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Investments (University of Melbourne)
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FNCE 30001 – Investments
Semester 1, 2021
Module 1: Financial Markets
Solutions to Tutorial Questions
Feel free to check your work in Excel (in fact, I encourage it), but please do the questions by
hand. Why? It forces you to think more, and hopefully more about the economic intuition and
to better understand what you are doing.
This tutorial will be unmarked.
1. Assume you bought a stock for $50 and it has increased to $75. You think it may go
higher, but you want to protect most of your current profit. What order would you place
to ensure a minimum gain of about $23 per share?
Emphasis here on the word “about”. I’d put a stop sell order at $73, since $73 - $50 is
$23. Keep in mind, that the stop sell order sets the price at which a trade is triggered,
it does not guarantee you will get that price with certainty, though it usually will be
very close.
2. On 15 August 2019 you purchased 100 shares in the Cara Cotton Company at $65 a
share. On 10 July 2020, you received a dividend of $3 per share, which you kept as cash.
On 15 July 2020, you sold your holdings for $61 a share.
Compute the realized holding period return.
The fact there are 100 shares doesn’t affect the return calculation, so we can ignore it:
Revenue − Cost
Cost
$61+$3 − $65
= −0.0154 𝑜𝑟 − 1.54%
Return =
$65
Return =
3. You might need to do some reading of your textbook to answer this question: What are
the key differences between common stock, preferred stock and corporate bonds?
All are claims to the cash flows of the firm. Both bonds are claims to predefined or
prespecified cash flows (called coupons and face value or par value). Preferred stock
are claims to an infinite stream of dividends. Common stock can pay dividends, but
they do not have to. By contract, payments to bond holders take precedence over
dividend payments to preferred equity holders, which in turn get precedence over
dividend payments to common stock holds. Failure to pay bond holders causes default
and ownership of the firm usually transfers to the bond holders at default. Failure to
pay dividends to preferred stock holders or common stock holders does not trigger
default. Common stock holders are residual claimants and are only entitled to payouts
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after all other stake holders are paid. Common stock holders usually get to vote for
members of the board of directors. Bond holders and preferred stock holders do not.
4. You bought 100 shares of DataPoint for $25 per share and it is currently selling for $40
per share. Assume that the stock eventually declines to $31. In answering the following,
please ignore brokerage commissions, margin interest costs, and other transaction costs.
a. Calculate your percentage Holding Period Return at the $31 price assuming that
you placed a stop-sell order at $40 per share and the order executed at that price.
If a stop-sell order were place, I would expect that I would have automatically
sold the stock for approximately $40 per share.
Revenue − Cost
Return =
Cost
Return =
$40×100 − $25×100 $4000 − $2500
=
= 60%
$25×100
$2500
b. Calculate your percentage Holding Period Return at the $31 price assuming you
did not place a stop-sell order.
Return =
$31×100 − $25×100 $3100 − $2500
=
= 24%
$25×100
$2500
c. Calculate your percentage Holding Period Return on your equity investment
assuming you bought 100 shares on 50% margin when the it was selling for $25
and you sold the stock for $40 per share.
Revenue − Cost
Return =
Cost
Revenue=Proceeds of the sale of 100 shares - repayment of the loan
Because we’re ignoring interest costs, the repayment of the loan is easy to
calculate:
Revenue=100×$40 - 50%×$25×100=$2750
Cost=50%×$25×100=$1250
2750 − 1250
Return =
= 1.2 𝑜𝑟 120%
1250
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5. You are running a large superannuation fund and have decided to create a new investment
portfolio. You have $10 billion to invest but must follow the investment guidelines of the
super fund. The guidelines state the following:
 No derivatives are allowed
 Minimum $2 billion in money-market investments
 Maximum $5 billion in bonds
Which of the following portfolios are suitable and why?
a. $2 billion of corporate bonds, $5 billion of preferred stock, $2 billion of
international bonds, $1 billion of bank bills
b. $2 billion of preferred stock, $3 billion of CDs, $5 billion of T-notes
c. $3 billion of CDs, $5 billion of T-notes, $2 billion of options
(a) Not suitable. It doesn’t have a minimum of $2 billion in money-market instruments. It
has only $1 billion.
(b) This looks all good. Both CDs and T-Notes are defined as Money-Market securities
(see section 2.1). Since the point of this question is to be pedantic, though money
market securities are a type of bond, the word “bond” is often reserved for nonmoney-market bond-like instruments. So, strictly speaking there are no bonds by this
definition and all all criteria are observed.
(c) Not suitable. The no option clause is violated.
6. You are pessimistic about Telecom shares and decide to sell short 100 shares at the
current market price of $50 per share.
a. How much in cash or securities must you put into your brokerage account if
the broker’s initial margin requirement is 50% of the value of the short
position?
Initial margin is 50% of $5000 or $2500
b. How high can the price of the stock go before you get a margin call if the
maintenance margin is 30% of the value of the short position?
Total assets are $7500 ($5000 from the sale of the stock and $2500 put up for margin).
Liabilities are 100P. Therefore, net worth is ($7500 – 100P). A margin call will be
issued when:
$7500  100P
= 0.30, when P = $57.69 or higher.
100P
c. Suppose the price drops to $45, ignoring transaction costs and lending fees,
what is your holding period return?
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Revenue from closing out the short sale at $45 is
$7500 (assets in the account) - $4500 to buy back the stock = $3000
Cost: you were our of pocket $2500.
𝐻𝑃𝑅 =
$3000 − $2500
= 20%
$2500
7. Consider the following limit order book. The last trade in the shares occurred at a price of
$50.
Limit buy orders
Price ($)
Shares
49.75
500
49.50
800
49.25
500
49.00
200
48.50
600
Limit sell orders
Price ($)
Shares
50.25
100
51.50
100
54.75
300
58.25
100
a. If a market buy order for 100 shares comes in, at what price will it be filled?
b. Assuming no new limit orders are placed, at what price would the next market
buy order be filled?
i. $50.25
ii. $51.50
8. Here is some price information on Fincorp shares. Suppose first that Fincorp trades in a
dealer market.
Bid
55.25
Ask
55.50
a. Suppose you have submitted an order to your broker to buy at market. At what
price will your trade be executed?
$55.50
b. Suppose you have submitted an order to sell at market. At what price will your
trade be executed?
$55.25
c. Suppose you have submitted a limit order to sell at $55.62. What will happen?
It will get placed on the ask side of the limit order book below the existing
best offer of $55.50.
d. Suppose you have submitted a limit order to buy at $55.37. What will happen?
It will get placed on the bid side of the limit order book above the existing best
offer of $55.25, and most probably get executed next.
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9. Let’s bring in some real-life details. It makes it a bit overly complicated – but that’s real
life – overly complicated. Let’s do one exercise like this by hand for your understanding,
and then if you ever get a job dealing with such things, a spreadsheet or computer
program can deal with this.
Suppose you purchased 1000 shares of BLD at $4.96 per share on Friday, 23 Aug. 2019.
You decided to buy on margin with a 70% Loan to Value using an Investment Loan from
a local bank. The bank will make a margin call if the loan to value increases to 5% over
the maximum loan to value of 70%. On Monday, 26 Aug. 2019, BLD closed at $3.94.
Interest on margin loans at the time were 6.53% (quoted as an APR or BEY) and
annualized, as Australian debt usually is on a 365-day basis. This means the daily interest
rate is the APR/365 (or 366 if a leap year).
Assume interest is compounded daily. Let’s further assume there are no fees for taking
out a margin loan and that you pay a flat broker’s commission for $15 per trade. We’ll
ignore the bid-ask spread and potential price impact and presume you both bought and
sold at the daily closing price. [BKM 3.8]
a. What is the 1-trading-day return on the stock (without buying on margin)?
Return =
Revenue − Cost
Cost
Revenue=$3.94×1000-$15=3925
Cost=$4.96×1000+$15 = 4975
Return =
3925 − 4975
= −0.2111 𝑜𝑟 − 21.11%
4975
b. Will you get a margin call on Monday? Assume that interest on your margin
loan has accrued on Saturday, Sunday and Monday. Show your work.
Method 1 – figure out the price at which you get a margin call:
𝐿𝑜𝑎𝑛 𝑡𝑜 𝑉𝑎𝑙𝑢𝑒 =
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘 − 𝐿𝑜𝑎𝑛
𝐿𝑜𝑎𝑛
= 1 − 𝑀𝑎𝑟𝑔𝑖𝑛 𝑃𝑒𝑟𝑐𝑒𝑛𝑡 = 1 −
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘
. 75 =
. 7 × 4.96 × 1000 1 +
0.0653
365
1000𝑃
750𝑃 = 3473.86
=
3473.86
1000𝑃
𝑃 = $4.63
Since the price dropped to $3.94, you will definitely get a margin call on Monday.
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Method 2 – figure out the percent margin or loan to value:
0.0653
. 7 × 4.96 × 1000 1 +
𝐿𝑜𝑎𝑛
365
𝐿𝑜𝑎𝑛 𝑡𝑜 𝑉𝑎𝑙𝑢𝑒 =
=
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘
$3.94×1000
3473.86
=
= 0.8817 ≫ 0.75
3940
c. Suppose ANZ requires you to get the loan to back to 70% following the
margin call. As you know from lecture and the text, upon a margin call, either
your position will be sold or you must provide more cash to the brokerage to
get the loan to value back down to 70%. How much cash would you have to
give them?
𝐿𝑜𝑎𝑛
𝐿𝑜𝑎𝑛 𝑡𝑜 𝑉𝑎𝑙𝑢𝑒 =
𝑉𝑎𝑙𝑢𝑒 𝑜𝑓 𝑆𝑡𝑜𝑐𝑘
See part b for the calculation of the loan size on Monday evening.
3473.86 − 𝑛𝑒𝑤 𝑐𝑎𝑠ℎ
3.94 × 1000
2,758 = 3473.86 − 𝑐𝑎𝑠ℎ
𝑐𝑎𝑠ℎ = $715.86
. 70 =
d. How much stock would you have to sell?
See part b for the calculation of the loan size on Monday evening.
N is the number of shares you need to sell:
3473.86 − (3.94 × 𝑁 − $15)
. 70 =
3.94 × (1000 − 𝑁)
. 70 =
3,488.86 − 3.94𝑁
3940 − 3.94𝑁
2,758 − 2.758𝑁 = 3,488.86 − 3.94𝑁
1.182𝑁 = 730.86
𝑁 = 618.3
𝑅𝑜𝑢𝑛𝑑 𝑢𝑝 𝑡𝑜 619 𝑏𝑒𝑐𝑎𝑢𝑠𝑒 𝑦𝑜𝑢 𝑐𝑎𝑛 𝑡 𝑠𝑒𝑙𝑙 𝑝𝑎𝑟𝑡𝑖𝑎𝑙 𝑠ℎ𝑎𝑟𝑒𝑠
e. Suppose, instead of fulfilling the demands of the margin call, you instead sell
the entire position. What is your 1 trading day return on investment? Assume
you are able to sell at $3.94.
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑃𝑒𝑟𝑖𝑜𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 =
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝐶𝑜𝑠𝑡
𝐶𝑜𝑠𝑡
𝐶𝑜𝑠𝑡 = (1 − .70)4.96 × 1000+$15 = $1,503
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Revenue is a bit trickier:
On Monday you owe the loan, again that is . 7 × 4.96 × 1000 1 +
.
= $3,473.86
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 = 3.94 × 1000 − $3,473.86 − $15 = 451.14.
$451.14 − $1,503
= −69.98%
𝐻𝑜𝑙𝑑𝑖𝑛𝑔 𝑃𝑒𝑟𝑖𝑜𝑑 𝑅𝑒𝑡𝑢𝑟𝑛 =
$1,503
Ouch!
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FNCE 30001 – Investments
Semester 1, 2021
Module 2: Risk
Solutions to Tutorial & Assignment Questions
Part B Assignment Answers are due 9 am Monday 15 March 2021
Feel free to check your work in Excel (in fact, I encourage it), but please do the questions by
