Introduction to Mathematics in Finance – HW 4
Swarna Ramineni sr3121 Answer 1)
Value at risk: It is a statistical technique to measure the amount of potential loss, the probability of the loss, and the time frame. Value at risk is used by risk managers in order to measure and control the level of risk which the firm undertakes. The risk manager's job is to ensure that risks are not taken beyond the level at which the firm can absorb the losses of a probable worst outcome. For example, a financial firm may determine that it has a 5% one month value at risk of $100 million. This means that there is a 5% chance that the firm could lose more than $100 million in any given month.
Conditional value at risk on the other hand is an extension of value at risk. It is derived by taking weighted average between the value at risk and losses exceeding the value at risk. The VaR model does allow managers to limit the likelihood of incurring losses caused by certain types of risk - but not all risks. The problem with relying solely on the VaR model is that the scope of risk assessed is limited, since the tail end of the distribution of loss is not typically assessed. Therefore, if losses are incurred, the amount of the losses will be substantial in value. Conditional value at risk does a better job at assessing the tail VaR and hence is a very useful tool for risk managers.
Answer 2) NAV as of Nov 1, 2013 is $169,018 Gross Leverage is 1.744 and Net Leverage is 0.7017 The latest 1 month values are as below
Ann Cash Rate 0.2%
Portfolio Cash GOOG AAPL SPY 50,413 100 200 -500 Date Cash GOOG AAPL SPY Gross leverage Net leverage NAV (Const. Number of shares)
11/1/2013 $ 50,413.00 $ 102,704.00 $ 104,006.00 $ (88,105.00) 1.744 0.7017 $ 169,018.00
10/31/2013 $ 50,412.72 $ 103,058.00 $ 104,540.00 $ (87,895.00) 1.737 0.7037 $ 170,115.72
10/30/2013 $ 50,412.45 $ 103,042.00 $ 104,980.00 $ (88,145.00) 1.739 0.7040 $ 170,289.45
10/29/2013 $ 50,412.17 $ 103,624.00 $ 103,336.00 $ (88,585.00) 1.751 0.7013 $ 168,787.17
10/28/2013 $ 50,411.90 $ 101,500.00 $ 105,976.00 $ (88,115.00) 1.741 0.7031 $ 169,772.90
10/25/2013 $ 50,411.07 $ 101,520.00 $ 105,192.00 $ (87,975.00) 1.742 0.7020 $ 169,148.07
10/24/2013 $ 50,410.79 $ 102,555.00 $ 106,382.00 $ (87,575.00) 1.726 0.7065 $ 171,772.79
10/23/2013 $ 50,410.51 $ 103,141.00 $ 104,992.00 $ (87,285.00) 1.725 0.7056 $ 171,258.51
10/22/2013 $ 50,410.24 $ 100,700.00 $ 103,974.00 $ (87,705.00) 1.747 0.6988 $ 167,379.24
10/21/2013 $ 50,409.96 $ 100,330.00 $ 104,272.00 $ (87,200.00) 1.739 0.6996 $ 167,811.96
10/18/2013 $ 50,409.13 $ 101,141.00 $ 101,778.00 $ (87,195.00) 1.746 0.6966 $ 166,133.13
10/17/2013 $ 50,408.86 $ 88,879.00 $ 100,900.00 $ (86,610.00) 1.800 0.6718 $ 153,577.86
10/16/2013 $ 50,408.58 $ 89,803.00 $ 100,222.00 $ (86,035.00) 1.788 0.6735 $ 154,398.58
10/15/2013 $ 50,408.30 $ 88,201.00 $ 99,736.00 $ (84,850.00) 1.777 0.6716 $ 153,495.30
10/14/2013 $ 50,408.03 $ 87,611.00 $ 99,208.00 $ (85,470.00) 1.794 0.6678 $ 151,757.03
10/11/2013 $ 50,407.20 $ 87,199.00 $ 98,562.00 $ (85,130.00) 1.794 0.6663 $ 151,038.20
10/10/2013 $ 50,406.92 $ 86,824.00 $ 97,928.00 $ (84,585.00) 1.789 0.6652 $ 150,573.92
10/9/2013 $ 50,406.65 $ 85,586.00 $ 97,318.00 $ (82,800.00) 1.765 0.6651 $ 150,510.65
