.75
102.75
93.20
186.39
507.20
Total
189.01
511.95
Total
B. Calculate the holding period return for each of the bonds over the period from end June
2011 end December 2011.
For TB 123
Holding period yield
Bond
value
$96.05
Par value (redemption
value)
97.19
Coupon rate (annual)
5.75%
Years till maturity
0.5
30-06Today
2011
Holding period yield
Maturity 31-12-2011
6.55%
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9.
For TB 127
Holding period yield
Bond
value
$95.12
Par value (redemption
value)
96.25
Coupon rate (annual)
4.75%
Years till maturity
0.5
30-06Today
2011
Holding period yield
Maturity 31-12-2011
5.56%
For TB 118
Holding period yield
Bond
value
$95.12
Par value (redemption
value)
97.90
Coupon rate (annual)
6.50%
Years till maturity
0.5
30-06Today
2011
Holding period yield
Maturity 31-12-2011
8.21%
For TB 129
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10.
Holding period yield
Bond
value
$95.12
Par value (redemption
value)
96.96
Coupon rate (annual)
5.50%
Years till maturity
0.5
30-06Today
2011
Maturity 31-12-2011
Holding period yield
6.70%
C. Calculate the modified duration, modified convexity and holding period return for an
equally-weighted portfolio of the bonds at both dates. Report on your findings. Compare and
contrast the return and volatility of the portfolio and the separate bonds at both dates.
At the end of June 2011
Macaulay Duration
Modified Duration
Convexity
Holding Return
0.99
0.97
1.43
= 5.63 %
At the end of December, 2011
Macaulay Duration
Modified Duration
Convexity
Holding Return
0.91
0.89
1.31
= 5.93 %
The calculated values suggests that for the equally weighted portfolio the the holding return get
increases for the end of the December 2011 because of the longer period than June 2011.
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11.
Part 2:
A. Use all available Government bond data (ie not just the bonds in your portfolio) to construct and
present a yield curve, spot curve and forward curve as at the end of June 2011 and the end of
December 2011. Your spot curve and forward curve estimation should go out no more than 5
years. Present and discuss your findings.
Suppose we have the following Treasury yields (based roughly on rba.gov.com) The face
value is taken as $100 for the maturity years of 5.
Maturity (yrs)
Coupon
Price
32nds
Yield
0.25
0
$99.46
2.17
0.50
0
$98.82
2.38
1.00
2.250
$99.66
99 21/32
2.61
1.50
2.250
$99.28
99 9/32
2.76
2.00
2.500
$99.15
99 5/32
2.96
2.50
2.875
$99.63
99 20/32
3.05
3.00
3.000
$99.53
99 17/32
3.19
3.50
3.125
$99.64
99 20/32
3.26
4.00
3.500
$100.58
100 19/32
3.37
4.50
3.375
$99.64
99 20/32
3.49
5.00
3.500
$99.77
99 25/32
3.58
Table: Value taken initially for the bonds for next 5 years and calculation of yield value
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12.
Fig: Curve represents the yielding curve for the given bond and coupan rate
The above curve reflects the yield for current securities with certain maturities.
Spot Curve
The one year spot rate is easily found by equalizing the cash flows.
y is the yield to maturity, z1 and z2 are the two zero rates (6mo and 1yr):
C1/(1+y/2) + (100+C2)/(1+y/2)^2 = C1/(1+z1/2) + (100+C2)/(1+z2/2)^2
1.1105 + 98.5364
=
1.1118 + 101.1250/(1+Z2/2)^2
Solving for z2, the 1yr zero rates:
99.6469
=
1.1118 + 101.1250/(1+Z2/2)^2
98.5351
=
101.1250/(1+Z2/2)^2
(1+Z2/2)^2
=
1.02628364
1+Z2/2 =
1.013056583
Z2/2
=
0.013056583
Z2
=
2.6113 Percent
(Square root)
In this case, the 1 year spot rate matches the yield; that isn't always the case.
