(a)
To calculate:
We have to collect the data as given in Connect and calculate each fund's excess return.
Explanation of Solution
Excess returns are calculated as shown below. Period chosen is from 2004 to 2008.
|
|
Total monthly returns |
|
|
|
Excess Returns |
|
|
|
Month |
Vanguard |
Value Fund |
S&P |
T-bills |
Vanguard |
Value Fund |
S&P |
| Jan-04 | 1.52 | -0.07 | 1.35 | 0.07 | 1.45 | -0.14 | 1.28 |
| Feb-04 | -0.53 | -0.99 | -1.32 | 0.06 | -0.59 | -1.05 | -1.38 |
| Mar-04 | -2.69 | -2.46 | -1.89 | 0.09 | -2.78 | -2.55 | -1.98 |
| Apr-04 | 0.22 | 2.52 | 1.70 | 0.08 | 0.14 | 2.44 | 1.62 |
| May-04 | 2.53 | 1.39 | 1.85 | 0.06 | 2.47 | 1.33 | 1.79 |
| Jun-04 | -1.93 | -7.46 | -3.22 | 0.08 | -2.01 | -7.54 | -3.30 |
| Jul-04 | 0.99 | -0.78 | 0.25 | 0.1 | 0.89 | -0.88 | 0.15 |
| Aug-04 | 1.52 | 2.49 | 1.00 | 0.11 | 1.41 | 2.38 | 0.89 |
| Sep-04 | 0.21 | 1.46 | 1.29 | 0.11 | 0.10 | 1.35 | 1.18 |
| Oct-04 | 5.22 | 4.66 | 4.46 | 0.11 | 5.11 | 4.55 | 4.35 |
| Nov-04 | 3.24 | 3.60 | 3.01 | 0.15 | 3.09 | 3.45 | 2.86 |
| Dec-04 | -2.36 | -3.48 | -2.24 | 0.16 | -2.52 | -3.64 | -2.40 |
| Jan-05 | 3.62 | 0.79 | 2.09 | 0.16 | 3.46 | 0.63 | 1.93 |
| Feb-05 | -2.33 | -2.86 | -1.83 | 0.16 | -2.49 | -3.02 | -1.99 |
| Mar-05 | -2.68 | -2.41 | -1.87 | 0.21 | -2.89 | -2.62 | -2.08 |
| Apr-05 | 4.08 | 7.13 | 3.22 | 0.21 | 3.87 | 6.92 | 3.01 |
| May-05 | 2.25 | 0.51 | 0.15 | 0.24 | 2.01 | 0.27 | -0.09 |
| Jun-05 | 3.55 | 5.28 | 3.82 | 0.23 | 3.32 | 5.05 | 3.59 |
| Jul-05 | -1.85 | -0.85 | -0.93 | 0.24 | -2.09 | -1.09 | -1.17 |
| Aug-05 | 0.38 | 1.34 | 0.80 | 0.3 | 0.08 | 1.04 | 0.50 |
| Sep-05 | -1.88 | 0.30 | -2.36 | 0.29 | -2.17 | 0.01 | -2.65 |
| Oct-05 | 2.87 | 5.04 | 4.39 | 0.27 | 2.60 | 4.77 | 4.12 |
| Nov-05 | 0.93 | 0.46 | -0.19 | 0.31 | 0.62 | 0.15 | -0.50 |
| Dec-05 | 3.41 | 3.30 | 2.41 | 0.32 | 3.09 | 2.98 | 2.09 |
| Jan-06 | 0.00 | -1.82 | 0.57 | 0.35 | -0.35 | -2.17 | 0.22 |
| Feb-06 | 0.71 | 0.34 | 1.65 | 0.34 | 0.37 | 0.00 | 1.31 |
| Mar-06 | 0.53 | -0.45 | 1.26 | 0.37 | 0.16 | -0.82 | 0.89 |
| Apr-06 | -3.00 | -5.84 | -3.01 | 0.36 | -3.36 | -6.20 | -3.37 |
| May-06 | -0.27 | -1.55 | 0.26 | 0.43 | -0.70 | -1.98 | -0.17 |