hand. Why? It forces you to think more, and hopefully more about the economic intuition and
to better understand what you are doing.
Part A: This part is unmarked
1. This is related to the utility formula: 𝑈 = 𝐸 [𝑟̃ ] − " 𝐴𝜎 " .
!
Utility formula data
Expected Return, 𝐸 [𝑟̃ ]
Standard deviation, 𝜎
.12
.3
.15
.5
.21
.16
.24
.21
Investment
1
2
3
4
a. Based on the formula for required risk premium above, which investment would
you select if you were risk averse with A=4?
Answer: It is clear that 3 and 4 dominate 1 and 2 because the expected returns are
lower and standard deviation higher for 1 and 2. I won’t waste time calculating
utility.
Whether 3 or 4 is preferred is a little less clear (but I bet it’s 3) because the
standard deviation is 3 is relatively smaller. Let’s see:
3: 𝑈 = 𝐸 [𝑟̃ ] − " 𝐴𝜎 " = .21 − 0.5 × 4 × 0.16" = .1588
!
4: 𝑈 = 𝐸 [𝑟̃ ] − 𝐴𝜎 " = .24 − 0.5 × 4 × 0.21" = .1518
"
!
3 wins.
b. Which do you choose if you are risk neutral?
Investment 4, because the return is the highest. You don’t care about risk – you
are risk neutral.
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2. Real and nominal return. Note that 1 + 𝑟#$%& =
Module 2 Tutorial Solutions
!'#!"#$!%&
!'(
, where i is the rate of
inflation. The return experienced by Australian investors over 1992 to 2012 averaged
10.43% per year and inflation was 2.66%.
What was the average real return from 1992 to 2012?
Answer: 1 + 𝑟#$%& =
!'#!"#$!%&
!'(
1 + 0.1043
1 + 0.0266
= 7.57%
1 + 𝑟#$%& =
That is a pretty amazing return.
𝑟#$%&
3. Scenarios 1, 2 and 3 are equally likely.
Consider two assets, A and B. A earns 4%, -5%, or 3%, in scenarios 1, 2, and 3.
B earns -5%, 3%, or 4%, in scenarios 1, 2, and 3.
a. Compute the expected rates of return and Std. Dev. for each asset, A and B.
Answer:
Expected Return for A = (4-5+3)/3 = 0.67%.
Expected Return for B =(-5+3+4)/3 = 0.67%.
Note that the Std. Deviation will also be the same for A and B, since they have
identical expected returns and possible returns in the 3 equally likely situations.
SD(asset A or asset B) =8) (4 − 0.67)" + ) (−5 − 0.67)" + ) (3 − 0.67)" = 4.03%
!
!
!
b. Now, consider a portfolio of assets A and B, where the investor holds a fraction of
his portfolio in each asset: 40% in A and 60% in B. What is the Standard
Deviation of this new diversified AB portfolio?
Now consider the new payoff table for each situation for Portfolio AB:
Situation 1: . 4 × 4% + .6 × −5% = −1.4%
Situation 2: . 4 × −5% + .6 × 3% = −0.2%
Situation 3: . 4 × 3% + .6 × 4% = 3.6%
Consider the new Expected rate of return (which should hopefully be the same, since
the portfolio is composed of two assets with the same Expected Return!):
(3.6-.2-1.4)/3 = (2/3)% = (0.67)%, which is the same as before.
Now consider the AB risk:
1
1
1
< (−1.4 − 0.67)" + (−.2 − 0.67)" + (3.6 − 0.67)" = 2.13%
3
3
3
The standard deviation of the portfolio of 2 imperfectly correlated assets is less.
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c. You will need to read section 5.3 again as this topic was not covered in lecture.
What’s the 5% value at risk (VaR) of your portfolio AB in dollar terms if you
invested $100,000 in portfolio AB?
From formula 5.10 in the text, we get that the the 5% VaR for return is:
𝑉𝑎𝑅 = 𝐸 [𝑟̃ ] − 1.64485𝜎
𝑉𝑎𝑅 = 0.0067 − 1.64485 × 0.0213 = −0.0283
And -2.83% of $100,000 is -$2830.
So, the 5% VaR of this $100,000 investment for this time frame is $2830.
4. XYZ share price and dividend history are as follows:
Year
Beginning-of-year Price
Dividend paid at end of
year
2017
$100
$4
2018
$110
$4
2019
$90
$4
2020
$95
$4
An investor buys three shares of XYZ at the beginning of 2017, buys another two shares
at the beginning of 2018, sells one share at the beginning of 2019 and sells all four
remaining shares at the beginning of 2020.
a. What are the arithmetic and geometric average time-weighted rates of return for
the investor?
5. Year
Return = [(capital gains + dividend)/price]
2017−2018
2018−2019
2019−2020
Arithmetic mean:
(110 – 100 + 4)/100 = 14.00%
(90 – 110 + 4)/110 = –14.55%
(95 – 90 + 4)/90 = 10.00%
(14.00% + 14.55% + 10.00%) = 3.15%
)
!
Geometric mean:
@(1 + 0.14)(1 − 0.1455)(1 + 0.10)A) − 1 = 0.0233 𝑜𝑟 2.33%
!
b. What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash
flows for the four dates corresponding to the turns of the year for 1 January 2017
to 1 January 2020. If your calculator cannot calculate internal rate of return you
will have to use a spreadsheet or trial and error.)
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Time
Cash flow
Explanation
0
−300
Purchase of three shares at $100 per share
1
−208
Purchase of two shares at $110,
plus dividend income on three shares held
2
110
Dividends on five shares,
plus sale of one share at $90
3
396
Dividends on four shares,
plus sale of four shares at $95 per share
Dollar-weighted return = internal rate of return = –0.1661%
5. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be
either $50 000 or $150 000, with equal probabilities of 0.5. The alternative riskless
investment in T-notes pays 5%.
a. If you require a risk premium of 10%, how much will you be willing to pay for the
portfolio?
The expected cash flow is: (0.5 × $50 000) + (0.5 × $150 000) = $100 000
With a risk premium of 10% the required rate of return is 15%. Therefore, if the
value of the portfolio is X then in order to earn a 15% expected return:
X(1.15) = $100 000 Þ X = $86 957
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b. Suppose the portfolio can be purchased for the amount you found in (a). What will the
expected rate of return on the portfolio be?
If the portfolio is purchased at $86 957, and and the expected payoff is $100 000, then
the expected rate of return, E(r), is:
= 0.15 = 15.0%
The portfolio price is set to equate the expected return with the required rate of return.
c. Now suppose you require a risk premium of 15%. What is the price you will be
willing to pay now?
If the risk premium over T-notes is now 15%, then the required return is:
5% + 15% = 20%
The value of the portfolio (X) must satisfy: X(1.20) = $100 000 Þ X = $83 333
d.
Comparing your answers to (a) and (c), what do you conclude about the relationship
between the required risk premium on a portfolio and the price at which the portfolio
will sell?
For a given expected cash flow, portfolios that command greater risk premiums must
sell at lower prices. The extra discount from expected value is a penalty for risk.
6. What is the mean-variance criterion? Use the mean-variance criterion to determine which
of the following investments are efficient and which are inefficient.
Investment
Expected Return
A
B
C
D
E
F
5.30%
12.40%
14.63%
37.47%
7.90%
3.83%
Standard Deviations of
Returns
9.30%
11.40%
8.47%
9.40%
47.20%
1.25%
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Mean-variance criterion states that investment X dominates Y if E[ r_X ] ≥ E[ r_Y ]
and Var[ r_X ] ≤ Var[ r_Y ] with one inequality holding. (That is, X does not
dominate Y if E[ r_X ] =E[ r_Y ] and Var[ r_X ] =Var[ r_Y ] .)
Investment
A
B
C
D
E
F
Inefficient
Yes, Dominated by C
Yes, Dominated by C & D
No
No
Yes, Dominated by B, C, & D
No
Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the assignment.
This is due 9 am Monday 15 March 2021.
Submit your answers as decimals rounded to the nearest .0001. So “0.392782” should
be entered as “0.3928”, and “55.236%” should be entered as “0.5524”.
You plan to invest $100 in a risky asset with an expected rate of return of 13% and a standard
deviation of 14% and a risk-free asset with a rate of return of 4%.
Question 1: What fraction of your money must be invested in the risky asset to form a
portfolio with an expected return of 8%?
The return is a weighted average of the returns of the two assets. Let w represent the proportion
of wealth invested in the risky asset. Then, 8% = w( 13% ) +( 1-w ) ( 4% ) w = 0.4444 Invest
44.44% in the risky asset, or 0.4444.
Question 2: What would be standard deviation of the portfolio formed in part (a) be?
Because the risk-free asset has both 0 standard deviation and 0 correlation with the risky asset,
the standard deviation of the portfolio is simply the weight on the risky asset times the risky
asset's standard deviation: w ⋅ σ_risky =0.4444( 14% ) =6.22% or 0.0622
Question 3: What fraction of your money must be invested in the risk-free asset to form
a portfolio with a standard deviation of 5%?
By the same logic, we can solve for the w such that 5% = w( 14% ) w = 0.3571 Invest 35.71%
in the risky asset and 64.29% in the risk-free asset, or 0.6429.
Question 4: What is the slope of the Capital Allocation Line (CAL) formed with the risky
asset and the risk-free asset?
The slope is given by E[ r_risky ] -E[ r_risk-free ] σ_risky = 0.13-0.04 0.14 =0.6429
Alternatively, we could have solved for it by E[ r_portfolio ] -E[ r_risk-free ] σ portfolio =
0.08-0.04 0.0622 =0.6429 since both the risky asset and the portfolio lie on the CAL.
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Question 5: What is the intercept of the CAL?
The intercept is just the risk-free rate: 4%, or .04
Question 6: Now suppose that the investor may still lend at a risk-free rate of 4%, but if
needed, needs to borrow at 9%. What is the slope of the CAL over the segment that
corresponds to borrowing?
The CAL will have a kink in it at point P . The slope of the CAL in the borrowing portion is
( 0.13-0.09 ) /0.14=0.2857 . The reward to variability ratio is lower than it was in the lending
portion (0.6429) since the investor must pay a higher rate to borrow.
E(r)
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FNCE 30001 – Investments
Semester 1, 2021
Module 3: Optimal Risky Portfolio
Solutions to Tutorial & Assignment Questions
Part B Assignment Answers are due 9 am Monday 22 March 2021
Feel free to check your work in Excel (in fact, I encourage it), but please do the questions
by hand. Why? It forces you to think more, and hopefully more about the economic intuition
and to better understand what you are doing.
Part A: This part is unmarked
1. An investor ponders various allocations to the optimal risky portfolio and risk-free T-notes
to construct his complete portfolio. How would the Sharpe ratio of the complete portfolio be
affected by this choice?
Answer: The Sharpe ratio of the portfolio will be unaffected. Changes to the weight of the
risky asset in the portfolio change the denominator and numerator of the portfolio Sharpe
ratio by equal amounts
2. Shares offer an expected rate of return of 10% with a standard deviation of 20% and gold
offers an expected return of 5% with a standard deviation of 25%.
a. In light of the apparent inferiority of gold to shares with respect to both mean return and
volatility, would anyone hold gold? If so, demonstrate graphically why one would do so.
b. How would your answer (a) change if the correlation coefficient between gold and shares
were 1.0? Draw a graph illustrating why one would or would not hold gold. Could these
expected returns, standard deviations and correlation represent an equilibrium for the security
market?
Answer:
a. Although it appears that gold is dominated by shares, gold can still be an attractive
diversification asset. If the correlation between gold and shares is sufficiently low, gold will
be held as a component in the optimal portfolio.
b.
If gold had a perfectly positive correlation with shares, gold would not be a part of
efficient portfolios. The set of risk/return combinations of shares and gold would plot as a