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13.
Maturity (yrs)
Coupon
Price
0.25
na
0.50
32nds
Yield
Spot Rate
$99.46
2.17
2.17
na
$98.82
2.38
2.38
1.00
2.250
$99.66
99 21/32
2.61
2.6113
1.50
2.250
$99.28
99 9/32
2.76
2.7440
2.00
2.500
$99.15
99 5/32
2.96
2.9448
2.50
2.875
$99.63
99 20/32
3.05
3.0359
3.00
3.000
$99.53
99 17/32
3.19
3.1784
3.50
3.125
$99.64
99 20/32
3.26
3.2497
4.00
3.500
$100.58
100 19/32
3.37
3.3651
4.50
3.375
$99.64
99 20/32
3.49
3.4889
5.00
3.500
$99.77
99 25/32
3.58
3.5835
Table: Calculation of the spot value and yield value based on the previous result.
Fig: Plotting of the regular yield curve (in blue) versus the spot curve.
This above figure represents the spot curve in yellow line that signifies that the spot curve is almost
follow the yield curve slop.
Forward Curve
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14.
The 6mo forward rate in 6 months can be though of as what we could borrow/lend at for 6 months,
6 months from now. By the law of no arbitrage, investing our money now for 1 year or now for
6months, with the next 6mo rate locked in, must result in the same present value. y is the yield to
maturity, z1 is the 6mo spot rate, and f1 is the 6mo forward rate 6months from now.
C1/(1+y/2) + (100+C2)/(1+y/2)^2 = C1/(1+z1/2) + (100+C2)/(1+z1/2)(1+f1/2
1.1105 + 98.5364
=
1.1118 + 101.1250/((1+z1/2)(1+f1/2))
Solving for f1:
99.6469
=
1.1118 + 101.1250/((1+z1/2)(1+f1/2))
98.5351
=
101.1250/((1+2.38/2)(1+f1/2))
98.5351
=
99.9357644
(1+f1/2)
=
1.014214487
f1/2
=
0.014214487
f1
=
2.8429
Maturity (yrs)
Coupon
Price
/(1+f1/2)
32nds
Yield
Spot Rate
6mo Fwd Rate
2.17
2.38
2.38
2.38
2.8429
0.00
0.50
na
$98.82
1.00
2.250
$99.66
99 21/32
2.61
2.6113
3.0209
1.50
2.250
$99.28
99 9/32
2.76
2.7440
3.5997
2.00
2.500
$99.15
99 5/32
2.96
2.9448
3.4176
2.50
2.875
$99.63
99 20/32
3.05
3.0359
3.9537
3.00
3.000
$99.53
99 17/32
3.19
3.1784
3.6984
3.50
3.125
$99.64
99 20/32
3.26
3.2497
4.2473
4.00
3.500
$100.58
100 19/32
3.37
3.3651
4.5875
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15.
4.50
3.375
$99.64
99 20/32
3.49
3.4889
5.00
3.500
$99.77
99 25/32
3.58
4.5123
3.5835
Table: Calculation of the Yield, spot and forward values
Fig: Yield Curve (Blue Line), Spot Curve (Yello Line) and Forward curve for sex months
(in Red color)
B. Review the predictive ability of the yield, spot and forward curves with comprehensive
reference to the relevant academic literature. Discuss the curves that you have estimated in Part
2A with reference to this literature. Does the June 2011 forward curve predict the 6 month spot
rate for December 2011? Comment.
Yield curve analyses the term to maturity for the particular bond given. The measurement of the
differences of the interest rate which also has the different term to maturity are being done in this
yield curve for the given bond rate, face value and discount rate. The maturity effect on the bond is
estimated using the yield curve. (Cox et al, 1993) For example, in case of Australian treasury bills
the yield curve is estimated based on the various debt security and term to the maturity. Mostly it is
found that the yield curve as upward sloping. (Tuckman, B, 2002) This describes that the upward
sloping is basically signifies that the short term to maturity generally estimated as the low interest
rates and on the other hand the longer term to maturity is estimated as the higher interest terms.