| Jun-06 | 1.09 | -2.36 | 0.45 | 0.4 | 0.69 | -2.76 | 0.05 |
| Jul-06 | 2.70 | 3.60 | 2.18 | 0.4 | 2.30 | 3.20 | 1.78 |
| Aug-06 | 2.63 | 2.87 | 2.70 | 0.42 | 2.21 | 2.45 | 2.28 |
| Sep-06 | 2.56 | 1.92 | 3.16 | 0.41 | 2.15 | 1.51 | 2.75 |
| Oct-06 | 0.75 | 1.94 | 1.99 | 0.41 | 0.34 | 1.53 | 1.58 |
| Nov-06 | 2.32 | 0.22 | 0.77 | 0.42 | 1.90 | -0.20 | 0.35 |
| Dec-06 | 1.86 | 2.18 | 1.51 | 0.4 | 1.46 | 1.78 | 1.11 |
| Jan-07 | -2.30 | -1.69 | -1.96 | 0.44 | -2.74 | -2.13 | -2.40 |
| Feb-07 | 0.81 | 0.89 | 1.16 | 0.38 | 0.43 | 0.51 | 0.78 |
| Mar-07 | 3.87 | 2.92 | 4.42 | 0.43 | 3.44 | 2.49 | 3.99 |
| Apr-07 | 3.65 | 3.37 | 3.39 | 0.44 | 3.21 | 2.93 | 2.95 |
| May-07 | -2.02 | -1.14 | -1.46 | 0.41 | -2.43 | -1.55 | -1.87 |
| Jun-07 | -4.74 | -0.79 | -3.13 | 0.4 | -5.14 | -1.19 | -3.53 |
| Jul-07 | 1.12 | 1.11 | 1.28 | 0.4 | 0.72 | 0.71 | 0.88 |
| Aug-07 | 2.54 | 4.91 | 3.88 | 0.42 | 2.12 | 4.49 | 3.46 |
| Sep-07 | 0.39 | 2.19 | 1.35 | 0.32 | 0.07 | 1.87 | 1.03 |
| Oct-07 | -5.86 | -3.90 | -3.87 | 0.32 | -6.18 | -4.22 | -4.19 |
| Nov-07 | 0.25 | 0.00 | -1.13 | 0.34 | -0.09 | -0.34 | -1.47 |
| Dec-07 | -5.39 | -10.34 | -6.04 | 0.27 | -5.66 | -10.61 | -6.31 |
| Jan-08 | -2.67 | -2.38 | -2.58 | 0.21 | -2.88 | -2.59 | -2.79 |
| Feb-08 | -0.89 | -0.06 | -0.90 | 0.13 | -1.02 | -0.19 | -1.03 |
| Mar-08 | 5.64 | 6.32 | 4.77 | 0.17 | 5.47 | 6.15 | 4.60 |
| Apr-08 | 1.44 | 3.00 | 1.51 | 0.17 | 1.27 | 2.83 | 1.34 |
| May-08 | -7.18 | -6.25 | -8.35 | 0.17 | -7.35 | -6.42 | -8.52 |
| Jun-08 | -1.80 | -0.40 | -0.90 | 0.17 | -1.97 | -0.57 | -1.07 |
| Jul-08 | 1.37 | 0.45 | 1.55 | 0.15 | 1.22 | 0.30 | 1.40 |
| Aug-08 | -7.50 | -11.63 | -9.42 | 0.12 | -7.62 | -11.75 | -9.54 |
| Sep-08 | -18.16 | -15.45 | -16.52 | 0.15 | -18.31 | -15.60 | -16.67 |
| Oct-08 | -7.16 | -7.93 | -6.96 | 0.08 | -7.24 | -8.01 | -7.04 |
| Nov-08 | 2.70 | 0.57 | 0.99 | 0.02 | 2.68 | 0.55 | 0.97 |
| Dec-08 | -10.76 | -4.49 | -8.22 | 0.09 | -10.85 | -4.58 | -8.31 |
Excess returns have been shown in the spread sheet.
(b)
To calculate:
In this part of the problem we have to calculate standard deviation of each fund.
