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1
1.The following are monthly percentage price changes for four market indexes.
Month DJ SP500
1 0.03 0.02 2 0.07 0.06 3 −0.02 −0.01 4 0.01 0.03 5 0.05 0.04 6 −0.06 −0.04
Russ500 Nicky
0.04 0.04 0.10 −0.02 −0.04 0.07 0.03 0.02 0.11 0.02 −0.08 0.06
(17 points) Compute the following.
a. Average monthly rate of return for each index
b. Standard deviation for each index
c. Covariance between the rates of return for the following indexes:
DJ– SP 500 SP500–Russ500 SP500–Nicky Russ500–Nicky
d. The correlation coefficients for the same four combinations
2.Xpord University has 10 million rupees in its consolidated fund. Xpord has hired your investment firm to manage the above fund. You work in the bond research department of the investment firm. And suppose that the portfolio manager has asked you to evaluate several bonds for potential purchase. Issuer Maturity Bond Rating Bond Annual Frequency of Coupon Coupon Payment Rate A 5% Annual B 8% Annual ? 6% Annual D 9% Semi- Annual Government Corporate Corporate Corporate S 5 3 4 Market Price (Rs.) 800 870 895 930 BBB AAA BB The economic team of your investment firm has updated its economic forecasts for the next five years and those details are given below. • Real risk-free rate :1% • Annual average of expected inflation rate for the next 5 years: 1% • AAA- Default risk premium:1% • BBB-Default risk premium: 2% • BB- Default risk premium: 3% • AAA-Liquidity risk premium: 0.6% • BBB-Liquidity risk premium :1% • BB-Liquidity risk premium :1.8% • Treasury Liquidity premium: 0% Maturity risk premium: 0.8 % per year The required rate of return will be calculated using following formula; Required rate of return = Real risk-free rafe + Inflation Premium + Default Risk Premium + Liquidity premium + Maturity Risk Premium The firm calculates the Inflation Premium based on following formula: Inflation Premium = (1-1) * Annual average of expected inflation Where, T-Maturity period in years
? The required rate of return will be calculated using following formula; Required rate of return = Real risk-free rate + Inflation Premium + Default Risk Premium + Liquidity premium + Maturity Risk Premium ? The firm calculates the Inflation Premium based on following formula; Inflation Premium = (T-1) * Annual average of expected inflation Where, T = Maturity period in years You can assume that the face value of bonds A, B, C and D is Rs.1000/= You are required to analyze the bonds and answer the following questions; i. The portfolio manager wants you to analyze above bonds and complete the following table (show all calculations). Further you are required to recommend witch bond/s to buy. Bond Current yield Require rate of Yield to Intrinsic value return maturity A B ? D ii. The portfolio manager is worried that the economy may enter a recession in 2021 with the covid -19 situation. Which bond would have the least risk in this scenario? Which bond would have the most risk? Why? The portfolio manager has asked you to identify any bonds that minimize the Reinvestment Risk. Which bond do you select? Justify your selection.
Expert Solution
1.
|
Month |
DJ |
SP500 |
Russ500 |
Nicky |
|
1 |
0.03 |
0.02 |
0.04 |
0.04 |
|
2 |
0.07 |
0.06 |
0.10 |
-0.02 |
|
3 |
-0.02 |
-0.01 |
-0.04 |
0.07 |
|
4 |
0.01 |
0.03 |
0.03 |
0.02 |
|
5 |
0.05 |
0.04 |
0.11 |
0.02 |
|
6 |
-0.06 |
0.04 |
-0.08 |
0.06 |
a) Average monthly rate of return for each index
Average monthly rate of return of DJ = (0.03+0.07-0.02+0.01+0.05-0.06)/6 = 0.0133
Average monthly rate of return of SP500 = (0.02+0.06-0.01+0.03+0.04+0.04)/6 = 0.0300
Average monthly rate of return of Russ500 = (0.04+0.10-0.04+0.03+0.11-0.08)/6 = 0.0267
Average monthly rate of return of DJ = (0.04-0.02+0.07+0.02+0.02+0.06)/6 = 0.0317
