# Application Chapter 6 - 9 Flashcards

## learning equations and formulas

Using the CAPM equation, when the risk-free rate is 2.5%, the expected return of the market is 12%, and the beta of asset i is 1.5, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (E(R(m)) - R(f))

0.025 + 1.5 (0.12 - 0.025) = 0.1675 (16.75%)

Using the CAPM equation, when the risk-free rate is 2%, the expected return of the market is 10%, and the beta of asset i is 1.25, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (E(R(m)) - R(f))

0.02 + 1.25 (0.1 - 0.02) = 0.12

Using the CAPM equation, when the risk-free rate is 3%, the expected return of the market is 15%, and the beta of asset i is 2, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (R(E(m)) - R(f))

0.03 + 2 (0.15 - 0.03) = 0.27 (or 27%)

Using the CAPM equation, when the risk-free rate is 4%, the expected return of the market is 20%, and the beta of asset i is 2.25, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (R(E(m) - R(f))

0.04 + 2.25 (0.20 - 0.04) = 0.4 (or 40%)

Using the CAPM equation, when the risk-free rate is 2%, the expected return of the market is 25%, and the beta of asset i is 1, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (E(R(m)) - R(f))

0.02 + 1 (0.25 - 0.02) = 0.25 (or 25%)

Using the CAPM equation, when the risk-free rate is 1.5%, the expected return of the market is 15%, and the beta of asset i is 1.15, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (R(E(m)) - R(f))

0.015 + 1.15 (0.15 - 0.015) = 0.17025 (or 17.03%)

Using the CAPM equation, when the risk-free rate is 2.5%, the expected return of the market is 10%, and the beta of asset i is 0.5, what is the expected return of asset i?

E(R(i)) = R(f) + Beta (E(R(m) - R(f))

0.025 + 0.5 (0.10 - 0.025) = 0.0625 (or 6.25%)

Risk-free rate is 2% and the beta of asset i is 1.25, if the action return of the market is 22%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 30%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta (R(mt) - R(f) + E(it)

(it) asset i in time period.

2% + 1.25(22% - 2%) = 0.27

to find the idiosyncratic return:

0.3 - 0.27 = 0.03

Risk-free rate is 2.5% and the beta of asset i is 1.5, if the action return of the market is 12%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 23%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta (R(mt) - R(f)) + E(it)

0.025+ 1.5 (0.12 - 0.025) = 0.1675 (or 16.75%)

Idiosyncratic return:

(23% is the actual return)

23% - 16.75% = 6.25%

Risk-free rate is 3% and the beta of asset i is 2, if the action return of the market is 15%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 29%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta (R(mt) - R(f)) - E(it)

0.03 + 2 (0.15 - 0.03) = 0.27 (27%)

Idiosyncratic return:

29% - 27% = 2%

Risk-free rate is 4% and the beta of asset i is 2.25, if the action return of the market is 20%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 22%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta (R(mt) - R(f)) - E(it)

0.04 + 2.25 (0.20 - 0.04) = 0.4 (40%)

Idiosyncratic return:

22% - 40% = -18%

Risk-free rate is 2% and the beta of asset i is 1, if the action return of the market is 25%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 41%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta (R(mt) - R(f)) - E(it)

0.02 + 1 (0.25 - 0.02) = 0.25 (or 25%)

Idiosyncratic return:

41% - 25% = 16%

Risk-free rate is 1.5% and the beta of asset i is 1.15, if the action return of the market is 15%, the ex post CAPM model would generate a return due to non-idiosyncratic effects of:

Then if the asset’s actual return is 28.30%, then the what would be attributable to idiosyncratic return, E(it):

R(it) = R(f) + Beta

A researcher wishes to test for statistically

significant factors in explaining asset returns. Using a confidence level of 90%,

how many statistically significant factors would the researcher expect to identify

by testing 50 variables, independent from one another, that had no true

relationship to the returns?

What if research were performed with a confidence level of

99.9% but with 100 researchers, each testing 50 different variables on different

data sets?

You are searching for the probability of a mistake.

1 researcher conducting the test with a 90% confidence level.

