Mehta, Shivam
WMS Assignment Cover Sheet
Family Name:
Mehta
Student ID Number:
1526130
Paper Code:
517-19B(Ham)
First Name:
Shivam
Tutorial:
Assignment Name:
Individual assignment
Due Date:
18/09/2019
Please fill in the above information and then save this page as the first page of your assignment to
be submitted.
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and student ID numbers on this form below:
Mehta, Shivam
Q.1. My pension plan will pay me $5,000 once a year for a 10-year period.
The first payment will come in exactly five years. The pension fund wants
to immunize its position.
a. What is the duration of its obligation to me? The current interest rate is
8% per year.
A.1.a) We know , Duration (D) = ∑𝑛𝑡=1(𝑤𝑡 *t)
Where, y is the interest rate,
𝑤𝑡 is the weight of each cash flow(CF) and is equal to
𝑤𝑡 = (CFt /(1+y)t )/Bond Price.
Bond price is the sum of all the present values of the cash flow.
So, to start, let us calculate the present values of all cash flows.
We know present value of a cash flow is given by,
P.V=CFt /(1+y)t
Here, since the first cash flow occurs in the 5th year, we can start the
process to calculate present value of cash flow from the 5th year and
continue till the 14th year. Also interest rate is given as 8% (y=0.08).
Tabular calculations for the Present value are done on the next page.
Mehta, Shivam
Year
5
6
7
8
9
10
11
12
13
14
Present value
5000/1.085
5000/1.086
5000/1.087
5000/1.088
5000/1.089
5000/1.0810
5000/1.0811
5000/1.0812
5000/1.0813
5000/1.0814
= 3402.916
= 3150.849
= 2917.452
= 2701.344
= 2501.245
= 2315.967
= 2144.414
= 1985.569
= 1838.490
= 1702.305
Sum of all present values .i.e. Bond price = 24660.551
Now , Calculating wt = Present value/ Bond price
Year
5
6
7
8
9
10
11
12
13
14
wt
3402.916/24660.551=0.138
3150.849/24660.551=0.128
2917.452/24660.551=0.118
2701.344/24660.551=0.110
2501.245/24660.551=0.101
2315.967/24660.551=0.094
2144.414/24660.551=0.087
1985.569/24660.551=0.081
1838.490/24660.551=0.075
1702.305/24660.551=0.069
Finally, Calculating duration as Duration (D) = ∑𝑛𝑡=1(𝑤𝑡 *t)
Where t starts from 5 as there are no cash flows in the first 4 years,
Mehta, Shivam
t*wt
5*0.138
6*0.128
7*0.118
8*0.110
9*0.101
10*0.094
11*0.087
12*0.081
13*0.075
14*0.069
=0.690
=0.767
=0.828
=0.876
=0.913
=0.939
=0.957
=0.966
=0.969
=0.966
Adding all t*wt to get net duration.
Net duration = 8.871 years.
Q.1.b) If the plan uses 3-year and 30-year zero-coupon bonds to construct
the immunized position, how much money ought to be placed in each
bond?
A.1.b) Now we know , net duration is 8.871 years.
In immunisation technique, we need the duration of the assets and
liabilities to be equal.
So, let us assume initially, ’ X ‘ proportion of funds are placed in the 3
year zero coupon bond. Therefore, (1-X) proportion of funds will b put in
the 30 year zero coupon bond. Since duration of the zero coupon bond is
equal to the time to maturity,
We have,
8.871= X*3 + (1-X)*30
Solving for X, we get,
As duration of assets and liabilities
needs to be equal.
Mehta, Shivam
X=0.7825 .
Therefore, initially we must allocate 78.25% of the total funds to the 3year zero coupon bond , and 21.75 % funds to the 30 year zero coupon
bond.
Total funds to be allocated is equal to the bond price, which has been
calculated as 24660.551.
Therefore, total allocation of the 3-year zero coupon bond is ,
Bond price= 0.7825*24660.551= $ 19296.881
X*
Allocation in 30 year zero coupon bond is, (1-X)*Bond price,
=0.2175*24660.551= $ 5363.670
Q.1.c) What will be the face value of the holdings in each zero?
A.1.c) The face value of any zero coupon bond is calculated as,
Value allocated in the bond * (1+Interest rate)Time to maturity
Interest rate is given as 8 percent .
Therefore, for the 3 year zero coupon bond,
Face value = 19296.881*(1+0.08)3
Solving to get face value of the 3 year zero coupon bond =
$24308.5125583.