10/8/2013 $ 50,406.37 $ 85,367.00 $ 96,188.00 $ (82,740.00) 1.771 0.6622 $ 149,221.37
10/7/2013 $ 50,406.09 $ 86,574.00 $ 97,550.00 $ (83,715.00) 1.776 0.6658 $ 150,815.09
10/4/2013 $ 50,405.27 $ 87,235.00 $ 96,606.00 $ (84,445.00) 1.791 0.6635 $ 149,801.27
10/3/2013 $ 50,404.99 $ 87,609.00 $ 96,682.00 $ (83,810.00) 1.777 0.6659 $ 150,885.99
10/2/2013 $ 50,404.71 $ 88,799.00 $ 97,912.00 $ (84,590.00) 1.779 0.6695 $ 152,525.71
10/1/2013 $ 50,404.44 $ 88,700.00 $ 97,592.00 $ (84,670.00) 1.782 0.6684 $ 152,026.44
Answer 3) a) &b) Ann Cash Rate 0.2%
Portfolio Cash GOOG AAPL SPY VAR Multiplier-> 1.644854 50,413 100 200 -500 Returns rolling 90day
Date Cash GOOG AAPL SPY Hitorical P/F % Changes (Returns) Vol for 1day Vol ANNUALIZED 95% VAR based on Vol for 1day
11/1/2013 $ 50,413.00 $ 102,704.00 $ 104,006.00 $ (88,105.00) -0.645% 1.41% 22.2% 2.31%
10/31/2013 $ 50,412.72 $ 103,058.00 $ 104,540.00 $ (87,895.00) -0.102% 1.41% 22.2% 2.31%
10/30/2013 $ 50,412.45 $ 103,042.00 $ 104,980.00 $ (88,145.00) 0.890% 1.41% 22.3% 2.32%
10/29/2013 $ 50,412.17 $ 103,624.00 $ 103,336.00 $ (88,585.00) -0.581% 1.42% 22.5% 2.34%
10/28/2013 $ 50,411.90 $ 101,500.00 $ 105,976.00 $ (88,115.00) 0.369% 1.42% 22.5% 2.34%
10/25/2013 $ 50,411.07 $ 101,520.00 $ 105,192.00 $ (87,975.00) -1.528% 1.43% 22.5% 2.34%
10/24/2013 $ 50,410.79 $ 102,555.00 $ 106,382.00 $ (87,575.00) 0.300% 1.42% 22.4% 2.33%
10/23/2013 $ 50,410.51 $ 103,141.00 $ 104,992.00 $ (87,285.00) 2.318% 1.42% 22.4% 2.33%
10/22/2013 $ 50,410.24 $ 100,700.00 $ 103,974.00 $ (87,705.00) -0.258% 1.40% 22.1% 2.30%
10/21/2013 $ 50,409.96 $ 100,330.00 $ 104,272.00 $ (87,200.00) 1.011% 1.40% 22.1% 2.30%
10/18/2013 $ 50,409.13 $ 101,141.00 $ 101,778.00 $ (87,195.00) 8.175% 1.40% 22.1% 2.30%
10/17/2013 $ 50,408.86 $ 88,879.00 $ 100,900.00 $ (86,610.00) -0.532% 1.11% 17.6% 1.83%
10/16/2013 $ 50,408.58 $ 89,803.00 $ 100,222.00 $ (86,035.00) 0.588% 1.11% 17.5% 1.83%
10/15/2013 $ 50,408.30 $ 88,201.00 $ 99,736.00 $ (84,850.00) 1.145% 1.11% 17.5% 1.82%
10/14/2013 $ 50,408.03 $ 87,611.00 $ 99,208.00 $ (85,470.00) 0.476% 1.11% 17.5% 1.82%
10/11/2013 $ 50,407.20 $ 87,199.00 $ 98,562.00 $ (85,130.00) 0.308% 1.11% 17.6% 1.83%
10/10/2013 $ 50,406.92 $ 86,824.00 $ 97,928.00 $ (84,585.00) 0.042% 1.11% 17.6% 1.83%
10/9/2013 $ 50,406.65 $ 85,586.00 $ 97,318.00 $ (82,800.00) 0.864% 1.11% 17.6% 1.83%
10/8/2013 $ 50,406.37 $ 85,367.00 $ 96,188.00 $ (82,740.00) -1.057% 1.11% 17.6% 1.83%
10/7/2013 $ 50,406.09 $ 86,574.00 $ 97,550.00 $ (83,715.00) 0.677% 1.11% 17.5% 1.82%
10/4/2013 $ 50,405.27 $ 87,235.00 $ 96,606.00 $ (84,445.00) -0.719% 1.11% 17.5% 1.82%
10/3/2013 $ 50,404.99 $ 87,609.00 $ 96,682.00 $ (83,810.00) -1.075% 1.10% 17.5% 1.82%
10/2/2013 $ 50,404.71 $ 88,799.00 $ 97,912.00 $ (84,590.00) 0.328% 1.10% 17.4% 1.81%
10/1/2013 $ 50,404.44 $ 88,700.00 $ 97,592.00 $ (84,670.00) 1.799% 1.10% 17.4% 1.81%
Answer 4) a) & b), Answer 4.5) a) & b)
Answer 5)
Covariance Matrix GOOG AAPL SPY
GOOG 0.0002030337 0.0000479498 0.0000517346
AAPL 0.0000479498 0.0003896443 0.0000484874
SPY 0.0000517346 0.0000484874 0.0000542938
Mean
GOOG AAPL SPY P/F
0.00174516 -0.000165898 0.000954297 0.001679059
P/F Weight P/F Weight P/F Weight P/F total
-0.5 -1 2.5 1
P/F Variance using Covariance matrix 0.000455915
P/F Variance using historical P/F returns 0.000482896