Explanation of Solution
Standard deviation on excel can be calculated by using formula as = stdev
Then we have to choose the data accordingly to calculate standard deviation.
|
|
Total monthly returns |
|
|
|
Excess Returns |
|
|
|
Month |
Vanguard |
Value Fund |
S&P |
T-bills |
Vanguard |
Value Fund |
S&P |
| Jan-04 | 1.52 | -0.07 | 1.35 | 0.07 | 1.45 | -0.14 | 1.28 |
| Feb-04 | -0.53 | -0.99 | -1.32 | 0.06 | -0.59 | -1.05 | -1.38 |
| Mar-04 | -2.69 | -2.46 | -1.89 | 0.09 | -2.78 | -2.55 | -1.98 |
| Apr-04 | 0.22 | 2.52 | 1.70 | 0.08 | 0.14 | 2.44 | 1.62 |
| May-04 | 2.53 | 1.39 | 1.85 | 0.06 | 2.47 | 1.33 | 1.79 |
| Jun-04 | -1.93 | -7.46 | -3.22 | 0.08 | -2.01 | -7.54 | -3.30 |
| Jul-04 | 0.99 | -0.78 | 0.25 | 0.1 | 0.89 | -0.88 | 0.15 |
| Aug-04 | 1.52 | 2.49 | 1.00 | 0.11 | 1.41 | 2.38 | 0.89 |
| Sep-04 | 0.21 | 1.46 | 1.29 | 0.11 | 0.10 | 1.35 | 1.18 |
| Oct-04 | 5.22 | 4.66 | 4.46 | 0.11 | 5.11 | 4.55 | 4.35 |
| Nov-04 | 3.24 | 3.60 | 3.01 | 0.15 | 3.09 | 3.45 | 2.86 |
| Dec-04 | -2.36 | -3.48 | -2.24 | 0.16 | -2.52 | -3.64 | -2.40 |
| Jan-05 | 3.62 | 0.79 | 2.09 | 0.16 | 3.46 | 0.63 | 1.93 |
| Feb-05 | -2.33 | -2.86 | -1.83 | 0.16 | -2.49 | -3.02 | -1.99 |
| Mar-05 | -2.68 | -2.41 | -1.87 | 0.21 | -2.89 | -2.62 | -2.08 |
| Apr-05 | 4.08 | 7.13 | 3.22 | 0.21 | 3.87 | 6.92 | 3.01 |
| May-05 | 2.25 | 0.51 | 0.15 | 0.24 | 2.01 | 0.27 | -0.09 |
| Jun-05 | 3.55 | 5.28 | 3.82 | 0.23 | 3.32 | 5.05 | 3.59 |
| Jul-05 | -1.85 | -0.85 | -0.93 | 0.24 | -2.09 | -1.09 | -1.17 |
| Aug-05 | 0.38 | 1.34 | 0.80 | 0.3 | 0.08 | 1.04 | 0.50 |
| Sep-05 | -1.88 | 0.30 | -2.36 | 0.29 | -2.17 | 0.01 | -2.65 |
| Oct-05 | 2.87 | 5.04 | 4.39 | 0.27 | 2.60 | 4.77 | 4.12 |
| Nov-05 | 0.93 | 0.46 | -0.19 | 0.31 | 0.62 | 0.15 | -0.50 |
| Dec-05 | 3.41 | 3.30 | 2.41 | 0.32 | 3.09 | 2.98 | 2.09 |
| Jan-06 | 0.00 | -1.82 | 0.57 | 0.35 | -0.35 | -2.17 | 0.22 |
| Feb-06 | 0.71 | 0.34 | 1.65 | 0.34 | 0.37 | 0.00 | 1.31 |
| Mar-06 | 0.53 | -0.45 | 1.26 | 0.37 | 0.16 | -0.82 | 0.89 |
| Apr-06 | -3.00 | -5.84 | -3.01 | 0.36 | -3.36 | -6.20 | -3.37 |
| May-06 | -0.27 | -1.55 | 0.26 | 0.43 | -0.70 | -1.98 | -0.17 |
| Jun-06 | 1.09 | -2.36 | 0.45 | 0.4 | 0.69 | -2.76 | 0.05 |