straight line with a negative slope. (See the following graph.) The graph shows that the shareonly portfolio dominates any portfolio containing gold. This cannot be an equilibrium; the
price of gold must fall and its expected return must rise. (NOTE: This answer is from the
textbook, but it actually requires a brief caveat: “This is true if the only benefit from gold is
as an investment; however, if gold serves other purposes other than as an investment, then
there is no reason to think that its price must fall as non-investment demand could keep the
price propped up.”)
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Expected Return (%)
12
10
Stocks
8
6
Gold
4
2
0
0
5
10
15
20
25
30
Standard Deviation (%)
3. Assume that you manage a risky portfolio with an expected rate of return of 17% and a
standard deviation of 27%. The T-note rate is 7%. Your client chooses to invest 70% of a
portfolio in your fund and 30% in a T-note cash fund.
a. What is the expected return and standard deviation of your client’s portfolio?
b. Suppose your risky portfolio includes the following investments in the given proportions:
Share A
Share B
Share C
27%
33%
40%
What are the investment proportions of your client’s overall portfolio, including the position
in T-notes?
c. What is the reward-to-volatility ratio (Sharpe Ratio) of your risky portfolio and your
client’s overall portfolio?
d. Suppose the client decides to invest in your risky portfolio a proportion (y) of his total
investment budget so that his overall portfolio will have an expected rate of return of 15%.
What is the proportion y?
e. Suppose the client prefers to invest in your portfolio a proportion (y) that maximises the
expected return on the overall portfolio subject to the constraint that the overall standard
deviation will not exceed 20%. What is the proportion y?
a. E(rP) = (0.3 × 7%) + (0.7 × 17%) = 14% per year
P = 0.7 × 27% = 18.9% per year
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b.
Security
Investment
proportions
30.0%
18.9%
23.1%
28.0%
T-notes
Share A
Share B
Share C
0.7  27% =
0.7  33% =
0.7  40% =
17 − 7
c. Your reward-to-variability ratio = S =
= 0.3704
27
Client's reward-to-variability ratio =
d.
14 − 7
= 0.3704
18.9
Mean of portfolio = (1 – y)rf + y rP = rf + (rP – rf )y = 7 + 10y
If the expected rate of return for the portfolio is 15%, then, solving for y:
15 = 7 + 10y  y =
15 − 7
= 0.8
10
Therefore, in order to achieve an expected rate of return of 15%, the client must invest
80% of total funds in the risky portfolio and 20% in T-notes.
e. Portfolio standard deviation = P = y × 27%
If the client wants a standard deviation of 20%, then:
y = (20%/27%) = 0.7407 = 74.07% in the risky portfolio
4. George Stephenson’s current portfolio of $2.0 million is invested as follows:
Summary of Stephenson’s current portfolio
Value
Short-term bonds
Per cent of
total
Expected
annual return
Annual
standard
deviation
$200 000
10%
4.6%
1.6%
Domestic large-cap
equities
600 000
30
12.4
19.5
Domestic small-cap
equities
1200 000
16.0
29.9
13.8%
23.1%
Total portfolio
$2 000 000
100%
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Stephenson expects to receive an additional $2.0 million soon and plans to invest the entire
amount in an index fund that best complements the current portfolio. Stephanie Wright, CFA,
is evaluating the four index funds shown in the following table for their ability to produce
portfolio that will meet two criteria relative to the current portfolio: (1) maintain or enhance
expected return and (2) maintain or reduce volatility.
Each fund is invested in an asset class that is not substantially represented in the current
portfolio.
Index fund characteristics
Index fund
Expected annual
return %
Expected
annual
standard
deviation %
Correlation of
returns with
current portfolio
Fund A
15
25
+0.80
Fund B
11
22
+0.60
Fund C
16
25
+0.90
Fund D
14
22
+0.65
State which fund Wright should recommend to Stephenson. Justify your choice by describing
how your chosen fund best meets both of Stephenson’s criteria. No calculations are required.
Answer: Fund D represents the single best addition to complement Stephenson’s current
portfolio, given his selection criteria. First, Fund D’s expected return (14.%) has the potential
to increase the portfolio’s return somewhat. Second, Fund D’s relatively low correlation with
his current portfolio (+0.65) indicates that Fund D will provide greater diversification
benefits than any of the other alternatives except Fund B. The result of adding Fund D should
be a portfolio with approximately the same expected return and somewhat lower volatility
compared to the original portfolio.
The other three funds have shortcomings in terms of either expected return enhancement or
volatility reduction through diversification benefits. Fund A offers the potential for increasing
the portfolio’s return, but is too highly correlated to provide substantial volatility reduction
benefits through diversification. Fund B provides substantial volatility reduction through
diversification benefits, but is expected to generate a return well below the current portfolio’s
return. Fund C has the greatest potential to increase the portfolio’s return, but is too highly
correlated to provide substantial volatility reduction benefits through diversification.
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5. You want to construct a portfolio consisting of the following two securities:
Stock
Expected
Return
20%
11%
A
B
Correlation
Standard Deviation
of Returns
25%
19%
0.50
Draw the efficient frontier.
First, use the covariance formula to find cov( r 1 , r 2 ) =0.02375 .
Then build a table with different combinations of weights and calculate the expected
return and standard deviation of the associated portfolio:
Weights
Security Security Exp.
Std.
1
2
Return
Deviation
0
1
0.11
0.19
0.1
0.9
0.119
0.185
0.2
0.8
0.128
0.182
0.3
0.7
0.137
0.182
0.4
0.6
0.146
0.185
0.5
0.5
0.155
0.191
0.6
0.4
0.164
0.199
0.7
0.3
0.173
0.209
0.8
0.2
0.182
0.221
0.9
0.1
0.191
0.235
1
0
0.2
0.25
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Part B: Assignment, worth 1.25 marks on a pass/fail basis
You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 22 March 2021.
Express all numerical answers as a decimal rounded to two places, e.g., 53% should
be entered as “0.53”.
1. Consider the following assets in which to potentially invest:
Fund ABC
Fund DEF
Government Bond
𝝈[r]
18%
25%
0%
E[r]
10%
9%
4%
𝝆
0.3
--
a) Question 1: Without doing any calculations, what which risky asset, ABC or
DEF, would you expect to hold more of in the optimal risky portfolio?
b) Question 2: What is the weight of ABC in the optimal risky portfolio?
c) Question 3: For an investor with a risk-aversion parameter A equal to 3, what is
weight of the risk-free asset in the optimal complete portfolio?
d) Question 4: For an investor with A>3, do you expect she will hold more, less, or
the same amount of the risk-free asset in her optimal complete portfolio
compared to the investor in (c)?
a) ABC. Notice that ABC has higher expected return and lower standard deviation, so we
would expect that it will have higher weight in the optimal risky portfolio.
b) The optimal risky portfolio is given by
Plugging in our numbers, we have
The optimal risky portfolio has approximately 79% invested in ABC and the remaining
21% invested in DEF.
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c) First, we must solve for the expected return and variance of the optimal risky portfolio
P, then we plug into the formula for the optimal complete portfolio.
So the optimal complete portfolio has approximately 70% in P and 30% in the risk free
asset.
d) Higher A means the investor is more risk averse, meaning she will hold more of the
risk-free asset and less of the risky portfolio
2. Given the following estimates:
State
Expansion
Normal
Recession
Probability
.3
.4
.3
Tech Firm (rT)
22%
10%
-8%
Discount Retailer (rD)
3%
5%
8%
Question 5: What is the correlation (not covariance!) between the returns of the two
firms?
Answer:
First, we need the expected returns.
E[r_T] = .3 * 22 + .4 * 10 + .3 * -8 = 8.2%
E[r_D] = .3 * 3 + .4 * 5 + .3 * 8 = 5.3%
Next, standard deviation
Sigma(r_T) = sqrt(0.3*(0.22-0.082)^2+0.4*(0.1-0.082)^2+0.3*(-0.08-0.082)^2)= 0.11712
Sigma(r_D) = sqrt(0.3* (0.03-0.053) ^2+0.4* (0.05-0.053) ^2+0.3*(0.08-0.053)^2)= 0.01952
Next we need the covariance.
𝕊
𝜎𝐴𝐵 = ∑ 𝑝(𝑠)(𝑟𝐴,𝑠 − 𝐸[𝑟̃𝐴 ])(𝑟𝐵,𝑠 − 𝐸[𝑟̃𝐵 ])
𝑠=1
𝜎𝐴𝐵 = .3(. 22 − .082)(. 03 − .053) + .4(. 10 − .082)(. 05 − .053)
+ .3(−.08 − .082)(. 08 − .053)
= −0.002286
Finally, we can calculate correlation.
𝜌𝐴𝐵
𝜎𝐴𝐵
𝜎𝐴 𝜎𝐵
−0.002286
=
= −0.99996
. 11712 × .01952
𝜌𝐴𝐵 =
7
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FNCE 30001 – Investments
Semester 1, 2021
Module 4: CAPM & Managed Funds
Solutions to Tutorial & Assignment Questions
Part B Assignment Answers are due 9 am Monday 29 March 2021
Part A: This part is unmarked
1. Consider the following table, which gives a security analyst’s expected return on two
shares for two particular market returns (the two scenarios are equally likely)
Scenario
1
2
Market Return
5%
20%
Aggressive share
2%
32%
Defensive share
3.5%
14%
a. What are the betas of the two shares?
E[r_M] = 12.5%, E[r_A] = 17%, E[r_D] = 8.75%
Var(r_M) = 0.5(-7.5%)^2 + 0.5(7.5%)^2 = .075^2
Cov(r_A,r_M) = 0.5(-7.5%)(-15%) + 0.5 (7.5%)(15%) = .075*.15
Cov(r_D,r_M) = 0.5(-7.5%)(-5.25%) + 0.5 (7.5%)(5.25%) = .075*.0525
Beta_A = (.075*.15)/(.075)^2 = .15/.075 = 2
Beta_A = (.075*.0525)/(.075)^2 = .0525/.075 = 0.7
The “Essentials” version of the textbook unfortunately doesn’t cover the characteristic line in
any detail, but we could also compute each share's beta by calculating the difference in its
return across the two scenarios divided by the difference in market return.
A =
2 − 32
= 2.00
5 − 20
D =
3.5 − 14
= 0.70
5 − 20
b. What is the expected rate of return on each share?
With the two scenarios equally likely, the expected rate of return is an average of the two
possible outcomes:
E(rA) = 0.5  (2% + 32%) = 17%
E(rB) = 0.5  (3.5% + 14%) = 8.75%
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2. Two investment advisers are comparing performance. One averaged a 19% return and the
other a 16% return. However, the beta of the first adviser was 1.5 while that of the second
was 1. Can you tell which adviser was a better selector of individual shares (aside from the
issue of general movements in the market)?
r1 = 19%; r2 = 16%; β1 = 1.5; β2 = 1.0
a. In order to determine which investor was a better selector of individual shares we look
at the abnormal return, which is the ex-post alpha; that is, the abnormal return is the
difference between the actual return and that predicted by the SML. Without information
about the parameters of this equation (i.e. the risk-free rate and the market rate of return)
we cannot determine which investment adviser is the better selector of individual shares.
3. A portfolio manager is using the CAPM for making recommendations to her clients. Her
research department has developed the information shown in the following:
Share X
Share Y
Market Index
Risk-free rate
Forecasted returns, standard deviations and betas
Forecasted return (%)
Standard deviation (%)
14
36
17
25
14
15
5
Beta
.8
1.5
1.0
a. Calculate the CAPM Expected Return and alpha for each share
E(rX) = 5% + 0.8(14% – 5%) = 12.2%
αX = 14% – 12.2% = 1.8%
E(rY) = 5% + 1.5(14% – 5%) = 18.5%
αY = 17% – 18.5% = –1.5%