Similarly, for the 30 year zero coupon bond,
Face value = 5363.670* (1+0.08)30
Solving to get face value of the 30 year old coupon bond = $53972.771
Q.2. Why do you think the change in the index of labor cost per unit of
output is a useful lagging indicator of the macroeconomy?
Mehta, Shivam
A.2. Firstly, let us define what is meant by a lagging indicator of the
macroeconomy. As the term suggests, a lagging indicator is a statistic that
changes value after the economic conditions have fluctuated. To simplify,
if the economy rises(i.e. there is growth in the economy), a lagging
indicator will also increase but only after some time has elapsed.
From a manufacturing standpoint, the labor cost per unit of output is
basically the wages that are paid to the labor for producing 1 unit of
product. If there is expansion in the economy, this index will rise, owing
to the increased demand for labor. If there is contraction in the economy,
this index will fall due to less demand for output. However, since this is a
lagging indicator, the labor cost will only rise after a certain period of
time, in case of expansion in economy and will fall only after certain time
as passed , in case of economic de-growth.
Now considering , if there is an expansion in the economy, since wages
will rise only after some time, the employers can considerably increase
the output without paying extra wage to the labor-force, thus they
produce more goods and at a low cost. This low cost of product-benefit, is
passed on to the consumers in terms of low price, so that they may buy
more product, thereby increasing the demand and hence stimulating the
economy.
Similarly, if there is a slowdown in the economy, the wages will fall only
after some time has elapsed. This means during the actual period of
slowdown, the labor force are being paid some excess wage . This extra
wage, will thereby promote and stimulate demand in the economy and
hence reduce the period of slowdown, as well as serve as an extra-pool of
income to help the labor during the later stages of economic slowdown,
when the wages get reduced.
Q.3. Your business plan for your proposed start-up firm envisions firstyear revenues of $240,000, fixed costs of $60,000, and variable costs
equal to one-third of revenue.
1) What are the expected profits based on these expectations?
Mehta, Shivam
A.3.1) Here, Revenue= R= $240,000
fixed costs=F= $60000
Variable costs =V= 1/3 *Revenue=240000/3= $80000
Let us assume, profit =P
P= R- (F+V)
P=240000-(60000+80000)
Therefore, profit , P= $100,000.
Based on given expected revenues, we can expect a profit of $100,000.
Q.3.2) What is the degree of operating leverage based on the estimate of
fixed costs and expected profits?
A.3.2) We know, Degree of operating leverage is mathematically defined
as , D.O.L= 1+ (Fixed Costs/Profits)
Putting values of fixed costs(F) and profits(P) , we get,
D.O.L =1+(60000/100000)=1.6
This means, for every 1 % change in sales, profits will change by 1.6
percent in the same direction.
Q.3.3) If sales are 10% below expectation, what will be the decrease in
profits?
A.3.3) As per above, since DOL = 1.6, it means, for every 1 % change in
sales, profits will change by 1.6 percent in the same direction.
Mehta, Shivam
For a 10 percent decrease in sales, we have 10*DOL=16 percent decrease
in profits.
Hence the profits will decrease by 16 percent if sales decrease by 10
percent.
Q.3.4) Show that the percentage decrease in profits equals DOL times the
10% drop in sales.
A.3.4) From the equation of DOL,
DOL= Percentage change in profits/Percentage change in sales
Mathematically, we multiply both sides of the equation by (percentage
change in sales)
This means , DOL*Percentage change in sales= Percentage change in
profits.
If sales drop by 10%,
Percentage change in profits= DOL*Percentage drop in sales
Percentage decrease in profits=DOL*10
It is important to note if sales decrease, the profits also decrease.
Q.3.5) Based on the DOL, what is the largest percentage shortfall in sales
relative to original expectations that the firm can sustain before profits
turn negative?
A.3.5) Since DOL=1.6, 1 percent decrease in sales causes 1.6 percent
decrease in profits.
We need to calculate what percent decrease in sales would cause 100
percent decrease in profits.( As the question specifies this. 100 percent
decrease in profits is the threshold for the profits turning negative)
Mehta, Shivam
We know DOL =Percentage change(decrease) in profits/ Percentage
change(decrease) in sales
DOL=1.6,
Percentage decrease in profits= 100 %
 1.6= 100/Percentage decrease in sales
 Percentage decrease in sales=100/1.6= 62.5 %
Hence 62.5 % shortfall in sales is the largest shortfall the firm can sustain
before the profits turn negative.