P/F SD using Covariance matrix 0.021352177
P/F SD using P/F historical returns 0.021974893 Confidence interval 97.5%
VAR Multiplier -1.96
VAR using historical P/F returns -0.0401641
VAR using P/F SD estimation -0.0430700
CVAR using VAR from historical P/F returns -0.0591416
CVAR using VAR calculated from SD estimation -0.0660504
Answer 6)
A B
Amount invested $ 2,000,000 $ 2,500,000
P/F Weights 0.4444 0.5556
Daily volatility 1% 1%
Correlation 0.3 0.3
Confidence interval 95%
VAR multiplier -1.6449
P/F Returns mean assumption 0
P/F daily Volatility 0.00809
P/F 10 Day Volatility 0.025579699
10 day 95% VAR -4.21%
Dollar value of 10 day 95% VAR $ -189,336.87
Answer 7)
The diameter of a golf ball is 1.5 in. So each golf ball will take a little less volume then a cube of edge 1.5 in. Volume of each golf ball is (1.5 in)3 = 3.375 in3 ≈ 3 in3
Volume of a Boeing 747 is = 200ft x 30ft x 30ft = 180,000ft3
Seats, cockpit equipment, etc will take up around ≈ 30,000ft3
Thus, total empty volume is around 150,000ft3 ≈ 225,000,000in3
Number of golf balls = (225,000,000/3) = 75,000,000
Answer 8) (Assumption that volatility is annualized) The above equations are fed into the excel solver and we get the portfolio $50,000 in A, $50,000 in B and $0 cash with the portfolio volatility of 0.0696. We can also arrive at this by solving the equations above and getting the variables WB WC and portfolio volatility as a function of WA and iteratively increasing its value from 0 until the other weights are positive too. We will find that the portfolio volatility will be directly proportional to the WA and hence we arrive at the optimal solution as below:
Answer 9)
Beta measures systematic risk based on how returns co-move with the overall market or a benchmark index
Stock Beta = Slope(AAPL, QQQ) = 1.22838073551628
Stock Beta = Cov(AAPL, QQQ)/Var(QQQ) = 1.22838073551629
QQQ QQQ QQQ AAPL AAPL AAPL
Date Adj Close Returns Date Adj Close Returns
11/29/2013 85.73 0.60% 11/29/2013 556.07 1.85%
11/27/2013 85.22 0.70% 11/27/2013 545.96 2.35%
11/26/2013 84.63 0.52% 11/26/2013 533.4 1.84%
11/25/2013 84.19 0.24% 11/25/2013 523.74 0.76%
11/22/2013 83.99 0.54% 11/22/2013 519.8 -0.26%
11/21/2013 83.54 1.02% 11/21/2013 521.14 1.19%
11/20/2013 82.7 -0.24% 11/20/2013 515 -0.88%
11/19/2013 82.9 -0.29% 11/19/2013 519.55 0.18%
11/18/2013 83.14 -0.98% 11/18/2013 518.63 -1.21%
11/15/2013 83.96 0.19% 11/15/2013 524.99 -0.60%
11/14/2013 83.8 0.31% 11/14/2013 528.16 1.45%
11/13/2013 83.54 1.21% 11/13/2013 520.63 0.12%
11/12/2013 82.54 0.16% 11/12/2013 520.01 0.18%
11/11/2013 82.41 -0.16% 11/11/2013 519.05 -0.29%
11/8/2013 82.54 1.35% 11/8/2013 520.56 1.57%
11/7/2013 81.44 -1.88% 11/7/2013 512.49 -1.62%
11/6/2013 83 -0.02% 11/6/2013 520.92 -0.28%
11/5/2013 83.02 0.11% 11/5/2013 522.4 -0.25%
11/4/2013 82.93 0.14% 11/4/2013 523.69 1.29%
11/1/2013 82.81 0.02% 11/1/2013 517.01 -0.51%
Answer 10)
Formula for forward interest rate is
The calculated forward interest rates are
Time to Maturity Spot interest rates Forward interest rates Time for Forward rate
1 1 1
2 1.5 2 2
3 2 3 3 2.5 1 to 3
Answer 11)
Bootstrapping equations:
The zero curve is the hart showing the zero rate as a function of time to maturity. Assumption is that the curve is linear between the points determined by bootstrap method.