b. Identify and justify which share would be more appropriate for an investor who wants to:
i.
Add this share to a well-diversified portfolio
For an investor who wants to add this share to a well-diversified equity portfolio, she should
recommend share X because of its positive alpha, while share Y has a negative alpha. In
graphical terms, share X’s expected return/risk profile plots above the SML, while share Y’s
profile plots below the SML. Also, depending on the individual risk preferences of her
clients, share X’s lower beta may have a beneficial impact on overall portfolio risk.
ii.
Hold this share as a single-share portfolio
For an investor who wants to hold this share as a single-share portfolio, she should
recommend share Y, because it has higher forecasted return and lower standard
deviation than share X. Share Y’s Sharpe ratio is:
(0.17 – 0.05)/0.25 = 0.48
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Share X’s Sharpe ratio is only:
(0.14 – 0.05)/0.36 = 0.25
The market index has an even more attractive Sharpe ratio:
(0.14 – 0.05)/0.15 = 0.60
However, given the choice between share X and Y, Y is superior. When a share is held
in isolation, standard deviation is the relevant risk measure. For assets held in isolation,
beta as a measure of risk is irrelevant. Although holding a single asset in isolation is not
typically a recommended investment strategy, some investors may hold what is
essentially a single-asset portfolio (e.g. the share of their employer). For such investors,
the relevance of standard deviation versus beta is an important issue.
4. The market price of a security, that is expected to pay a constant dividend in perpetuity, is
$40. Its expected rate of return is 13%. The risk-free rate is 7% and the market risk premium
is 8%. What will the market price of the security be if its beta doubles (and all other variables
remain unchanged)?
The value of a perpetuity is (we learned this is week 1 or 2)
𝑃=
̃]
𝐸[𝐷
𝐸[𝑟̃ ]
$40 =
What’s its beta?
̃]
𝐸[𝐷
. 13
̃ ] = $5.20
𝐸[𝐷
𝐸[𝑟̃𝑖 ] − 𝑟𝑓 = 𝛽𝑖 (𝐸[𝑟̃𝑀 ] − 𝑟𝑓 )
13% − 7% = 𝛽𝑖 (8%)
𝛽𝑖 = .75
Double the beta is 1.5, so the new expected return is:
𝐸[𝑟̃𝑖 ] = 𝑟𝑓 + 𝛽𝑖 (𝐸[𝑟̃𝑀 ] − 𝑟𝑓 )
𝐸 [𝑟̃𝑖 ] = 7% + 1.5(8%)
New Price:
𝐸[𝑟̃𝑖 ] = 19%
𝑃=
𝑃=
̃]
𝐸[𝐷
𝐸[𝑟̃ ]
$5.20
= $27.37
. 19
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5. The risk-free rate is 3.9% and the standard deviation of the market portfolio is 17.0%.
a) If the average investor has a risk aversion coefficient of 1.7, what is the equilibrium
value of the market risk premium? What is the expected rate of return on the market?
b) Recalculate your answer to part a if the average investor has a risk aversion
coefficient equal to 2.8. Are your answers consistent with each other?
6. You invest $10,000 in the New Fund at a NAV of $20 per share at the beginning of the
year (i.e. $10,000 is everything that comes out of your bank account or from under your
mattress). The fund changes an entry fee of 3%. The securities in which the fund invested
rose in value 12% during the year. The fund’s Management Expense Ratio (MER) expense
ratio was 1.2%, paid throughout the year. What is your return if you sell the fund at the end of
the year? There is no buy-sell spread.
You invest $10,000. The entry fee costs you 0.03 × $10,000 = $300. So, you have only
$9700 left to invest at $20 per share, i.e. $9700 ÷ $20 = 485 shares.
The securities grew by 12%, but the management fee comes out continuously. So, you only
earn 12% − 1.2% = 10.8%.
Your fund grows in value to $9700 × 1.108 = $10,747.60
𝐻𝑃𝑅 =
𝑅𝑒𝑣𝑒𝑛𝑢𝑒 − 𝐶𝑜𝑠𝑡 $10,747.60 − $10,000
=
= .07476 𝑜𝑟 7.476%
$10,000
𝐶𝑜𝑠𝑡
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Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the assignment.
This is due 9 am Monday 29 March 2021 via the LMS Quizzes Page.
1. If the CAPM holds, in equilibrium which of the situations below are possible? Consider
each situation independently.
a. (Question 1)
Portfolio
𝑟𝑓
Market
A
𝐸[𝑟]
8%
16
12
𝛽
0
1.0
.25
Not Possible
𝐸[𝑟̃𝑖 ] = 𝑟𝑓 + 𝛽𝑖 (𝐸[𝑟̃𝑀 ] − 𝑟𝑓 )
𝐸[𝑟̃𝑖 ] = 8% + .25(16% − 8%) = 10%
And A has an expected return of 12%, therefore the CAPM does not hold. Note that it does
not matter whether the Market or A is mispriced or both are mispriced. Either way CAPM
doesn’t hold (and if this opportunity existed, it would potentially present an arbitrage
opportunity).
b. (Question 2)
Portfolio
𝑟𝑓
Market
A
𝐸[𝑟]
10%
18
16
𝛽
0
1.0
1.5
Not possible. Given these data, the SML is: E(r) = 10% + β(18% – 10%)
A portfolio with beta of 1.5 should have an expected return of:
E(r) = 10% + 1.5 × (18% – 10%) = 22%
The expected return for portfolio A is 16% so that portfolio A plots below the SML (i.e. has
an alpha of –6%) and hence is an overpriced portfolio. This is inconsistent with the CAPM.
c. (Question 3)
Portfolio
𝑟𝑓
Market
A
𝐸[𝑟]
10%
18
16
𝜎
0%
24
12
Not possible. The reward-to-variability ratio for portfolio A is better than that of the
market, which is not possible according to the CAPM, since the CAPM predicts that the
market portfolio is the most efficient portfolio. Using the numbers supplied:
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16 − 10
= 0.5
12
SA =
18 − 10
= 0.33
SM = 24
These figures imply that portfolio A provides a better risk-reward trade-off than the
market portfolio.
d. (Question 4)
Portfolio
𝐸[𝑟]
𝜎
10%
0%
𝑟𝑓
Market
18
24
A
20
22
Not possible. Portfolio A clearly dominates the market portfolio. It has a lower standard
deviation with a higher expected return.
e. (Question 5)
Portfolio
𝐸[𝑟]
𝜎
10%
0%
𝑟𝑓
Market
18
24
A
16
22
Possible. Portfolio A's ratio of risk premium to standard deviation is less attractive than
the market's. This situation is consistent with the CAPM. The market portfolio should
provide the highest reward-to-variability ratio.
f. (Question 6)
Portfolio
𝐸[𝑟]
𝜎
A
30%
35
B
40%
25
Possible. If the CAPM is valid, the expected rate of return compensates only for
systematic (market) risk as measured by beta, rather than the standard deviation, which
includes non-systematic risk. Thus, portfolio A's lower expected rate of return can be
paired with a higher standard deviation, as long as portfolio A's beta is lower than that of
portfolio B.
g. (Question 7)
Portfolio
𝐸[𝑟]
𝛽
10%
0
𝑟𝑓
Market
18
1.0
A
16
.9
Not possible. The SML is the same as in question 12. Here, the required expected return
for portfolio A is: 10% + (0.9 × 8%) = 17.2%
This is still higher than 16%. Portfolio A is overpriced, with alpha equal to: –1.2%.
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h. (Question 8)
Portfolio
A
B
𝐸[𝑟]
20%
24%
Module 4 Tutorial Solutions
𝛽
1.4
1.2
Not possible. Portfolio A has a higher beta than portfolio B, but the expected return
for portfolio A is lower.
2. Given the following information about Stocks 1 to 4:
Betai
Actual E(ri)
Stock 1
-0.10
6.29%
Stock 2
0.67
9.08%
Stock 3
1.95
27.24%
Stock 4
2.20
24.80%
The risk-free rate of return is 6.1%, and you estimate the expected return on the market
portfolio is 14.6%.
Question 9: According to the CAPM, which of the stocks are overpriced?
Question 10: According to the CAPM, which of the stocks are underpriced
First, find CAPM expected returns:
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FNCE 30001 – Investments
Semester 1, 2021
Module 5: Index Models
Solutions to Tutorial & Assignment Questions
Part B Assignment Answers are due 9 am Monday 5 April 2021
Feel free to check your work in Excel (in fact, I encourage it), but please do the questions
by hand. Why? It forces you to think more, and hopefully more about the economic intuition
and to better understand what you are doing.
Part A: This part is unmarked
1. Consider a single-index model. The alpha of a stock is 0%. The return on the market index
is 12%. The risk-free rate of return is 5%. The stock earns a return that exceeds the risk-free
rate by 7% and there are no firm-specific events affecting the stock’s performance. Find the
beta of the stock.
𝑟𝑖,𝑡 − 𝑟𝑓,𝑡 = 𝛼𝑖 + 𝛽𝑖,𝑀 (𝑟𝑀,𝑡 − 𝑟𝑓,𝑡 ) + 𝑒𝑖,𝑡
. 07 = 0 + 𝛽𝑖,𝑀 (. 12 − .05) + 0
𝛽𝑖,𝑀 = 1
2. The standard deviation of the market index portfolio is 20%. Share A has a beta of 1.5 and
a residual standard deviation of 30%.
a. What should make for a larger increase in the share’s variance: an increase of 0.15 in its beta
from 1.5 to 1.65 or an increase of 3% in its residual standard deviation from 30% to 33%?
Starting point:
Increase Beta to 1.65:
2
𝜎𝑟2𝑖 = 𝛽𝑖,𝑀
𝜎𝑟2𝑀 + 𝜎𝑒2𝑖
𝜎𝑟2𝐴 = 1.52 × .22 +. 32 = .18
𝜎𝑟𝐴 = √𝜎𝑟2𝐴 = 0.42426
𝜎𝑟2𝐴 = 1.652 × .22 +. 32 = .1989
Leave beta at 1.5, but increase residual standard deviation to .33:
𝜎𝑟2𝐴 = 1.52 × .22 +. 332 = .1989
The impact of the changes is the same in this particular example.
b. An investor who currently holds the market-index portfolio decides to reduce the portfolio
allocation to the market index to 90% and to invest 10% in share A. Which of the changes
above (a change beta from 1.5 to 1.65 or a change in the residual standard deviation from 30%
to 33%) would have a greater impact on the portfolio's standard deviation?
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You can answer this question without math, but you can also prove it with math. The following
hint will be very helpful for mathematically proving the result:
𝛽𝐴 =
𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 )
𝜎𝑀2
We can do this with and without math. The without math is more important. Here’s the
intuition:
A higher beta means that stock A has a higher correlation with the market, so the diversification
benefit is lower, and the increase in the portfolio risk will be greater.
To see this mathematically, let’s compare the portfolio risk when stock A has 𝛽𝐴 = 1.65 and .30
residual risk to the same portfolio when stock A has 𝛽𝐴 = 1.5 and .33 residual risk.
2
𝜎𝑝2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝑀
𝜎𝑀2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴𝐵
We have everything but the covariance. We can get the covariance from 𝛽𝐴 . Page 123, Example
6.3 in your text notes that:
𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 )
𝛽𝐴 =
𝜎𝑀2
When 𝛽𝐴 = 1.5, 𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 ) = 1.5 × .04 = .06
When 𝛽𝐴 = 1.65, 𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 ) = 𝟏. 𝟔𝟓 × .04 = .066
When 𝛽𝐴 = 1.5 and 𝜎𝑒𝑖 = 0.33:
2
𝜎𝑝2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝑀
𝜎𝑀2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴𝐵
𝜎𝑝2 =. 12 × .1989 +. 92 × .04 + 2 × .1 × .9 × .06 = .045189
𝜎𝑝 = √𝜎𝑝2 = .21258
When 𝛽𝐴 = 1.65 and 𝜎𝑒𝑖 = 0.3:
2
𝜎𝑝2 = 𝑤𝐴2 𝜎𝐴2 + 𝑤𝑀
𝜎𝑀2 + 2𝑤𝐴 𝑤𝐵 𝜎𝐴𝐵
𝜎𝑝2 =. 12 × .1989 +. 92 × .04 + 2 × .1 × .9 × .066 = .046269
𝜎𝑝 = √𝜎𝑝2 = .21510
3. Consider the following results for two stocks, A and B.
𝑟𝐴 − 𝑟𝑓 = 0.04 + 0.4(𝑟𝑀 − 𝑟𝑓 ) + 𝑒𝐴
𝑟𝐵 − 𝑟𝑓 = −0.05 + 0.9(𝑟𝑀 − 𝑟𝑓 ) + 𝑒𝐵
𝜎𝑀 = 0.35
Regression 𝑅𝐴2 = 0.40
Regression 𝑅𝐵2 = 0.15
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a. Find the standard deviation of each stock.
Note that:
2
𝛽𝑖,𝑀
𝜎𝑟2𝑀
𝑅 = 2 2
𝛽𝑖,𝑀 𝜎𝑟𝑀 + 𝜎𝑒2𝑖
2
And
2
𝜎𝑖2 = 𝛽𝑖,𝑀
𝜎𝑟2𝑀 + 𝜎𝑒2𝑖
2
𝛽𝑖,𝑀
𝜎𝑟2𝑀
𝑅 =
𝜎𝑖2
2
𝜎𝑖2
2
𝛽𝑖,𝑀
𝜎𝑟2𝑀
=
𝑅2
. 42 ×. 352
𝜎𝐴 = √𝜎𝐴2 = √
= √. 049 = .2214
.4
. 92 ×. 352
𝜎𝐵 = √𝜎𝐵2 = √
= √. 6615 = .8133
. 15
b. Separate the variance of each stock into the systematic and firm-specific components of
variance (not standard deviation).
2
𝜎𝑖2 = 𝛽𝑖,𝑀
𝜎𝑟2𝑀 + 𝜎𝑒2𝑖
Systematic variance:
Firm-specific variance:
2
𝛽𝐴,𝑀
𝜎𝑟2𝑀 =. 42 ×. 352 = .0196
2
𝛽𝐵,𝑀
𝜎𝑟2𝑀 =. 92 ×. 352 = .0992
2
𝜎𝑒2𝑖 = 𝜎𝑖2 − 𝛽𝑖,𝑀
𝜎𝑟2𝑀
𝜎𝑒2𝐴 = .049 − .0196 = .0294
𝜎𝑒2𝐴 = .6615 − .0992 = .5623
c. Find the covariance between each stock and the market index.