| Jul-06 | 2.70 | 3.60 | 2.18 | 0.4 | 2.30 | 3.20 | 1.78 |
| Aug-06 | 2.63 | 2.87 | 2.70 | 0.42 | 2.21 | 2.45 | 2.28 |
| Sep-06 | 2.56 | 1.92 | 3.16 | 0.41 | 2.15 | 1.51 | 2.75 |
| Oct-06 | 0.75 | 1.94 | 1.99 | 0.41 | 0.34 | 1.53 | 1.58 |
| Nov-06 | 2.32 | 0.22 | 0.77 | 0.42 | 1.90 | -0.20 | 0.35 |
| Dec-06 | 1.86 | 2.18 | 1.51 | 0.4 | 1.46 | 1.78 | 1.11 |
| Jan-07 | -2.30 | -1.69 | -1.96 | 0.44 | -2.74 | -2.13 | -2.40 |
| Feb-07 | 0.81 | 0.89 | 1.16 | 0.38 | 0.43 | 0.51 | 0.78 |
| Mar-07 | 3.87 | 2.92 | 4.42 | 0.43 | 3.44 | 2.49 | 3.99 |
| Apr-07 | 3.65 | 3.37 | 3.39 | 0.44 | 3.21 | 2.93 | 2.95 |
| May-07 | -2.02 | -1.14 | -1.46 | 0.41 | -2.43 | -1.55 | -1.87 |
| Jun-07 | -4.74 | -0.79 | -3.13 | 0.4 | -5.14 | -1.19 | -3.53 |
| Jul-07 | 1.12 | 1.11 | 1.28 | 0.4 | 0.72 | 0.71 | 0.88 |
| Aug-07 | 2.54 | 4.91 | 3.88 | 0.42 | 2.12 | 4.49 | 3.46 |
| Sep-07 | 0.39 | 2.19 | 1.35 | 0.32 | 0.07 | 1.87 | 1.03 |
| Oct-07 | -5.86 | -3.90 | -3.87 | 0.32 | -6.18 | -4.22 | -4.19 |
| Nov-07 | 0.25 | 0.00 | -1.13 | 0.34 | -0.09 | -0.34 | -1.47 |
| Dec-07 | -5.39 | -10.34 | -6.04 | 0.27 | -5.66 | -10.61 | -6.31 |
| Jan-08 | -2.67 | -2.38 | -2.58 | 0.21 | -2.88 | -2.59 | -2.79 |
| Feb-08 | -0.89 | -0.06 | -0.90 | 0.13 | -1.02 | -0.19 | -1.03 |
| Mar-08 | 5.64 | 6.32 | 4.77 | 0.17 | 5.47 | 6.15 | 4.60 |
| Apr-08 | 1.44 | 3.00 | 1.51 | 0.17 | 1.27 | 2.83 | 1.34 |
| May-08 | -7.18 | -6.25 | -8.35 | 0.17 | -7.35 | -6.42 | -8.52 |
| Jun-08 | -1.80 | -0.40 | -0.90 | 0.17 | -1.97 | -0.57 | -1.07 |
| Jul-08 | 1.37 | 0.45 | 1.55 | 0.15 | 1.22 | 0.30 | 1.40 |
| Aug-08 | -7.50 | -11.63 | -9.42 | 0.12 | -7.62 | -11.75 | -9.54 |
| Sep-08 | -18.16 | -15.45 | -16.52 | 0.15 | -18.31 | -15.60 | -16.67 |
| Oct-08 | -7.16 | -7.93 | -6.96 | 0.08 | -7.24 | -8.01 | -7.04 |
| Nov-08 | 2.70 | 0.57 | 0.99 | 0.02 | 2.68 | 0.55 | 0.97 |
| Dec-08 | -10.76 | -4.49 | -8.22 | 0.09 | -10.85 | -4.58 | -8.31 |
|
Average | -0.59 | -0.60 | -0.55 | ||||
|
Standard Deviation | 4.05 | 4.21 | 3.81 |
Standard deviation for the two funds is 4.05 and 4.21.
(d)
To Calculate:
In this question we have to calculate Sharpe, Jensen and Treynor ratio for each fund.