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16.
Based on the literature it can be explained that the for maximization of the investment return will
only be happen when the investment will be done for the longer term means higher yields. (M.
Felton, 1994) In this case there is no immediate liquidity requirement and the yield curve shape and
its level will be continuous and constant that means it will not change.
The shape and the level of the yield curve are used to develop the interest rate of forecast with the
use of the mathematical expectations theory model. The expectations theory that is applied for the
yield curve generation explains that the for a longer time the interest rate on the yield curve can be
calculated as the multiplication of the all interest rate of the shorter time intervals that is comprised
for the longer time to maturity. (De Boor, 1978) If the increasement is done to the investment yield
for the longer period then the yield curve shapes like a flat curve this means throughout same rate
for all kind of maturities, if the shorter period has the high interest rate and the longer period have
the low interest rate than the slop will be downward, and if there is upward sloping and as well as
flat sloping combining in a one yield curve then this yield curve is called as humped yield curve.
For the pricing bond generally the spot rate curve is estimated. It is also called spot rate treasury
curve. These spot curves can be constructed from on the run or off the run or combination of both
treasuries. (Hull, 2005) Based on the coupon strips for treasury the spot treasury curve can be
calculated. There number of bond which have the multiple cash flows or say coupon rates at
number of points in the bonds lives. Therefore, it is not the correct evaluation of the maturity and
bonds based on the single interest rate because there are different interest rates based on the
discounted cash flows. Therefore while making the fair bond valuation there is need to have the
good practice to matching the discount to the each payment for the corresponding treasury spot rate
for the present value of the each bond price.
Based on the forward curve, the valuation of time value of money is estimated. This curve
represents the price at which the market is ready to transact the future price. (James, J, 2000) It
explains the forward rate of the price for the bonds or treasury bonds. But still there is one issue as
found in the earlier researches that forward curve doesn’t represent the actual future price. Based on
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17.
the above curve yield, spot and forward curve, it can be explained easily that the yield curve has the
upward slope so in this case the possibility is of having the longer period with higher interest rate
and the shorter period with the low interest rate. (W. H., Flannery, 1992) For the treasury curve
called as spot curve, it can be seen that its slope is almost same as the yield price slope. The
conclusion can be drawn for the spot rate that based on the multiple cash flows and similar as the
yield value the slope of the curve is same. In the last figure, there have been constructed the entire
three yield, spot and forward curve. It can be seen from the figure that the forward is much higher
for the larger term to maturity but also in between it goes down and then up.
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18.
Part 3:
A:
Describe the shape and level of the current Yield Curve in Australian Commonwealth Government
Securities (i.e.: at the time of your dealing sessions – use data gathered from the trading system in
the FMTS, plot the curve and discuss it).
Maturity (yrs)
Coupon
1
2
3
5
7
9
10
11
15
Price
6.5
6.25
6.25
6
5.25
5.75
5.75
5.5
4.75
32nds
$99.46
$98.82
$110.48
$112.26
$112.61
$119.18
$123.78
$123.80
$118.01
Yield
2.17
2.38
2.61
2.76
2.96
3.05
3.19
3.26
3.37
110 15/32
112 8/32
112 5/8
119 6/32
123 25/32
123 13/16
118
Table: Data calculation for the current trading system in the FMTS and the yield and bond price
Fig: Treasury Yields curve for the current trading system in the FMTS
As shown in the figure the yield curve is on the upward side which means the longer term maturity
has the higher interest rate and the shorter term maturity has the lower interest rate. As seen from
the above graph that the maturity for the 15 years and so the coupon rate is higher as 6.00 on an
average for the complete maturity period. In case of the fixed interest fund managers the implication
of this yield curve will lower down the maturity period because for the fixed interest the maturity
period will be lower if the fixed rate is greater than the average of the yield rate otherwise reverse
will be true.