The following hint may be helpful:
𝛽𝐴 =
𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 )
𝜎𝑀2
𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 ) = 𝛽𝐴 𝜎𝑀2
𝐶𝑜𝑣(𝑟̃𝐴 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 ) = .4 × .1225 = 0.049
𝐶𝑜𝑣(𝑟̃𝐵 − 𝑟𝑓 , 𝑟̃𝑀 − 𝑟𝑓 ) = .9 × .1225 = 0.1103
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4. Consider two portfolios, one composed of four securities and the other of ten securities. All
the securities have a beta of 1 and idiosyncratic risk of 30%. Each portfolio distributes weight
equally among its component securities. If the standard deviation of the market index is 20%,
calculate the total risk of both portfolios.
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Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 5 April 2021 via the LMS Quizzes Page.
Round all answers to two decimal points. E.g., “0.517” should be entered as “0.52”.
Watch the video: “Calculating Betas”. In practice, there are two ways one could estimate beta
from historical data. The first is to calculate the covariance of asset i’s and the market’s excess
returns over and above the risk-free rate and divide that by the variance of the market’s excess
returns, both calculated over a fixed period of time. The second is to take those same
observations and run a regression where the independent variable is i’s excess return and the
dependent variable is the market’s excess return.
Using the spreadsheet Assignment5_CalculatingBetas.xlsx, calculate the beta of
Commonweath Bank (CBA) using the 60 monthly observations from 1 Jan 2015 to 1 Dec
2019 using both methods. To do this, you first need to calculate the monthly excess returns for
BHP and the ASX 200.
For the first method, use the “covariance.s” and “var.s” functions in Excel for the 60 monthly
observations of excess returns for CBA and the ASX 200 index.
Question 1: Using this method, what is the calculation of beta for CBA over the specified
time period?
Solve for the covariance, variance, and beta using excel as:
0.000988732, 0.000990833, 0.997879025 or 1.00 after rounding
For the second method, you will need the “Analysis ToolPak – VBA” add-in for Excel. (Google
search how to install if you don’t have it already.) Once you’ve done that, go to:
“Data” tab > “Data Analysis” > “Regression”
Use CBA’s excess returns for “Input Y Range” and the ASX 200’s excess returns for “Input X
Range”. You will get an output similar to what was shown in the slides for the HP regression.
The beta is the coefficient on the ASX 200’s excess returns.
Question 2: Based on the regression results, what is the calculation of beta for CBA over
the specified time period?
Using a regression, you should find the coefficient on the ASX200 excess return to be
1.00 after rounding.
Question 3: What is the R-Square value from the regression?
0.409292392, or 0.41 after rounding
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Semester 1, 2021
Module 6: Multifactor Models
Tutorial Questions
Feel free to check your work in Excel (in fact, I encourage it), but please do the questions
by hand. Why? It forces you to think more, and hopefully more about the economic intuition
and to better understand what you are doing.
Note: There is no marked assignment for this module.
1. Suppose two factors are identified in the Australian economy: the growth rate of industrial
production, IP, and the inflation rate, IR. IP is expected to be 4% and IR, 6%. A share with a
beta of 1 on IP and 0.4 on IR is expected to provide a rate of return of 14%. If industrial
production actually grows by 5%, while the inflation rate turns out to be 7%, what is your
best guess for the rate of return on the share?
The revised estimate of the expected rate of return of the share would be the old estimate plus
the sum of the unexpected changes in the factors times the sensitivity coefficients, as follows:
Revised estimate = 14% + [(1  1) + (0.4  1)] = 15.4%
2. Suppose there are two independent economic factors, M1 and M2, The risk-free rate is 7%
and all shares have independent firm-specific components with a standard deviation of 50%.
Portfolios A and B are both well diversified.
Portfolio
A
B
Beta on M1
1.8
2
Beta on M2
2.1
-0.5
Expected Return
40%
10%
What is the expected return-beta relationship in this economy? (Hint: solve for the risk
premia on the two factors. You have two equations and two unknowns – the risk premium for
each factor.)
E(rP) = rf + βP1[E(r1) − rf] + βP2[E(r2) – rf]
We need to find the risk premium for these two factors:
1 = [E(r1) − rf] and
2 = [E(r2) − rf]
To find these values, we solve the following two equations with two unknowns:
40% = 7% + 1.81 + 2.12
10% = 7% + 2.01 + (−0.5) 2
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The solutions are: 1 = 4.47% and 2 = 11.88%
Thus, the expected return-beta relationship is:
E(rP) = 7% + 4.47βP1 + 11.88βP2
3. Note: This is a long question that only needs a short answer. Suppose I propose that factor
X explains airline stock returns well. I run a time-series regression of each airline’s return on
X,
𝑟𝑎,𝑡 − 𝑟𝑓,𝑡 = 𝛼 + 𝛽𝑋𝑡 + 𝜖𝑡
where 𝑟𝑎,𝑡 is the return on one airline stock at time t, 𝑟𝑓,𝑡 is the risk-free return at time t, and 𝑋𝑡
is the risk premium of the factor. I find that factor X, all by itself, explains on average 32% of
airline stock return variance over time (an average R2=0.32). I form portfolios of airline stocks
based on the 𝛽 from the regression above, and I find the following out-of-sample average
monthly returns for each of the 𝛽 portfolios:
Low β
2
3
4
High β
Return
1.56
1.12
0.98
1.25
1.42
t-stat
(2.98)
(2.10)
(2.05)
(1.98)
(3.45)
p-value
0.003
0.036
0.041
0.048
0.001
Are these findings consistent with X being a risk factor that explains airline stock returns?
Why?
No. If investors are rational and risk-averse and X is a risk factor then the reward for bearing
risk must be positive. Further, we would expect that covariance with X, which is reflected in
𝛽 would be compensated. High risk exposure (that is, high 𝛽) should be associated with high
return. We do not see that here: higher 𝛽‘s are not consistently associated with higher return
and, therefore, these findings are not consistent with X being a risk factor that explains airline
stock returns.
4. Consider the following data for a single-factor economy. All portfolios are well diversified.
Portfolio
A
B
𝐸 [𝑟̃ ]
10%
9%
𝛽
1
2⁄
3
If the risk-free rate is 4%, does an arbitrage opportunity exist? If so, what would an arbitrage
strategy be?
The simplest approach here would be to combine A (B) with the risk-free asset so that the
resulting portfolio has the same beta as B (A):
A portfolio C of 2/3 A and 1/3 r_f has E[r] = 2/3(10) + (1/3) 4 = 8% and a beta of 2/3.
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Since B and C have the same beta but different expected returns, there’s an arbitrage
opportunity:
•
•
•
Long (e.g.) $10,000 of B
Short 2/3($10,000) of A
Borrow 1/3($10,000) at the risk-free rate
5. Assume both portfolios A and B are well diversified, with 𝐸 [𝑟̃𝐴 ] = 14% and 𝐸 [𝑟̃𝐵 ] =
14.8%. If the economy has only one factor and 𝛽𝐴 = 1 and 𝛽𝐵 = 1.1, what must be the riskfree rate?
[Hint: this is just an algebra problem, with two equations and two unknowns – the risk-free
rate and the return on the factor.]
Substituting the portfolio returns and betas in the expected return−beta relationship, we
obtain two equations in the unknowns, the risk-free rate (rf ) and the factor return (F):
14.0% = rf + 1  (F – rf)
14.8% = rf + 1.1  (F – rf)
From the first equation we find that F = 14%. Substituting this value for F into the second
equation, we get:
14.8% = rf + 1.1  (14% – rf)  rf = 6%
6. Assume that you are using a two-factor APT model to find the expected return on a stock.
The factors, their betas, and their assumed risk premiums are shown in the table below. The
risk-free rate is 4.8%.
Factor
Factor Beta
A
B
1.7
0.9
Assumed Factor Risk
Premium
2.0%
10.5%
a) What is the expected return on the stock if it is fairly priced?
b) Now suppose that the factor risk premiums you used are found to be incorrect. The
true factor risk premiums are shown below. Recalculate the expected return on the
stock based on the true factor risk premiums.
Factor
Factor Beta
A
B
1.7
0.9
True Factor Risk
Premium
3.5%
9.0%
c) Compare your answers to parts a and b. If you based the expected return on the
assumed factor risk premiums rather than the true ones, would you have overpriced or
underpriced the stock?
a) The expected return on the stock would be 4.8 + 1.7(2.0) + 0.9(10.5) = 17.65%.
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b) The corrected expected return should be 4.8 + 1.7(3.5) + 0.9(9) = 18.85%.
c) If you required a return of 17.65% on the stock, you required too little and would have
been willing to pay too much. You would have overpriced the stock.
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Semester 1, 2021
Module 8: Fixed Income Fundamentals
Tutorial Questions
Part A is UNMARKED. It will be discussed in your tutorial during the
week of 3 May.
Part B is graded pass/fail purely for the attempt at answering. You must fill
in an answer for full credit. Part B Assignment Answers are due 9 am
Monday 3 May via the LMS
Part A: This part is unmarked
In all questions assume everything is risk free. Assume the annualized rates are the
same for all maturities (a flat yield curve).
A1. Suppose the yield on a one-year zero is 1%. What is the value of
a) A zero-coupon bond that matures in a year and has a face value of $25?
25
= $24.75
(1 + 0.01)1
b) A zero-coupon bond that matures in two years and has a face value of $25?
25
= $24.51
(1 + 0.01)2
c) A zero-coupon bond that matures in three years and has a face value of $25?
25
= $24.26
(1 + 0.01)3
d) A zero-coupon bond that matures in four years and has a face value of $1025?
1025
= $985.00
(1 + 0.01)4
e) If you held all 4 of these bonds in a portfolio, what is the value of your portfolio?
24.75+24.51+24.26+985=1058.52 (or 1058.53, depending on your rounding).
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A2. Complete the following table of the prices of zero-coupon bonds. Assume that all zero-coupon
bonds have a par value of $100.
Term (years)
Price if zero
Price if zero
rate is 6.25%pa rate is 6.50%pa
Change in
price ($)
Change in
price (%)
2
$88.58131
$88.16593
– $0.41538
– 0.469%
9
$57.94815
$56.73532
– $1.21283
– 2.093%
2-year bond
$𝟏𝟎𝟎
= $𝟖𝟖. 𝟓𝟖𝟏𝟑𝟏
(𝟏. 𝟎𝟔𝟐𝟓)𝟐
$𝟏𝟎𝟎
= $𝟖𝟖. 𝟏𝟔𝟓𝟗𝟑
𝑷=
(𝟏. 𝟎𝟔𝟓)𝟐
Price change = − $0.41538
−$0.41538
% Price change =
= −𝟎. 𝟒𝟔𝟗%
$𝟖𝟖. 𝟓𝟖𝟏𝟑𝟏
𝑷=
9-year bond
$𝟏𝟎𝟎
= $𝟓𝟕. 𝟗𝟒𝟖𝟏𝟓
(𝟏. 𝟎𝟔𝟐𝟓)𝟗
$𝟏𝟎𝟎
𝑷=
= $𝟓𝟔. 𝟕𝟑𝟓𝟑𝟐
(𝟏. 𝟎𝟔𝟓)𝟗
Price change = − $1.21283
−$1.21283
% Price change =
= −𝟐. 𝟎𝟗𝟑%
$𝟓𝟕. 𝟗𝟒𝟖𝟏𝟓
𝑷=
Comments:
(1)
Inverse relationship between interest rates and prices.
(2)
Magnitude of the % price response is much greater for the longer-term bond.
A3. A 3-year zero-coupon bond issued by Spinifex Sands Ltd is priced at $77.7414 per $100 par
value. The market believes that there is a 99% chance that Spinifex Sands will be able to make full
payment on the bond when it matures and a 1% chance that it will default. It is believed that, in
the event of default, there is an 80% chance that the company will pay 60% of what it owes and a
20% chance that it will pay zero. Calculate the promised interest rate (pa) and the expected interest
rate (pa).
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The promised rate is given by:
1T
z0T
 Par 
=