Q.4. A mutual fund manager achieve 12.7% annual return during last
year. Is it possible that this is associated with inferior performance? You
are expected to provide comprehensive analysis of performance
evaluation.
A.4. The term annual return is usually a base calculation of the absolute
returns earned without taking into account a plethora of factors, the
most of which is the risk of the portfolio(here, the mutual fund).
Before moving on, let us denote return earned by mutual fund= rp=12.7 %
Let the market return be rm.
Let the T-bill rate be rf
In performance evaluation , we use a number of ratios to understand
whether a given return is better or worse based on the ideal
environment.
First, let us define some ratios and their applications.
SHARPE’s Ratio : It is a reward to risk ratio which measures excess return
to entire portfolio risk(or standard deviation)
Mehta, Shivam
Mathematically , Sharpe’s ratio = (rp-rf)/∂p , where ∂p is the standard
deviation of the portfolio.
This ratio is used when we need to choose various portfolios which
competing for the entire risky portfolio.
Next we define Treynor’s Ratio : It is also a reward to risk ratio, which
measures excess return to systematic risk of the portfolio.
Mathematically, Treynor’s ratio = (rp-rf)/βp, where βp is the systematic risk
of portfolio. It is important to note βm = 1, where βm is the systematic risk
of the market.
This ratio is used when we need to select from portfolios that will be
mixed together to form a large investor portfolio.
Finally we define Information ratio: It is the ratio of alpha to the residual
or diversifiable risk.
Mathematically , Information ratio = αp/∂(ep) , where ∂(ep) is the residual
risk and αp is Jenson’s Alpha, defined as (rp –(rf + βp(rm-rf)). Jenson’s alpha
basically signifies the excess return generated by the portfolio compared
to the predicted return by CAPM. It is important to note, αm =0
We use information ratio, when we need to evaluate a portfolio to be
mixed with a benchmark portfolio.
We are given, rp =12.7% , βm = 1 , αm =0 and let us assume, rf =6%
Now continuing our analysis, let us look at few possible scenario’s, where
rm<rp.
Case
1
Portfolio
Standard
Deviation
50%
Market
standard
deviation
30%
Beta of
portfolio
Market
return
Residual
risk
1.7
10%
5
Mehta, Shivam
2
20%
30%
0.9
10%
2
In Case 1, let us calculate the respective ratios
Sharpe ratio for portfolio = (rp- rf)/Portfolio standard deviation
Sp=(12.7-6)/50 =0.1218
Similarly, Sharpe’s ratio for market,
Sm =(10-6)/30 = 0.1333
Next we calculate Treynor’s ratio for portfolio
Tp=(rp-rf)/βp
Substituting values, Tp=(12.7-6)/1.7 => Tp= 3.941
Similarly, for market, Tm=(10-6)/1 => Tm=4
Finally we calculate the information ratios,
αp = rp –(rf + βp(rm-rf))
Putting values,
αp = 12.7-(6+1.7(10-4))
 αp = -0.1
Hence information ratio= αp/∂(ep)
I.R =-0.1/5= -0.02
I.R of market =0
Mehta, Shivam
As a result, analysing these ratios, we find that the market performed
better as compared to the mutual fund, given the standard deviation
and residual risk.
Now we look at case 2 and calculate the same ratios for the portfolio
and the market.
Sharpe ratio for portfolio = (rp- rf)/Portfolio standard deviation
Sp=(12.7-6)/20 =0.335
Similarly, Sharpe’s ratio for market,
Sm =(10-6)/30 = 0.1333
Next we calculate Treynor’s ratio for portfolio
Tp=(rp-rf)/βp
Substituting values, Tp=(12.7-6)/0.9 => Tp= 7.44
Similarly, for market, Tm=(10-6)/1 => Tm=4
Finally we calculate the information ratios,
αp = rp –(rf + βp(rm-rf))
Putting values,
αp = 12.7-(6+0.9(10-4))
 αp = 3.1
Hence information ratio= αp/∂(ep)
I.R =3.1/2= 1.55
I.R of market =0
Mehta, Shivam
This actually shows the fund manager did well and outperformed the
market, given the standard deviation and residual risk.
Concluding all results, we find that the given return of 12.7% may be
associated with inferior performance by the fund manager. However ,
there is no decision to be made unless the context of the return, i.e.
risk(including beta, standard deviation, unsystematic risk ) is provided.