Answer 12)
Duration (D) is a measure of the average life of a bond. It is also an approximation to the ratio of the proportional change in the bond price (B) to the absolute change in its yield (y). For cash flows(ci) at time ti we have:
D= ∑_(i=1)^n▒〖t_i [(c_i e^(-yt_i ))/B] 〗
Convexity (C) is a measure of curvature in the relationship between bond prices and bond yields.
C= (∑_(i=1)^n▒〖c_i t_i^2 e^(-yt_i ) 〗)/B
The relationship between change in yield (Δy) and Bond Price (B) :
∆B/B= -D∆y+1/2 C〖(∆y)〗^2
Answer 13) Yield to maturity is calculated using the rate function in excel: The rate function does not use continuous compounding assumption and hence when e apply continuous compounding assumption we get a different result below:
Using iterative calculation applying continuous compounding (excel snapshot):
We will use YTM = 5.33724% for all future purposes Bond Duration = 8.7697
Convexity = 88.6476
Bond Price at 5.33724% = $90.4844
Bond price at 5.34724% = $90.40508
DV01 = $0.07931 per bond
For the total investment of $10,000,000 par value worth bonds, DV01 is = $7,931.21
If $10,000,000 is invested at present price the DV01 of the portfolio is = $8,765.28
YTM Time Cash Flow Present Value Weight Time X Weight Time^2 X Weight
5.33724% 0.5 2.125 2.0690 0.02287 0.0114 0.0057
Bond Price 1 2.125 2.0146 0.02226 0.0223 0.0223 90.4844 1.5 2.125 1.9615 0.02168 0.0325 0.0488 2 2.125 1.9099 0.02111 0.0422 0.0844 2.5 2.125 1.8596 0.02055 0.0514 0.1284 3 2.125 1.8106 0.02001 0.0600 0.1801 3.5 2.125 1.7629 0.01948 0.0682 0.2387 4 2.125 1.7165 0.01897 0.0759 0.3035 4.5 2.125 1.6713 0.01847 0.0831 0.3740 5 2.125 1.6273 0.01798 0.0899 0.4496 5.5 2.125 1.5844 0.01751 0.0963 0.5297 6 2.125 1.5427 0.01705 0.1023 0.6138 6.5 2.125 1.5021 0.01660 0.1079 0.7014 7 2.125 1.4625 0.01616 0.1131 0.7920 7.5 2.125 1.4240 0.01574 0.1180 0.8852 8 2.125 1.3865 0.01532 0.1226 0.9807 8.5 2.125 1.3500 0.01492 0.1268 1.0780 9 2.125 1.3145 0.01453 0.1307 1.1767 9.5 2.125 1.2798 0.01414 0.1344 1.2765 10 2.125 1.2461 0.01377 0.1377 1.3772 10.5 2.125 1.2133 0.01341 0.1408 1.4784 11 102.125 56.7753 0.62746 6.9021 75.9226
Total 146.75 90.4844 1 8.7697 88.6476
Bond price when discounting is applied at 10bps increase in yield i.e. at yield 5.43724% = $89.69487134
Bond Price using the Convexity and duration is using the following formula:
∆B/B= -D∆y+1/2 C〖(∆y)〗^2
The bond price calculated is = $(90.4844 – 0.7895) = $89.6949
Both the results are consistent
Answer 15)
The YTM of Bond A is more than the Bond B YTM and hence the bond A is cheaper than bond B
Answer 14) YTM using Continuous compounding assumption and Face value of $100 assumption: -------- YTM (continuous compounding) = 8.126%
YTM (using excel function RATE() – no compounding) = 8.294%
Face Value FV $100
Bond Price PV $50
Coupon rate r 3.75% coupon payment c $1.88
Time to maturity T 30
YTM 8.294%
Convexity = 281.7337
Duration = 13.2352
Bond Price at YTM 8.126% = $50.00246
Bond Price at YTM 8.136% = $49.93635
DV01 = $0.066109
DV01 for $20,000,000 par value worth bonds = $13,221.75
DV01 for $20,000,000 present value worth bonds = $26,443.5 