 P0 
−1
13
 $100 
=

 $77.7414 
= 8.755% pa
−1
The expected rate is given by:
 Par 1 − d (1 − r )  
=

P
0


1T
z0*T
−1
 $100 1 − 0.01 (1 − 0.8  0.6 )  
=

$77.7414


13
−1
13
 $100  0.9948 
=

 $77.7414 
= 8.566% pa
−1
A4. “If the yield on a 5-year zero-coupon government bond is 8.4% pa, then the yield on a 6-year
zero-coupon government bond must be at least 6.95% pa.” Assuming interest rates are nonnegative (not such a great assumption since the GFC), do you agree? Why or why not?
Assuming that interest rates can’t be negative, then the statement is true.
If the 5-year zero-coupon bond is held to maturity, every dollar invested in the bond will become
$(1.084)5 = $1.49674 in 5 years’ time. Therefore, even if the one-year interest rate in Year 6 turns
out to be zero, the minimum we can be sure of having after 6 years is $1.49674, which is equivalent
to a 6-year rate of (1.49674)1/6 – 1 = 6.95% pa.
To make this point another way, suppose that the six-year zero rate was only 6.8% pa. Then if the
6-year bond is held to maturity, every dollar invested will become $(1.068)6 = $1.48398 in 6 years’
time. In this case, no-one would buy the 6-year bond because they are certain to earn more by
buying the 5-year bond and holding it to maturity, then waiting another year. Even if the one-year
interest rate in Year 6 is zero, they will still do better with the 5-year bond.
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Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 3 May 2021 via the LMS Quizzes Page.
B1. [Question 1] Suppose an Australian company issues commercial paper (a zero) with a
maturity in exactly 270 days and a face value of $10,000. To two decimal places, what is the
price today, if the bond’s yield is 3%?
Australian Money Market Securities use a 365-day year. So:
365
𝑛
365
0.03 = 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 ×
270
270
𝑟𝑝𝑒𝑟𝑖𝑜𝑑 = 0.03 ×
= 0.02219
365
𝑟𝐵𝐸𝑌 = 𝑟𝑝𝑒𝑟𝑖𝑜𝑑 ×
𝑇
𝑃=∑
𝑡=1
𝑃=
𝐶𝐹𝑡
(1 + 𝑟𝑓 )
𝑡
$10,000
= $9782.90
(1 + 0.02219)1
B2. [Question 2] A 5-year zero with a face value of $1,000,000 is sold for $650,000.
Expressed as a three-digit decimal (e.g., “7.5%” would be “0.075”), what is the annualized
yield to maturity of this zero-coupon bond?
𝑃=
𝐶𝐹𝑇
(1 + 𝑟𝑌𝑇𝑀 )𝑇
$650,000 =
(1 + 𝑟𝑌𝑇𝑀 )5 =
$1,000,000
(1 + 𝑟𝑌𝑇𝑀 )5
$1,000,000
= 1.53846
$650,000
5
𝑟𝑌𝑇𝑀 = √1.53846 − 1 = 0.090
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B3. [Question 3] Definitions are important – an example of other weirdness with bond
definitions. Recall that US Money market dealers, who deal in bonds with less than 1 year to
maturity, use something called “bank discount rates”.
Suppose a US dealer agrees to a quoted rate of 6% for a term of 90 days for a zero with a
$100,000 face value. To two decimal places, what is the price of the bond?
𝑛
)
360
90
)
𝑃 = $100,000 × (1 − 0.06 ×
360
𝑃 = $100,000 × (1 − 0.015)
𝑃 = $98,500
𝑃 = 𝐹𝑎𝑐𝑒𝑉𝑎𝑙𝑢𝑒 × (1 − 𝑞 ×
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FNCE 30001 – Investments
Semester 1, 2021
Module 9: Coupon Bonds & Term Structure
Tutorial Questions
Part B, the marked Quiz part of this assignment is graded pass/fail purely
for the attempt at answering. You must fill in an answer for full credit. Part
B Assignment Answers are due 9 am Monday 10 May via the LMS
Part A: This part is unmarked
A1: Given the following $100 par value risk-free, zero-coupon bonds:
Bond
A
B
C
D
Years to Maturity
1
2
3
4
Yield to Maturity
5%
6%
6.5%
7%
a. What is the 1-year forward rate 3 years from now?
1
(1.065)3 (1 + 𝑓3,4 ) = (1.07)4
𝑓3,4 = 8.51%
b. If the expectations hypothesis is correct, what is the market's expectation of the
one-year interest rate three years from now?
If the expectations hypothesis is correct, then the expected one-year interest rate is the same
as the forward rate. Please see the answer to A1a.
c. If you believe liquidity preference theory are expected future rates higher or lower or equal
to the forward rate?
You expect lower because today’s prices, and by extension the forward rates, add in a price
discount for illiquidity. This discount translates to a return premium.
A2. Given the table of bonds in A1, you would like to invest $1K in one year from time 1 to
time 3, but you would like to guarantee the rate you could get on that loan today at a rate of
7.258%. Assume no traded forward contracts exist, how can you lock in the interest rate
today?
–
Borrow
$1,000
1+𝑟0,1
(= $952.38) today from a 1-year bond at a rate of 𝑟0,1.
1
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$1,000
–
Invest
–
Notice that no money goes into your pocket at time zero and none leaves
–
Pay back $1,000 at time 1 (this is your “investment” in one year), and
–
Receive
1+𝑟0,1
=
(= $952.38) today in a 3-year loan at a rate of 𝑟0,3.
$1,000
1+𝑟0,1
3
(1 + 𝑟0,3 ) (= $1,150.43) at time 3.
$1,000
2
2
(1 + 𝑟0,1 )(1 + 𝑓1,3 ) = $1,000(1 + 𝑓1,3 )
1 + 𝑟0,1
Another view: the forward you are interested, f_13, corresponds to an interest rate
arranged today with which you invest in 1 year and withdraw in 3 years.
The starting point for this problem: you want to have a negative cash flow of $1,000
in 1 year. How can you do that? Well, you can borrow the present value of $1,000,
which is 1000/1.05=952.38.
Then you want to receive money in three years. Well, you can invest that 952.38
today for three years.
That means today, your net cash flows are +952.38 (borrowed) -952.38 (invested for
3 years) = 0. In one year, you pay $1000, and in three years, you receive $1,150.43.
A3:
(a)
What is the price of a 10% 5-year coupon bond trading at par?
By definition, the price must be the par value (ie $100 in this case).
(b)
Calculate the price of this bond if the yield is 6% pa.
$10 
1 
$100
1 −

+
0.06  ( 1.06 ) 5  ( 1.06 ) 5


= $42.123638 + $74.725817
P=
= $116.849455
(c)
Calculate the price of this bond if the yield is 12% pa.
$10 
1 
$100
1 −

+
P=
0.12  ( 1.12 ) 5  ( 1.12 ) 5


= $36.047762 + $56.742686
= $92.790448
2
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A4. The ratio of a bond’s coupon rate to its price is rather confusingly called its “current
yield”: current yield = coupon/price. (I say “confusingly” since this is a different calculation
from the current yield-to-maturity.)
If the coupon rate of a bond is less than the current yield, is the bond trading at a premium or
discount?
If the coupon rate is lower than the current yield, then the price today (currently!) must be
lower than the face value, therefore the bond is trading at a discount. To see this, consider:
𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑦𝑖𝑒𝑙𝑑 =
𝑐𝑜𝑢𝑝𝑜𝑛
,
𝑃
𝐶𝑜𝑢𝑝𝑜𝑛 𝑅𝑎𝑡𝑒 =
Coupon rate lower than the current yield means:
𝑐𝑜𝑢𝑝𝑜𝑛 𝑐𝑜𝑢𝑝𝑜𝑛
<
𝑃
𝐹𝑉
the bond is trading at a discount
𝑐𝑜𝑢𝑝𝑜𝑛
𝐹𝑉
𝑐𝑜𝑢𝑝𝑜𝑛 𝑐𝑜𝑢𝑝𝑜𝑛
<
𝐹𝑉
𝑃
𝑃 < 𝐹𝑉
A5. “The liquidity premium theory maintains that if today’s term structure slopes downwards, then
we can definitely say that the market expects interest rates to fall. But if today’s term structure
slopes upwards, then we can’t necessarily say anything about what the market expects interest rates
to do.” Is this statement true? Explain.
Assuming that liquidity premiums are never negative, the statement is correct.
In this case, liquidity premiums impart an upward bias to the rates that would be produced under
the pure expectations hypothesis. If, despite this upward bias, the observed term structure is
downward sloping then the underlying expectations must also be downward sloping.
But if the term structure slopes upwards, we can’t say for sure, as shown in these diagrams:
3
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Interest rate pa
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Observed
term
structure
Gap due to
liquidity
premiums
Underlying
expectations
Interest rate pa
Term (years)
Observed
term
structure
Gap due to
liquidity
premiums
Underlying
expectations
Term (years)
Part B: Assignment, worth 1.25 marks on a pass/fail basis.
4
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You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 10 May 2021 via the LMS Quizzes
Page.
On Monday 22 February 2021 you agreed to purchase 2.25% November 2022 Australian
government bonds at a yield of 1.06% pa. The par value is $75 million. The maturity date is 21
November 2022.
Using the (Reserve Bank) pricing formula in the module slides:
Question 1: What is f, the number of days from the pricing date to the next coupon payment?
Question 2: What is h, the number of days in the half-year ending on the next coupon payment
date?
Question 3: What is C, the half-yearly coupon payment, per $100 of par value?
Question 4: What is ytm, the half-yearly yield to maturity?
Question 5: What is n, the number of remaining coupons?
Question 6: Expressed as a whole number, what is the price of the bonds ($75m par value)?
5
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The Reserve Bank formula is:
P0 =