b) Standard deviation for each index
As the distribution is Sample, Stock’s Standard Deviation = √{1/(n-1)*∑(Ri-Rmean)2}
where, Ri is expected return of index i
Rmean is arithmetic mean of return
n is number of observation
Std dev of DJ:
|
Expected Return (%)R(i) |
R(mean) |
R(i)-R(mean) |
{R(i)-R(mean)}2 |
|
0.03 |
0.0133 |
0.0167 |
0.00028 |
|
0.07 |
0.0133 |
0.0567 |
0.00321 |
|
-0.02 |
0.0133 |
-0.0333 |
0.00111 |
|
0.01 |
0.0133 |
-0.0033 |
0.00001 |
|
0.05 |
0.0133 |
0.0367 |
0.00134 |
|
-0.06 |
0.0133 |
-0.0733 |
0.00538 |
Stock’s Standard Deviation = Square root of{sum [{R(i)-R(mean)}2] / (n-1)} = Square root (0.0113/5) Stock’s Standard Deviation of DJ = 0.0476
Std dev of SP500:
|
Expected Return (%)R(i) |
R(mean) |
R(i)-R(mean) |
{R(i)-R(mean)}2 |
|
0.02 |
0.0300 |
-0.0100 |
0.00010 |
|
0.06 |
0.0300 |
0.0300 |
0.00090 |
|
-0.01 |
0.0300 |
-0.0400 |
0.00160 |
|
0.03 |
0.0300 |
0.0000 |
0.00000 |
|
0.04 |
0.0300 |
0.0100 |
0.00010 |
|
0.04 |
0.0300 |
0.0100 |
0.00010 |
Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}2] / (n-1) = Square root of (0.0028)/5 Stock’s Standard Deviation of SP500 = 0.0237
Std dev of RUSS500:
|
Expected Return (%)R(i) |
R(mean) |
R(i)-R(mean) |
{R(i)-R(mean)}2 |
|
0.04 |
0.0267 |
0.0133 |
0.00018 |
|
0.10 |
0.0267 |
0.0733 |
0.00538 |
|
-0.04 |
0.0267 |
-0.0667 |
0.00444 |
|
0.03 |
0.0267 |
0.0033 |
0.00001 |
|
0.11 |
0.0267 |
0.0833 |
0.00694 |
|
-0.08 |
0.0267 |
-0.1067 |
0.01138 |
Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}2] / (n-1) = Square root of (0.0283)/5 Stock’s Standard Deviation of Russ500 = 0.0753
Std dev of Nicky:
|
Expected Return (%)R(i) |
R(mean) |
R(i)-R(mean) |
{R(i)-R(mean)}2 |
|
0.04 |
0.0317 |
0.0083 |
0.00007 |
|
-0.02 |
0.0317 |
-0.0517 |
0.00267 |
|
0.07 |
0.0317 |
0.0383 |
0.00147 |
|
0.02 |
0.0317 |
-0.0117 |
0.00014 |
|
0.02 |
0.0317 |
-0.0117 |
0.00014 |
|
0.06 |
0.0317 |
0.0283 |
0.00080 |
Stock’s Standard Deviation = Square root of sum [{R(i)-R(mean)}2] / (n-1) = Square root of (0.0053)/5 Stock’s Standard Deviation of Nicky = 0.0325
c) Covariance between the rates of return for the following indexes:
Cov (i,j) = √[1/(n-1)*∑{(Xi-Xmean)* (Yj-Ymean)}]
where, Xi is expected return of index i
Xmean is arithmetic mean of return of index i
Yi is expected return of index j
Ymean is arithmetic mean of return of index j
n is number of observation
DJ– SP 500
|
Expected Return (%) X(i) |
X(mean) |
X(i) - X(mean) |
Expected Return (%) Y(j) |
Y(mean) |
Y(j) - Y(mean) |
{X(i) - X(mean)}*{Y(j) - Y(mean)} |
|
0.03 |
0.0133 |
0.01667 |
0.02 |
0.0300 |
-0.01000 |
-0.000167 |
|
0.07 |
0.0133 |
0.05667 |
0.06 |
0.0300 |
0.03000 |
0.001700 |
|
-0.02 |
0.0133 |
-0.03333 |
-0.01 |
0.0300 |
-0.04000 |
0.001333 |
|
0.01 |
0.0133 |
-0.00333 |
0.03 |
0.0300 |
0.00000 |
0.000000 |
|
0.05 |
0.0133 |
0.03667 |
0.04 |
0.0300 |
0.01000 |
0.000367 |
|
-0.06 |
0.0133 |
-0.07333 |
0.04 |
0.0300 |
0.01000 |
-0.000733 |
Cov(DJ, SP500) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (0.0025)/5
Cov(DJ, SP500) = 0.0005
SP500–Russ500
|
Expected Return (%) X(i) |
X(mean) |
X(i) - X(mean) |
Expected Return (%) Y(j) |
Y(mean) |
Y(j) - Y(mean) |
{X(i) - X(mean)}*{Y(j) - Y(mean)} |
|
0.02 |
0.0300 |
-0.01000 |
0.04 |
0.0267 |
0.01333 |
-0.000133 |
|
0.06 |
0.0300 |
0.03000 |
0.10 |
0.0267 |
0.07333 |
0.002200 |
|
-0.01 |
0.0300 |
-0.04000 |
-0.04 |
0.0267 |
-0.06667 |
0.002667 |
|
0.03 |
0.0300 |
0.00000 |
0.03 |
0.0267 |
0.00333 |
0.000000 |
|
0.04 |
0.0300 |
0.01000 |
0.11 |
0.0267 |
0.08333 |
0.000833 |
|
0.04 |
0.0300 |
0.01000 |
-0.08 |
0.0267 |
-0.10667 |
-0.001067 |
Cov(SP500–Russ500) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (0.0045)/5
Cov(SP500–Russ500) = 0.0009
SP500–Nicky
|
Expected Return (%) X(i) |
X(mean) |
X(i) - X(mean) |
Expected Return (%) Y(j) |
Y(mean) |
Y(j) - Y(mean) |
