Investment Management
assignments
Chapter 5
Exercise 16
Put option:
- exercise price = $110
- maturity = 1 year
- price = $12
a) What is the probability distribution of the HPR on the put option?
put = right to sell
if prices ↑, profit = 0 and loss is 100%
if prices ↓, profit >0.
Total investment = $100 in stock + $12 in put , according to spreadsheet = $112 total inv.
Excellent: prices go up, HPR of -100%
Good: prices go up, HPR of -100%
Poor: prices go down, ending value is 112 – 93.25 = $18.75
9ending value – put value) / put value x 100 = 6.75/12 x 100 = 56.25%
Crash: prices go down, ending value is 112 – 48 = $64
9ending value – put value)/put value x 100 = 964-12)/12 x 100 = 433.33%
b) what is probability distribution of the HPR on a portfolio consisting of one share of the
index and a put option?
In this case you own both a share and a put option.
if prices go up, you lose put but gain because of shares
if prices go down, you lose share but gain because of put
Ending price + put + dividend gives
excellent: $131
Good: $114
Poor: $93.25 + 18.75 = $112
Crash: $48 + 64 = $112
,9Ending value – total investment) /total investment x 100 gives HPR for all situations:
9there’s a mistake in the exercise, poor should have a HPR of 0% as well. 112 – )
c) buying a put gives insurance in the sense that HPR will always be zero or higher, as seen in
exercise b. If you don’t make profit based on increased share value, you make profit because
of the put option.
CFA 3
what is E[R] for stocks X and Y, given the numbers
E[R] = Σ probability * return
for X, E[R] = 0.2 * -20% + 0.5 * 18% + 0.3 *50% = 20%
for Y, E[R] = 0.2 * -15% + 0.5 * 20% + 0.3 *10% = 10%
CFA 4
what are the std. deviations for X and Y?
σ = sqrt. 9Σ p9x – μ))2), in which x is any value of the table and p the probability corresponding
σX = sqrt [0.2 x 9-20 -20)2] + [0.5 x 918-20)2] + [0.3 x 950 – 20)2] = 24.33%
σY = sqrt [0.2 x 9-15 – 10)2] + [0.5 x 920 – 10)2 + [0.3 x 910 – 10)2] = 13.23%
CFA 5
If you invest 9000 in X and 1000 in Y, this leads to an E[R] of
9000 x 0.2 + 1000 x 0.1 = 1900
percentual, the HPR is 1900 /10000 = 19%
Chapter 6
Exercise 10:
S&P 500 has average about 8% more than treasury bill risk premium
std. deviation of S&P was about 20% per year.
current T-bill rate is 5%
so, E[RI] = 5 + 8 = 13%
For every situation, multiply weight with expected return, sum it up to get E[R P] .
calculating variance can be done by taking the weights of both options and multiplying
them with their std. deviation, then square the outcome of that 9since V = σ 2)
however, since the σ of risk-free rates is 0, you only need to take the weight of the risky
, option and do what I described above.
11) utility level U = E[R] – 0.5 * Aσ2
take every expected return and variance as given above, plug and chug with A = 2
example: for WB = 0.6 and WI = 0.4, U = 0.082 – 0.5 * 2 * 0.0064 = 0.00756
12) same formula and method. For the weights we looked at in 11, we now get a utility level
of 0.0724.
Conclusion: if risk-aversion level A increases, this decreases the level of utility
13 – 19
risky portfolio with E[RP] of 18% and σ of 28%. T-bill rate is 8%
13)
70% in fund and 30% risk-free
0.7 x 18% + 0.3 x 8% = 15% E[RF]
14)
stock A: 0.7 x 0.25 = 17.5%
stock B: 0.7 x 0.32 = 22.4%
stock C: 0.7 x 0.43 = 30.1%
15) Sharpe ratio of portfolio:
90.18 – 0.08) / 0.28 = 0.3571
Client only invests 70% in portfolio, so get’s only 70% of the risk premium and 70% of the
std. deviation sharpe ratio comes out at the same of 0.3571
proof: 915% - 8%)/ 0.7 x 28% = 0.3571
16) slope of capital allocation line equals the Sharpe ratio, so slope = 0.3571
client takes smaller risk and therefore gets lower return.
assignments
Chapter 5
Exercise 16
Put option:
- exercise price = $110
- maturity = 1 year
- price = $12
a) What is the probability distribution of the HPR on the put option?
put = right to sell
if prices ↑, profit = 0 and loss is 100%
if prices ↓, profit >0.