Even if the given portfolio outperforms the market, there may also
exist another portfolio Q, with specified returns and risks, for which
the ratios(or one of the ratio’s) is better than that of our portfolio. In
this case, it will be important to specify the purpose of the mutual
fund, i.e. If we need to compare these portfolios in a absolute sense,
we would prefer the portfolio with a higher Sharpe’s ratio. If the
mutual fund is part of a pension fund we need to look at the Treynor’s
ratio. Finally if the mutual fund is needed to add a position to an
existing portfolio, we need to pick the portfolio that has a higher
information ratio.
Q.5. You are an independent investment adviser who is assisting Alfred
Darwin, the head of the Investment Committee of General Technology
Corporation, to establish a new pension fund. Darwin asks Irish about
international equities and whether the Investment Committee should
consider them as an additional asset for the pension fund.
1) Explain the rationale for including international equities in General’s
equity portfolio. Identify and describe three relevant considerations in
formulating your answer.
A.5.1) International equity investment is an interesting choice.
Contrary to many passive fund managers, active managers prefer to
exploit the international markets in search of better opportunities.
Mehta, Shivam
Firstly, inclusion of international equity makes the entire portfolio
more diversified. It is well established that returns in different
countries are not perfectly correlated. This is because each country has
its own economic structures. Adding foreign equity markets to any
country’s individual index decreases the overall volatility of the given
portfolio and as a result will provide a better risk-return trade-off.
Consider an example wherein the US economy experiences a
slowdown and goes in a downtrend, since returns from other markets
are not fully correlated to US returns, investment abroad increases
diversity and hedges the position.
Secondly, International markets have become more accessible in
recent years. Accurate information about global markets and returns
can be obtained for most of the countries. All investors can take part in
investment in equity abroad. For example , instruments such as ADR’s
allow investors a claim on a given number of shares of foreign stock.
Also, many companies offer several mutual funds with international
focus. There is a proper classification of the country as ‘Developed’ and
‘Emerging’ based on the FTSE criteria which affords the investors more
information about their investment and helps mitigate some risk and
uncertainty about investing abroad.
Finally, International equity allows the investors to invest in emerging
markets. While investing in developed countries has its own
advantages, the growth rate in these markets is relatively low.
However, the emerging markets make up more than 20% percent of
the world GDP. The important part to consider here is that emerging
markets, although come with high risk on investment, but afford the
investors an opportunity to be a part of a high growth story that
presents itself in these markets. For example, China had an average
growth rate of 15% from 2011 to 2015. Another factor in favour of
investing in emerging markets is that since market capitalisation as a
Mehta, Shivam
percentage of GDP is relatively lower in these countries, the markets
can grow significantly even if the growth in GDP is not as rapid.
Q.5. 2) List three possible arguments against international equity
investment and briefly discuss the significance of each.
A.5.2) While international equity comes with a lot of opportunities, it also
has certain pitfalls which may make the investors wary of putting their
capital abroad.
Firstly, there is a highly evident exchange rate risk. For the investment in
international markets to reap gains, there are two uncertainties’ instead
of the usual one that investors have to keep in mind. Firstly, the
performance of the investment in the local currency. This is the risk that
the managers already bear. In addition, investors need to be cautious of
exchange rate fluctuations . The exchange rate determines the rate at
which the investment can be brought back into their home country ,
ideally with profit. However, if there are unfavourable movements in the
exchange rate, despite making a profit in the local currency, investors
may bleed money due to exchange rate fluctuations. While the exchange
rate may be hedged by purchasing forward contracts, this may lead to
increased costs by the investors and this hedging may be imperfect due to
uncertainty in the equity rate of the foreign country.
Secondly, when it comes to investing abroad, investors may face political
risks. Due to instability in the foreign country, the investment’s returns
could suffer. This instability may arise from a bunch of factors like change
in government, change in foreign policy, tax laws, internal conflicts,
external conflicts etc. This political information is very difficult to acquire
at the time of investment and can virtually nullify the investment made
by the investor. For example, even though emerging markets may offer
Mehta, Shivam
double digit returns, due to less stability in these countries, investors may
find themselves facing losses on their investments.
Finally, foreign investors may be more susceptible to liquidity risks. These
are more common in emerging markets. It might be difficult to cash out
at a reasonable price due to lack of buyers. This may also imply the
brokerage costs are high. As a result, investors might have to sell a lower
price or in some cases watch their investment go entirely worthless due
to absence of any buyers.
There are other risks associated with investment in foreign markets such
as interest rate risk , which may become unfavourable due to changes in
government policy , and risk related to high rates of inflation, which again
may be caused by changes in government policy.