C 
1
Par


1
C
+
−
+


( 1 + ytm ) f / h  ytm  ( 1 + ytm ) n−1  ( 1 + ytm ) n−1 
1
where:
ytm
n
C
Par
f
h
is the yield to maturity per half-year
is the number of half-yearly coupons to be received
is the half-yearly coupon
is the par (face) value of the bond
is the number of days from the pricing date to the next coupon payment
is the number of days in the half-year ending on the next coupon payment date
Coupons are paid on 21 May and 21 November each year. The transaction date is Monday 22
February 2021. Hence, following Australian conventions, the pricing date is 3 working days later:
25 February 2021. The previous coupon date was 21 November 2020. The next coupon date is
Friday 21 May 2021, which is a trading day.
ytm
n
C
Par
f
h
=
=
=
=
=
=
½ × 0.0106 = 0.0053 per half-year
4
½ × 2.25% × $100 = $1.125
$100
3 (February) + 31 (March) + 30 (April) + 21 (May) = 85 days
9 (November) + 31 (December) + 31 (January) + 28 (February) + 31
(March) + 30 (April) + 21 (May) = 181 days
𝑃=
=
1
85 {1.125
181
1.0053
1
85
1.0053181
= 102.636
+
1.125
1
100
[1 −
]+
}
3
(1.0053)
0.0053
1.00533
{102.891}
𝑃𝑟𝑖𝑐𝑒($75𝑚 𝑃𝑎𝑟 𝑉𝑎𝑙𝑢𝑒) =
75,000,000
∗ 102.636 = 76,977,000
100
Note: If you use Excel and plug in the price directly from the calculation (without rounding), you
will get $76,977,110
6
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FNCE 30001 – Investments
Semester 1, 2021
Module 10: Managing Fixed Income Portfolios
Tutorial and Assignment Questions
Part B, the marked Quiz part of this assignment is graded pass/fail purely
for the attempt at answering. You must fill in an answer for full credit. Part
B Assignment Answers are due 9 am Monday 17 May via the LMS.
You will need to use a spreadsheet program like Excel or Google Sheets to solve this
assignment.
Part A: This part is unmarked
A1. A junior portfolio manager has been asked to establish a fund that will be worth $175 million
in four years’ time. Her supervisor has suggested to her that an appropriate investment would be
5-year 15% coupon bonds at a yield of 7.6% pa. Although the junior manager has some knowledge
of bonds, she does not understand the reason for this suggestion.
(a)
Explain the reason to the junior manager in simple terms.
“Duration” is a measure of the weighted average time period it takes for cash flows to arrive. The
weights are the present value of each cash flow. In turn, duration is closely related to a measure of
the sensitivity of a bond’s price to changes in required yield.
As shown in the table below, the duration of a 5-year 15% bond priced to yield 7.6% pa is almost
exactly 4 years.
yield =
0.076
Time
1
2
3
4
5
Totals
D=
Cash flows
15
15
15
15
115
t × cash
15
30
45
60
575
PV(cash)
13.9405204
12.955874
12.0407751
11.1903114
79.7327023
129.860183
PV(t × cash)
13.9405204
25.911748
36.1223253
44.7612458
398.663512
519.399351
3.999681333
It can be shown that if duration matches the investment horizon, then the future value of the
investment is “immunised” against parallel shifts in the yield curve. Hence, as suggested by the
supervisor, an investment in this 5-year bond is appropriate for an investment horizon of 4 years.
(b)
How much should she invest to establish the fund? What annual coupon interest will this
investment produce? If the par value of one bond is $10 million, how many bonds should
be bought?
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Given that the target sum in 4 years’ time is $175 million, the amount to be invested today is $175m
/ (1.076)4 = $130,553,634.
To find the annual coupon interest (C):

Par
0.15  Par 
1
1 −
+
5
0.076  ( 1.076 )  ( 1.076 ) 5


= 0.605274  Par + 0.693328  Par
$130,553,634 =
$130,553,634
1.298602
= $100,533,985
Par =
C = 0.15  $100,533,985
= $15,080,098
If “one bond” has a par value of $10 million, then the annual coupon interest is $1.5 million per
bond. Using the standard bond pricing formula, the price of one such bond is:

$1.5m 
1
$10m
1 −
+
5
0.076  ( 1.076 )  ( 1.076 ) 5


= $12,986,018
P=
The number of these bonds purchased at the start of Year 1 is therefore $130,553,634 / $12,986,018
= 10.0534001.
(c)
Immediately after the fund is established, yields increase by 100 basis points. Show that,
if no further yield shifts occur, the fund will achieve the target in four years’ time.
Yields increase by 100 basis points – that is, by 1% – so yields are 8.6% pa. Therefore, the price
of one bond becomes:

$1.5m 
1
$10m


P=
1−
+
0.086  ( 1.086 ) 5  ( 1.086 ) 5


= $12,515,430
Therefore, the bond holding after the increase in yield is worth 10.0534001 × $12,515,430 =
$125,822,625.
The investor should rebalance the portfolio now but we will ignore this. We will also ignore the
need to rebalance on future coupon dates.
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At the end of Year 1
The coupon interest received = 10.0534001 × $1,500,000 = $15,080,100.
The ex-interest price of one bond is:

$1.5m 
1
$10m
1 −

P=
+
0.086  ( 1.086 ) 4  ( 1.086 ) 4


= $12,091,757
Therefore, the number of new bonds purchased is $15,080,100 / $12,091,757 = 1.2471389 bonds.
The total bond holding therefore increases to 10.0534001 + 1.2471389 = 11.3005390 bonds. The
value of the bond holding is 11.3005390 × $12,091,757 = $136,643,372.
At the end of Year 2
The coupon interest received = 11.3005390 × $1,500,000 = $16,950,809.
The ex-interest price of one bond is:

$1.5m 
1
$10m
1 −

P=
+
0.086  ( 1.086 ) 3  ( 1.086 ) 3


= $11,631,648
Therefore, the number of new bonds purchased is $16,950,809 / $11,631,648 = 1.4573007 bonds.
The total bond holding therefore increases to 11.3005390 + 1.4573007 = 12.7578397 bonds. The
value of the bond holding is 12.7578397 × $11,631,648 = $148,394,701.
At the end of Year 3
The coupon interest received = 12.7578397 × $1,500,000 = $19,136,760.
The ex-interest price of one bond is:

$1.5m 
1
$10m


P=
1−
+
0.086  ( 1.086 ) 2  ( 1.086 ) 2


= $11,131,969
Therefore, the number of new bonds purchased is $19,136,760 / $11,131,969 = 1.7190813 bonds.
The total bond holding therefore increases to 12.7578397 + 1.7190813 = 14.476921 bonds. The
value of the bond holding is 14.476921 × $11,131,969 = $161,156,636.
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At the end of Year 4
The coupon interest received = 14.476921 × $1,500,000 = $21,715,382.
The ex-interest price of one bond is:
$11.5m
1.086
= $10,589,319
P=
At this point the portfolio is liquidated, as follows:
Sale of bonds = 14.476921 × $10,589,319 = $153,300,735
Coupon interest received = $21,715,382
Hence, cash held = $153,300,735 + $21,715,382 = $175,016,117.
Therefore, the objective of having at least $175 million has been achieved.
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The above is summarised in the following table:
Progress of the investment if yields increase from 7.6% pa to 8.6% pa.
Date = investment period expired (years)
0
1
2
3
Bond term left (years)
5
4
3
2
Coupon interest received ($)
$0
$15,080,100
$16,950,809
$19,136,760
Price of 1 bond ($); Par = $10m
$12,515,430
$12,091,757
$11,631,648
$11,131,969
No. of extra bonds bought
0
1.2471389
1.4573007
1.7190813
No. of bonds held
10.0534001
11.3005390
12.7578397
14.476921
Value of bonds held ($)
$125,822,625
$136,643,372
$148,394,701
$161,156,636
Bond price is the present value of the remaining cash flows; par value for one bond is $10m.
At the end of Year 4, the cash holding is $21,715,382 + $153,300,735 = $175,016,117.
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4
1
$21,715,382
$10,589,319
0
14.476921
$153,300,735
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A2. Consider a 7-year 12% coupon bond, with a par value of $100, and which has just paid a
coupon. The yield curve is flat at 9.25% pa. Coupons are paid annually.
(a)
Calculate the duration. Use the duration to make a first approximation of the percentage
capital gain or loss if the yield increases by 25 basis points.
See spreadsheet on the next page. Duration is 5.2349 years. First approximation is a capital loss of
1.19792%.
(b)
Calculate the convexity adjustment. Use this adjustment to make a second approximation
of the percentage capital gain or loss if the yield increases by 25 basis points.
See spreadsheet. Second approximation is a capital loss of 1.18810%.
(c)
Calculate the exact percentage capital gain or loss if the yield increases by 25 basis points.
See spreadsheet. Exact change is a capital loss of 1.18817%.
(d)
Assuming yields do not change, what will be the duration of the bond three months later?
If there is no coupon payment and no change in yield, then duration falls 1-for-1 with term to
maturity. In the next three months there is no coupon payment and the question tells us that there
has been no change in yield. Hence, in three months’ time, the duration will be 0.25 years less than
it is now. That is, duration will be 5.2349 – 0.25 = 4.9849 years.
yield =
0.0925
multiples
Time
Cash flows
1
2
3
4
5
6
7
Totals
D=
X=
Delta i
ADJ
0.095
PV(cash)
12
12
12
12
12
12
112
10.98398
10.05399
9.202735
8.423556
7.710349
7.057527
60.29314
113.7253
tx
t x (t+1) PV(cash x
New price
PV(cash)
multiples)
10.98398
2
21.96796
10.9589
20.10798
6
60.32393
10.00813
27.6082
12
110.4328
9.139846
33.69422
20
168.4711
8.346892
38.55174
30
231.3105
7.622732
42.34516
42
296.4162
6.961399
422.052
56
3376.416
59.33613
595.3433
4265.338
112.374
5.23492484
15.711715
0.0025
9.8198E-05
1st
approx %
2nd
approx %
-1.19792%
Exact %
-1.18817%
-1.18810%
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(e)
What will be the duration of the bond
(i)
immediately before the next coupon payment?
(ii)
immediately after the next coupon payment?
IMMEDIATELY BEFORE
Time
Cash flows
0
1
2
3
4
5
6
t x cash
12
12
12
12
12
12
112
PV(cash)
0
12
24
36
48
60
672
12
10.9839817
10.0539878
9.20273485
8.42355592
7.71034867
65.8702556
124.244865
PV(t x cash)
0
10.9839817
20.1079756
27.6082045
33.6942237
38.5517434
395.221534
526.167663
Totals
D=
4.234924835
IMMEDIATELY AFTER
Time
Cash flows
t x cash
PV(cash)
PV(t x cash)
0
0
0
0
1
12
12
10.9839817
10.9839817
2
12
24
10.0539878
20.1079756
3
12
36
9.20273485
27.6082045
4
12
48
8.42355592
33.6942237
5
12
60
7.71034867
38.5517434
6
112
672
65.8702556
395.221534
Totals
112.244865
526.167663
D=
4.687676934
A3. What is the “convexity” of a coupon bond? Why do investors have a positive view of
convexity?
The bond price is negatively related to the yield (first derivative). The curve is convex to the origin
(second derivative). The more convex the curve, the greater is the gain if yields fall and the smaller
is the loss if yields rise.
If the current yield-to-maturity increases or decreases, the high-convexity bond is the better one to
have (and hence, contrary to the diagram, the bond prices today would not be equal – the higher
convexity bond would have a higher price).
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Bond Price
High-convexity
bond
Low-convexity
bond
Current
price
Current yield
to maturity
Yield
Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 17 May 2021 via the LMS Quizzes
Page.
Given the following bond portfolio, comprised of bonds paying annual coupons:
Bond
Yield
FV per Bond
A
B
0.5%
0.6%
$100
$100
Number of Coupon Rate
Bonds Held
10
2%
20
2.5%
Years
Maturity
3
5
to
Question 1: What is the price per $100 FV of Bond A?
Question 2: What is the price per $100 FV of Bond B?
Question 3: What is the total value of the portfolio? (For each bond, multiply the Number of
Bonds Held by the price.)
To solve Questions 4 and 5, you should fill out a table that looks something like this:
Year Bond 1 Bond 2 Portfolio Cash Flows
0
-[Answer to Question 3]
1
?
?
?
…
…
…
….
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Question 4: What are cash flows of the portfolio in Year 3?
Question 5: What is the yield of the portfolio? (Hint: Use the IRR function in Excel on the
“Portfolio Cash Flows” column.)
Bond
Yield
A
B
FV per
Bond
0.50%
0.60%
Year
Bond 1
0
1
2
3
4
5
Coupon
Rate
100
100
Bond 2
20
20
1020
50
50
50
50
2050
Yield
2%
2.50%
Years to Bonds
Maturity Held
3
5
10
20
Price
Value of
Bond
Holding
$104.46 $1,044.55
$109.33 $2,186.63
$3,231.18
Portfolio
Cash Flows
-$3,231.18
70
70
1070
50
2050
0.5773%
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FNCE 30001 – Investments
Semester 1, 2021
Module 11: Portfolio Performance Evaluation
Tutorial and Assignment Questions
Part B, the marked Quiz part of this assignment is graded pass/fail purely
for the attempt at answering. You must fill in an answer for full credit. Part
B Assignment Answers are due 9 am Monday 24 May via the LMS.
Part A: This part is unmarked
A1. Conventional wisdom says one should measure a manager’s investment performance over
an entire market cycle. What arguments support this contention? What arguments contradict
it?
Support: A manager could be a better forecaster in one scenario than another. For example, a
high-beta manager will do better in up markets and worse in down markets. Therefore, we
should observe performance over an entire cycle. Also, to the extent that observing a manager
over an entire cycle increases the number of observations, it would improve the reliability of
the measurement.
Contradict: If managerial skill (ability to generate alpha) doesn’t vary over the business
cycle, and if we adequately control for exposure to the market (i.e. adjust for beta), then
market performance should not affect the relative performance of individual managers. It is
therefore not necessary to wait for an entire market cycle to pass before you evaluate a
manager.
A2. See Chapter 18.6 “Performance Attribution Procedures” in the textbook.
Consider the following information regarding the performance of a money manager in a recent
month. The table presents the actual return of each sector of the manager’s portfolio in column
(1), the fraction of the portfolio allocated to each sector in column (2), the benchmark or neutral
sector allocations in column (3) and the returns of sector indexes in column (4).
(1) Actual
(2) Actual
(3) Benchmark
(4) Index return
return
weight
weight
Equity
2.0%
0.60
0.50
2.5% (ASX200)
Bonds
1.0%
0.30
0.40
1.2%(Aggregate bond index)
Cash
0.5%
0.10
0.10
0.5%
a. What was the manager’s return in the month? What was her over- or
underperformance?
b. What was the contribution of security selection to relative performance?
c. What was the contribution of asset allocation to relative performance?
a. Actual: (0.60 x 2.0%) + (0.30 x 1.0%) + (0.10 x 0.5%) = 1.55%
Bogey: (0.50 x 2.5%) + (0.40 x 1.2%) + (0.10 x 0.5%) = 1.78%
It underperformed: = 1.78% – 1.55% = 0.23%
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b. Security selection:
Market
Equity
Bonds
Cash
Manager’s
Portfolio
Index
Excess
portfolio
Contribution
performance performance performance
weight
2.0%
2.5%
−0.5%
0.60
−0.30%
1.0%
1.2%
−0.2%
0.30
−0.06%
0.5%
0.5%
0.0%
0.10
0.00%
Contribution of security selection:
−0.36%
c. Asset allocation:
Market
Equity
Bonds
Cash
Actual
weight
0.60
0.30
0.10
Summary
Security selection
Asset allocation
Excess performance
Benchmark
Excess weight Index return
weight
0.50
0.10
2.5%
0.40
−0.10
1.2%
0.10
0.00
0.5%
Contribution of asset allocation:
Contribution
0.25%
-0.12%
0.00%
0.13%
−0.36%
0.13%
−0.23%
A3. [Use Excel or Google Sheets or another spreadsheet program!]
The Bigger and Better Australia Fund (BBAF) is an open-ended mutual fund that invests in a
wide range of assets. It is fully invested at all times and revalues its funds under management
monthly. As at 31 December 2019, funds under management stood at $84.590m. BBAF’s
month-by-month record (in thousands of dollars) for 2020 is shown in the table below.
Month
(end)
Funds under Divi- Capital Distribut- Redempt- New Fees
management dends gains
ions
ions
inflows
&
losses
January
100637
53
6300
0
4367
14230 169
February
108986
41
−1051
0
215
9775 201
March
101294
59
−5891
153
4188
2699 218
April
106319
388
2460
0
3137
5517 203
May
107980
33
1394
0
1810
2257 213
June
108209
40
783
461
924
1007 216
July
104572
65
569
0
5184
1129 216
August
98657
80
−1960
0
4177
351 209
September
95467
18
1313
163
8037
3876 197
October
93435
358
−1425
0
2883
2109 191
November
100554
35
3963
0
1520
4828 187
December
99561
58
−756
451
1600
1957 201
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(a)
Calculate the after-fee time-weighted rate of return (pa) on BBAF in 2008.
To calculate the after-fee time-weighted rate of return (TWR):
Month
(end)
January
February
March
April
May
June
July
August
September
October
November
December
Funds under
management
Dividends
100637
108986
101294
106319
107980
108209
104572
98657
95467
93435
100554
99561
53
41
59
388
33
40
65
80
18
358
35
58
Capital Fees Returns =
gains
Dividends
&
+ gains &
losses
losses −
fees
6300 169
6184
−1051 201
−1211
−5891 218
−6050
2460 203
2645
1394 213
1214
783 216
607
569 216
418
−1960 209
−2089
1313 197
1134
−1425 191
−1258
3963 187
3811
−756 201
−899
Rate of
return
(pm)
1 plus
rate of
return
(pm)
0.073106
−0.012033
−0.055512
0.026112
0.011418
0.005621
0.003863
−0.019977
0.011494
−0.013177
0.040788
−0.008940
1.073106
0.987967
0.944488
1.026112
1.011418
1.005621
1.003863
0.980023
1.011494
0.986823
1.040788
0.991060
TWR = 1.073106 × 0.987967 × 0.944488 × 1.026112 × 1.011418 × 1.005621 × 1.003863
× 0.980023 × 1.011494 × 0.986823 × 1.040788 × 0.991060 − 1
= 5.856% pa
(b)
Calculate the dollar-weighted rate of return (pa) on BBAF in 2020.
To calculate the dollar-weighted rate of return (DWR):
Month
(end)
January
February
March
April
May
June
July
August
September
October
November
December
Funds under Distribut- Redemptmanagement
ions
ions
100637
108986
101294
106319
107980
108209
104572
98657
95467
93435
100554
99561
0
0
153
0
0
461
0
0
163
0
0
451
4367
215
4188
3137
1810
924
5184
4177
8037
2883
1520
1600
New
inflows
Dist +
Reds −
NIs
14230
9775
2699
5517
2257
1007
1129
351
3876
2109
4828
1957
−9863
−9560
1642
−2380
−447
378
4055
3826
4324
774
−3308
94
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Opening
and
closing
values
−84590
99561
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The DWR is the value of r in:
−
+
$9863 $9560
$1642
$2380
$447
$378
$4055
−
+
−
−
+
+
1 + r ( 1 + r ) 2 ( 1 + r ) 3 ( 1 + r )4 ( 1 + r )5 (1 + r )6 (1 + r )7
$3826
(1 + r )
8
+
$4324
(1 + r )
9
+
$774
(1 + r )
10
−
$3308
(1 + r )
11
+
$94
(1 + r )
12
+
$99561
(1 + r )
12
− $84590 = 0
which (using Excel’s IRR function) solves to give r = 0.374834% pm = (1.00374834)12 − 1
pa = 4.592% pa.
A4. In 2020 the return on the Safety First Fund was 10%, while the return on the market
portfolio was 12% and the risk-free return was 3%. Comparative statistics are shown in the
table below.
Statistic
Standard deviation of return
Beta
Residual standard deviation σ(eP)
Safety First Fund
7.5%
0.75
4.0%
Market Portfolio
15%
1.00
0%
Calculate and comment on:
(a)
(b)
(c)
(d)
The Sharpe ratio
The Treynor ratio
Jensen’s alpha
The information ratio
(a)
The Sharpe ratio
Sharpe ratio =
rP − r f
P
10% − 3%
7.5%
= 0.93
=
The Sharpe ratio for the market portfolio is:
Sharpe ratio =
rM − r f
M
12% − 3%
15%
= 0.60
=
Because 0.93 > 0.60, this indicates outperformance by the Safety First Fund.
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(b)
The Treynor ratio
Treynor ratio =
rP − r f
P
10% − 3%
0.75
= 9.33%
The market risk premium is rM − r f = 12% − 3% = 9%.
=
Because 9.33% > 9.00%, this indicates outperformance by the Safety First Fund.
(c)
Jensen’s alpha
Jensen’s alpha is the excess return according to the CAPM.
(
)
 P = rP −  r f +  P rM − r f 
= 10% −  3% + 0.75  ( 12% − 3%) 
= 0.25%
Because 0.25% > 0, this indicates outperformance by the Safety First Fund.
(d)
The information ratio
Information ratio =
P
 ( eP )
0.25%
4%
= 0.0625
=
The information ratio is small but positive.
(Part B follows on next page)
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Part B: Assignment, worth 1.25 marks on a pass/fail basis.
You will receive full marks if you make an honest attempt at completing the
assignment. This is due 9 am Monday 24 May 2021 via the LMS Quizzes
Page.
Please download the template answer sheet and put all the answers on the template. This is
very similar to what you will have to do for the final exam.
Step 1: If you haven’t done so already, please download the app “Scannable” for Apple/iOS
devices from the Apple app store or “Genius Scan” for Android devices from the Google
Play store. You will use these to create one PDF file of your answers.
Step 2: Please read the directions for scanning. Please use white paper and black/dark ink.
Step 3: Download the answer template. Other than the multiple-choice question, answers
must be written on the template in the space provided for each question.
Step 4: Print the template to write your answers for Part B or import the PDF to a tablet, where
you can write directly on the PDF electronically (if you write your answers electronically on a
tablet, you obviously do not need to worry about scanning).
Step 5: Write your name and Student ID number in the marked boxes on the first page.
Your name and ID number will be read by computer program (Gradescope), so
please make sure you write your name and ID # very clearly on the first page of the
template.
Step 6: Write up good attempts at each of the questions on the assignment for full credit.
Please use white paper and black/dark ink.
Step 7: Upload your scanned answers as one PDF file to the appropriate Canvas Quiz
question.
Please use the apps “Scannable” or “Genius Scan” do not upload photos.
The multiple-choice question:
A plan sponsor with a portfolio manager who invests in small-capitalisation U. S. stocks
should have the plan sponsor's performance measured against which one of the following?
Note: each of these are US stock indices.
•
•
•
•
The Dow Jones Industrial Average –an index of 30 of the largest listed companies in
the US.
The S&P 500 –an index of 500 of the largest companies in the US. It is meant to be
representative of the entire US stock market.
The Wilshire 5000 –an index of all stock listed in the US (regardless of whether there
are exactly 5000 or not).
The Russell 2000 –an index of the stocks ranked from the 3000th largest to the 1001st
largest companies in the US.
The best answer is the Russell 2000. It is an index of small(er) stock.
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B1. Could portfolio A show a higher Sharpe ratio than that of another portfolio B and at the
same time a lower 𝑀2 measure? Explain.
No. The M2 is an equivalent representation of the Sharpe measure, with the added difference
of providing a risk-adjusted measure of performance that can be easily interpreted as a
differential return relative to a benchmark. Thus, it provides the same information as the
Sharpe measure but in a different format.
B2. You’ve been provided with the following data, covering one year, concerning the portfolios
of two equity managers (manager A and manager B). Although the portfolios consist primarily
of common stocks, cash reserves are included in the calculation of both portfolio betas and
performance. By way of perspective, selected data for the financial markets are included in the
following table.
Manager A
Manager B
S&P 500
Lehman bond index
91-day Treasury bills
Total return
24.0%
30.0
21.0
31.0
12.0
Beta
1.0
1.5
a. First calculate the alphas and Treynor ratios of the two managers and then compare
the risk adjusted performance of these two managers relative to each other and to the
S&P500.
αA = 24% – [12% + 1.0(21% – 12%)] = 3.0%
αB = 30% – [12% + 1.5(21% – 12%)] = 4.5%
TA = (24 – 12)/1 = 12
TB = (30 – 12)/1.5 = 12
TS&P500 = (21 – 12)/1 = 9
Both managers performed better than the market. We see this in the positive alphas
and in the Treynor measures that are better than the S&P500’s Treynor measure. As
an addition to a passive diversified portfolio, both A and B are candidates because
they both have positive alphas and superior reward for risk. Comparing A and B to
each other, the only difference is leverage. Note that B has 50% more risk than A
(Beta B = 1.5 and Beta A = 1.0) and B has a 50% higher alpha and no difference in
the Treynor measure. Both A & B exhibited the same level of skill or stock picking
ability, the only difference is leverage.
b. Explain why the conclusions drawn from this calculation may be misleading.
(i) One year of data is too small a sample.
(ii) The portfolios may have significantly different levels of diversification. If both
have the same risk-adjusted return, the less diversified portfolio has a higher exposure
to risk because of its higher diversifiable risk. Since the above measure adjusts for
systematic risk only, it does not tell the entire story.
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