Explanation of Solution
All these ratios are calculated as:
Sharpe ratio=
Excess Return / standard devation
Treynor=Excess return / Beta
Jensen=αP = E(rP) − {rf + βP [E(rM) − rf]}
|
|
Total monthly returns |
|
|
|
Excess Returns |
|
|
|
Month |
Vanguard |
Value Fund |
S&P |
T-bills |
Vanguard |
Value Fund |
S&P |
| Jan-04 | 1.52 | -0.07 | 1.35 | 0.07 | 1.45 | -0.14 | 1.28 |
| Feb-04 | -0.53 | -0.99 | -1.32 | 0.06 | -0.59 | -1.05 | -1.38 |
| Mar-04 | -2.69 | -2.46 | -1.89 | 0.09 | -2.78 | -2.55 | -1.98 |
| Apr-04 | 0.22 | 2.52 | 1.70 | 0.08 | 0.14 | 2.44 | 1.62 |
| May-04 | 2.53 | 1.39 | 1.85 | 0.06 | 2.47 | 1.33 | 1.79 |
| Jun-04 | -1.93 | -7.46 | -3.22 | 0.08 | -2.01 | -7.54 | -3.30 |
| Jul-04 | 0.99 | -0.78 | 0.25 | 0.1 | 0.89 | -0.88 | 0.15 |
| Aug-04 | 1.52 | 2.49 | 1.00 | 0.11 | 1.41 | 2.38 | 0.89 |
| Sep-04 | 0.21 | 1.46 | 1.29 | 0.11 | 0.10 | 1.35 | 1.18 |
| Oct-04 | 5.22 | 4.66 | 4.46 | 0.11 | 5.11 | 4.55 | 4.35 |
| Nov-04 | 3.24 | 3.60 | 3.01 | 0.15 | 3.09 | 3.45 | 2.86 |
| Dec-04 | -2.36 | -3.48 | -2.24 | 0.16 | -2.52 | -3.64 | -2.40 |
| Jan-05 | 3.62 | 0.79 | 2.09 | 0.16 | 3.46 | 0.63 | 1.93 |
| Feb-05 | -2.33 | -2.86 | -1.83 | 0.16 | -2.49 | -3.02 | -1.99 |
| Mar-05 | -2.68 | -2.41 | -1.87 | 0.21 | -2.89 | -2.62 | -2.08 |
| Apr-05 | 4.08 | 7.13 | 3.22 | 0.21 | 3.87 | 6.92 | 3.01 |
| May-05 | 2.25 | 0.51 | 0.15 | 0.24 | 2.01 | 0.27 | -0.09 |
| Jun-05 | 3.55 | 5.28 | 3.82 | 0.23 | 3.32 | 5.05 | 3.59 |
| Jul-05 | -1.85 | -0.85 | -0.93 | 0.24 | -2.09 | -1.09 | -1.17 |
| Aug-05 | 0.38 | 1.34 | 0.80 | 0.3 | 0.08 | 1.04 | 0.50 |
| Sep-05 | -1.88 | 0.30 | -2.36 | 0.29 | -2.17 | 0.01 | -2.65 |
| Oct-05 | 2.87 | 5.04 | 4.39 | 0.27 | 2.60 | 4.77 | 4.12 |
| Nov-05 | 0.93 | 0.46 | -0.19 | 0.31 | 0.62 | 0.15 | -0.50 |
| Dec-05 | 3.41 | 3.30 | 2.41 | 0.32 | 3.09 | 2.98 | 2.09 |
| Jan-06 | 0.00 | -1.82 | 0.57 | 0.35 | -0.35 | -2.17 | 0.22 |
| Feb-06 | 0.71 | 0.34 | 1.65 | 0.34 | 0.37 | 0.00 | 1.31 |
| Mar-06 | 0.53 | -0.45 | 1.26 | 0.37 | 0.16 | -0.82 | 0.89 |
| Apr-06 | -3.00 | -5.84 | -3.01 | 0.36 | -3.36 | -6.20 | -3.37 |
| May-06 | -0.27 | -1.55 | 0.26 | 0.43 | -0.70 | -1.98 | -0.17 |
| Jun-06 | 1.09 | -2.36 | 0.45 | 0.4 | 0.69 | -2.76 | 0.05 |
| Jul-06 | 2.70 | 3.60 | 2.18 | 0.4 | 2.30 | 3.20 | 1.78 |
| Aug-06 | 2.63 | 2.87 | 2.70 | 0.42 | 2.21 | 2.45 | 2.28 |
| Sep-06 | 2.56 | 1.92 | 3.16 | 0.41 | 2.15 | 1.51 | 2.75 |
| Oct-06 | 0.75 | 1.94 | 1.99 | 0.41 | 0.34 | 1.53 | 1.58 |
| Nov-06 | 2.32 | 0.22 | 0.77 | 0.42 | 1.90 | -0.20 | 0.35 |
| Dec-06 | 1.86 | 2.18 | 1.51 | 0.4 | 1.46 | 1.78 | 1.11 |
| Jan-07 | -2.30 | -1.69 | -1.96 | 0.44 | -2.74 | -2.13 | -2.40 |
| Feb-07 | 0.81 | 0.89 | 1.16 | 0.38 | 0.43 | 0.51 | 0.78 |