{X(i) - X(mean)}*{Y(j) - Y(mean)} |
|
0.02 |
0.0300 |
-0.01000 |
0.04 |
0.0317 |
0.00833 |
-0.000083 |
|
0.06 |
0.0300 |
0.03000 |
-0.02 |
0.0317 |
-0.05167 |
-0.001550 |
|
-0.01 |
0.0300 |
-0.04000 |
0.07 |
0.0317 |
0.03833 |
-0.001533 |
|
0.03 |
0.0300 |
0.00000 |
0.02 |
0.0317 |
-0.01167 |
0.000000 |
|
0.04 |
0.0300 |
0.01000 |
0.02 |
0.0317 |
-0.01167 |
-0.000117 |
|
0.04 |
0.0300 |
0.01000 |
0.06 |
0.0317 |
0.02833 |
0.000283 |
Cov(SP500–Nicky) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (-0.0030)/5
Cov(SP500–Nicky) = -0.0006
Russ500–Nicky
|
Expected Return (%) X(i) |
X(mean) |
X(i) - X(mean) |
Expected Return (%) Y(j) |
Y(mean) |
Y(j) - Y(mean) |
{X(i) - X(mean)}*{Y(j) - Y(mean)} |
|
0.04 |
0.0267 |
0.01333 |
0.04 |
0.0317 |
0.00833 |
0.000111 |
|
0.10 |
0.0267 |
0.07333 |
-0.02 |
0.0317 |
-0.05167 |
-0.003789 |
|
-0.04 |
0.0267 |
-0.06667 |
0.07 |
0.0317 |
0.03833 |
-0.002556 |
|
0.03 |
0.0267 |
0.00333 |
0.02 |
0.0317 |
-0.01167 |
-0.000039 |
|
0.11 |
0.0267 |
0.08333 |
0.02 |
0.0317 |
-0.01167 |
-0.000972 |
|
-0.08 |
0.0267 |
-0.10667 |
0.06 |
0.0317 |
0.02833 |
-0.003022 |
Cov(Russ500–Nicky) = sum [{X(i)-X(mean)}*{Y(j)-Y(mean)}] / (n-1) = (-0.01027)/5
Cov(Russ500–Nicky) = -0.0021
d) The correlation coefficients for the same four combinations
Correlation coefficient between the two stocks = Covariance / Product of standard deviations = Cov (RA , RB) / (sigmaA x sigmaB )
where, A is one index
B is other index
sigmaA is std dev of A
sigmaB is std dev of B
DJ– SP 500
Correlation coefficient = Cov(DJ– SP 500)/ (std dev of DJ * std dev of SP500 )
= 0.0005 / (0.0476*0.0237) = 0.4432
SP500–Russ500
Correlation coefficient = Cov(SP500–Russ500)/ (std dev of SP500 * std dev of Russ500 )
= 0.0009 / (0.0237*0.0753) = 0.5043
SP500–Nicky
Correlation coefficient = Cov(SP500–Nicky)/ (std dev of SP500 * std dev of Nicky )
= -0.0006 / (0.0237*0.0325) = -0.7790
Russ500–Nicky
Correlation coefficient = Cov(Russ500–Nicky)/ (std dev of Russ500 * std dev of Nicky )
= -0.0021 / (0.0753*0.0325) = -0.8581
2.Given: We have assumed face value of all the bonds be Rs 1000
For Current Yield= (Coupon Payment)/Current Market price of the Bond
Yield to Maturity(YTM)= (Coupon+(Face Value-Market value)/maturity)/(Face Value-Marke
t Value/2)
YTM for semi annual coupon payments= (Coupon/2+(Face Value-Market value)/maturity*2)/(Face Value-Market Value/2)
n= maturtiy period
C= Coupon payment
Given:
Required rate of return = Real Risk-Rree Rate+Inflation premium+Default Risk Premium+Liquidity Premium+Maturity Risk premium
Calculation as folllows:
Answer 1)
Answer2) a)The Bond A which is the government bond has the least risk
Reason: We do not have default risk premium for this bond which means it has no probability of default thus it is safeguarded from market risk also the government bonds are assumed to be the safest among the categories of bonds.
b)The Bond D has the highet risk
Reason: The credit rating of Bond D is also not good compare to other bonds and we have the highest default risk premium for this bond which means the probability of default on this bond is high. So, if we enter the recession the market will not favour the company which has issued the Bond D.
Default risk premium is the additional amount that we receive for the bond
Answer3) In this scenario, I would select the Bond C.
Reason: Though the yield of the Bond C is lesser than the Bond B and Bond D, the Bond C bond has the good credit rating with low liquidity premium which means the bond is liquid and the cash flows from the bond will not be an issue as the frequency of bond trading is good, also, the duration of the bond is not long means the bond is safeguarded from the long term market risk.
PLEASE SEE THE ATTACHED FILE.
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