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19.
Based on your view of the yield curve, and other sources, what is your view on Interest Rates in
Australia for the following time horizons?
6 Months – The interest rate in Australia for the six months time horizons is lower.
12 Months – For 12 months time horizon the interest rate in Australia will be higher.
As the curve suggests that for the shorter period the interest rate will be lower and for the longer
period the interest rate will be higher.
B: (4 Marks)
With respect to your trading portfolio:
1. Report on your Portfolio composition in Market Value Terms and Face Value Terms:
(a) At the commencement of trading
At commencement of trading the Face Value is less than the market value which means the
original price is less than the actual price during the investment this means the investment value
is in profit.
(b) After Dealing Session 2
After the dealing of the session 2, the face value is again lower the actual market value, so it shows
that the invested money to the stock market is still in profit.
2 What is the level of interest rate risk in your portfolio, as measured by?
(a) Portfolio Modified Duration at the commencement of trading
The level of interest rate risk at the commencement of the trading is lower but increases soon
after the trading starts.
(b) Portfolio Modified Duration after Dealing Session 2
After the dealing session 2, the level of the interest rate goes increases and after some time it
becomes medium risk for the investment.
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20.
3. How has the interest rate risk profile of your portfolio changed? What are the implications for
your portfolio in the current yield curve environment?
The interest rate of risk profile for the portfolio goes increasing with the maturity time but soon
after some time the level of interest risk goes down and the risk reduces. In the current yield curve
environment the portfolio signifies that the investment return will be higher over the time. As the
yield is on average of 3.50% which means the return will be almost 30 percent higher of the
invested money. So the current yield curve environment is in favour of the investment.
4. Comment on whether you achieved your set objectives with regard to interest rate risk:
If you achieved your objective, what trading actions/strategy contributed to your success?
Yes the set objectives achieved because of the balanced portfolio and using the options of
forwarding rate which shows the yield rate was higher for the given maturity period.
If you did not achieve your objective, what would you do differently?
In case the objectives were not achieved then the investment need to revise based on the forward
rate and should wait for the right time for investment.
Was your trading consistent with the market view developed in Part 2?
Yeah, little bit it seems from the part -2 yield graph that the trading was consistent as the yield
rate was almost similar to the earlier rate except of some places where it was higher than the
previous.
C:
Compare and contrast your performance from one dealing session to another. Discuss what you
could do to improve future dealing sessions. What lessons did you learn and how did you perform
as a team?
In one dealing session it was normally slow but in another dealing session it got improved because
of the higher rate of return of interest. For future dealing session it is needed to more focus on the
longer period of maturity instead of the shorter period of maturity. The longer period always gives
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21.
the fair result than the shorter period. From this I learnt lot of things like how to make the portfolio,
how to invest money, how to buy government bonds and how to perform yield curve, stop curve,
and forward curve based on the coupon rate for the multiple time intervals. Based on overall
conclusion the performance as a team, it was quite satisfactory.
Document: JOB # MAY 221
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22.
References:
1. Cox, Raymond A. K., and Daniel E. Vetter. "Money Market Returns and Risk, 19381989." Journal of Midwest Finance 22 (1993): 50-54.
2. M. Felton. "Performance from Riding the Yield Curve, 1980-1992." Journal of Business and
Economic Perspectives 20 (1994): 128-32.
3. De Boor, C.: A Practical Guide to Splines, Springer Verlag (1978).
4. Hull, J.: Options, Futures and Other Derivatives Prentice Hall (2005).
5. James, J., and Webber, N.: Interest Rate Modelling, Wiley (2000).
6. W. H., Flannery, B. P., Teukolsky, S. A., and Vetterling, V. T.: Numerical Recipes in C:
The Art of Scientiﬁc Computing, Cambridge University Press (1992).
7. Tuckman, B.: Fixed Income Securities, Wiley (2002).
Document: JOB # MAY 221
Author: Stuart Thomas
Save Date: 27/11/2013
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