| Mar-07 | 3.87 | 2.92 | 4.42 | 0.43 | 3.44 | 2.49 | 3.99 |
| Apr-07 | 3.65 | 3.37 | 3.39 | 0.44 | 3.21 | 2.93 | 2.95 |
| May-07 | -2.02 | -1.14 | -1.46 | 0.41 | -2.43 | -1.55 | -1.87 |
| Jun-07 | -4.74 | -0.79 | -3.13 | 0.4 | -5.14 | -1.19 | -3.53 |
| Jul-07 | 1.12 | 1.11 | 1.28 | 0.4 | 0.72 | 0.71 | 0.88 |
| Aug-07 | 2.54 | 4.91 | 3.88 | 0.42 | 2.12 | 4.49 | 3.46 |
| Sep-07 | 0.39 | 2.19 | 1.35 | 0.32 | 0.07 | 1.87 | 1.03 |
| Oct-07 | -5.86 | -3.90 | -3.87 | 0.32 | -6.18 | -4.22 | -4.19 |
| Nov-07 | 0.25 | 0.00 | -1.13 | 0.34 | -0.09 | -0.34 | -1.47 |
| Dec-07 | -5.39 | -10.34 | -6.04 | 0.27 | -5.66 | -10.61 | -6.31 |
| Jan-08 | -2.67 | -2.38 | -2.58 | 0.21 | -2.88 | -2.59 | -2.79 |
| Feb-08 | -0.89 | -0.06 | -0.90 | 0.13 | -1.02 | -0.19 | -1.03 |
| Mar-08 | 5.64 | 6.32 | 4.77 | 0.17 | 5.47 | 6.15 | 4.60 |
| Apr-08 | 1.44 | 3.00 | 1.51 | 0.17 | 1.27 | 2.83 | 1.34 |
| May-08 | -7.18 | -6.25 | -8.35 | 0.17 | -7.35 | -6.42 | -8.52 |
| Jun-08 | -1.80 | -0.40 | -0.90 | 0.17 | -1.97 | -0.57 | -1.07 |
| Jul-08 | 1.37 | 0.45 | 1.55 | 0.15 | 1.22 | 0.30 | 1.40 |
| Aug-08 | -7.50 | -11.63 | -9.42 | 0.12 | -7.62 | -11.75 | -9.54 |
| Sep-08 | -18.16 | -15.45 | -16.52 | 0.15 | -18.31 | -15.60 | -16.67 |
| Oct-08 | -7.16 | -7.93 | -6.96 | 0.08 | -7.24 | -8.01 | -7.04 |
| Nov-08 | 2.70 | 0.57 | 0.99 | 0.02 | 2.68 | 0.55 | 0.97 |
| Dec-08 | -10.76 | -4.49 | -8.22 | 0.09 | -10.85 | -4.58 | -8.31 |
|
Average | -0.59 | -0.60 | -0.55 | ||||
|
Sharpe ratio | -0.15 | -0.14 | -0.14 | ||||
|
Alpha | -0.03 | -0.04 | 0.00 | ||||
|
Treynor | -0.57 | -0.58 | -0.55 |
| Alpha | -0.03 | -0.04 | 0.00 |
| Treynor | -0.57 | -0.58 | -0.55 |
Want to see more full solutions like this?
Chapter 18 Solutions
Connect 1-Semester Access Card for Essentials of Investments
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- To invest in a project, a company needs $50 million. Given its flotation costs of 7%, how much does the company need to raise? Multiple choice question. $53.76 million $46.50 million $50.00 million $53.50 millionarrow_forwardWhile determining the appropriate discount rate, if a firm uses a weighted average cost of capital that is unique to a particular project, it is using the Blank______. Multiple choice question. economic value added method pure play approach subjective approach security market line approacharrow_forwardWhat are flotation costs? Multiple choice question. They are the costs incurred to issue new securities in the market. They are the costs incurred to insure the payment due to bondholders. They are the costs incurred to meet day to day expenses. They are the costs incurred to keep a project in the business.arrow_forward
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