Total investment = $100 in stock + $12 in put , according to spreadsheet = $112 total inv.
Excellent: prices go up, HPR of -100%
Good: prices go up, HPR of -100%
Poor: prices go down, ending value is 112 – 93.25 = $18.75
9ending value – put value) / put value x 100 = 6.75/12 x 100 = 56.25%
Crash: prices go down, ending value is 112 – 48 = $64
9ending value – put value)/put value x 100 = 964-12)/12 x 100 = 433.33%
b) what is probability distribution of the HPR on a portfolio consisting of one share of the
index and a put option?
In this case you own both a share and a put option.
if prices go up, you lose put but gain because of shares
if prices go down, you lose share but gain because of put
Ending price + put + dividend gives
excellent: $131
Good: $114
Poor: $93.25 + 18.75 = $112
Crash: $48 + 64 = $112
,9Ending value – total investment) /total investment x 100 gives HPR for all situations:
9there’s a mistake in the exercise, poor should have a HPR of 0% as well. 112 – )
c) buying a put gives insurance in the sense that HPR will always be zero or higher, as seen in
exercise b. If you don’t make profit based on increased share value, you make profit because
of the put option.
CFA 3
what is E[R] for stocks X and Y, given the numbers
E[R] = Σ probability * return
for X, E[R] = 0.2 * -20% + 0.5 * 18% + 0.3 *50% = 20%
for Y, E[R] = 0.2 * -15% + 0.5 * 20% + 0.3 *10% = 10%
CFA 4
what are the std. deviations for X and Y?
σ = sqrt. 9Σ p9x – μ))2), in which x is any value of the table and p the probability corresponding
σX = sqrt [0.2 x 9-20 -20)2] + [0.5 x 918-20)2] + [0.3 x 950 – 20)2] = 24.33%
σY = sqrt [0.2 x 9-15 – 10)2] + [0.5 x 920 – 10)2 + [0.3 x 910 – 10)2] = 13.23%
CFA 5
If you invest 9000 in X and 1000 in Y, this leads to an E[R] of
9000 x 0.2 + 1000 x 0.1 = 1900
percentual, the HPR is 1900 /10000 = 19%
Chapter 6
Exercise 10:
S&P 500 has average about 8% more than treasury bill risk premium
std. deviation of S&P was about 20% per year.
current T-bill rate is 5%
so, E[RI] = 5 + 8 = 13%
For every situation, multiply weight with expected return, sum it up to get E[R P] .
calculating variance can be done by taking the weights of both options and multiplying
them with their std. deviation, then square the outcome of that 9since V = σ 2)
however, since the σ of risk-free rates is 0, you only need to take the weight of the risky
, option and do what I described above.
11) utility level U = E[R] – 0.5 * Aσ2
take every expected return and variance as given above, plug and chug with A = 2
example: for WB = 0.6 and WI = 0.4, U = 0.082 – 0.5 * 2 * 0.0064 = 0.00756
12) same formula and method. For the weights we looked at in 11, we now get a utility level
of 0.0724.
Conclusion: if risk-aversion level A increases, this decreases the level of utility
13 – 19
risky portfolio with E[RP] of 18% and σ of 28%. T-bill rate is 8%
13)
70% in fund and 30% risk-free
0.7 x 18% + 0.3 x 8% = 15% E[RF]
14)
stock A: 0.7 x 0.25 = 17.5%
stock B: 0.7 x 0.32 = 22.4%
stock C: 0.7 x 0.43 = 30.1%
15) Sharpe ratio of portfolio:
90.18 – 0.08) / 0.28 = 0.3571
Client only invests 70% in portfolio, so get’s only 70% of the risk premium and 70% of the
std. deviation sharpe ratio comes out at the same of 0.3571
proof: 915% - 8%)/ 0.7 x 28% = 0.3571
16) slope of capital allocation line equals the Sharpe ratio, so slope = 0.3571
client takes smaller risk and therefore gets lower return.