(1 - 0.9) x 50 = 5

If there are 100 researchers conducting the test with a 99.9% confidence level

(((1 - 0.999) x 50) x 100) = 5

20 researchers wishes to test for statistically

significant factors in explaining asset returns. Using a confidence level of 95%,

how many statistically significant factors would the researchers expect to identify

by testing 100 variables, independent from one another, that had no true

relationship to the returns?

What if research were performed with a confidence level of

90% but with 25 researchers, each testing 35 different variables on different

data sets?

1 - 0.95 = 0.05

0.05 x 100 = 5

20 x 5 = 100

1 - 0.9 = 0.1

- 1 x 35 = 3.5
- 5 x 25 = 87.5

20 researchers wishes to test for statistically

significant factors in explaining asset returns. Using a confidence level of 95%,

how many statistically significant factors would the researchers expect to identify

by testing 100 variables, independent from one another, that had no true

relationship to the returns?

What if research were performed with a confidence level of

90% but with 25 researchers, each testing 35 different variables on different

data sets?

1 - 0.95 = 0.05

0.05 x 100 = 5

20 x 5 = 100

1 - 0.9 = 0.1

- 1 x 35 = 3.5
- 5 x 25 = 87.5

Nine-month riskless securities trade for $97,000,

and 12-month riskless securities sell for $P (both with $100,000 face values and

zero coupons). A forward contract on a three-month, riskless, zero-coupon bond,

with a $100,000 face value and a delivery of nine months, trades at $99,000.

What is the arbitrage-free price of the 12-month zero-coupon security (i.e., P)?

100,000/P = 100,000/97,000)(100,000/99,000)

100,000/P = 1.041341

P = 100,000/1.041341

P = 96,030.00

Therefore, the 12-month bond must sell for $96,030.00 to prevent arbitrage.

Nine-month riskless securities trade for $98,000,

and 12-month riskless securities sell for $P (both with $100,000 face values and

zero coupons). A forward contract on a three-month, riskless, zero-coupon bond,

with a $100,000 face value and a delivery of nine months, trades at $98,980.

What is the arbitrage-free price of the 12-month zero-coupon security (i.e., P)?

100,000/P = (100,000/98,000)(100,000/98,980)

$97,000.40

A three-year riskless security trades at a yield of

3.4%, whereas a forward contract on a two-year riskless security that settles in

three years trades at a forward rate of 2.4%. Assuming that the rates are

continuously compounded, what is the no-arbitrage yield of a five-year riskless

security?

F(T-t) = (T x R(T) - t x R(t)) / (T - t)

manipulation:

F (T-t) x (T - t) = (T x R(T) - t x R(t))

F(T-t) x (T - t) + t R(t) = T x R(T)

R(T) = F(T-t) x (T - t) + t x R / T

R(t) = 3.4% F(T-t) = 2.4% T = 5 t = 3

R(T) = 0.024% x (5 - 3) + 3 x 0.034% / 5 = 0.03 (3%)

A three-year riskless security trades at a yield of

3.4%, whereas a forward contract on a two-year riskless security that settles in

three years trades at a forward rate of 2.4%. Assuming that the rates are

continuously compounded, what is the no-arbitrage yield of a five-year riskless

security?

F(T-t) = (T x R(T) - t x R(t)) / (T - t)

manipulation:

F (T-t) x (T - t) = (T x R(T) - t x R(t))

F(T-t) x (T - t) + t (R(t)) = T x R(T)

R(T) = F(T-t) x (T - t) + t x R(t) / T

R(t) = 3.4% F(T-t) = 2.4% T = 5 t = 3

R(T) = 0.024% x (5 - 3) + 3 x 0.034% / 5 = 0.03 (3%)

A three-year riskless security trades at a yield of

7%, whereas a forward contract on a two-year riskless security that settles in

three years trades at a forward rate of 5%. Assuming that the rates are

continuously compounded, what is the no-arbitrage yield of a five-year riskless

security?

F(T-t) = (T x R(T) - t x R(t)) / (T - t)

manipulation:

F(T-t) x (T - t) = T x R(T) - t x R(t)

(F(T-t) x (T - t)) + (t x R(t) = T x R(T)

R(T) = (F(T-t) x (T-t)) + (t x R(t)) / T

R(T-t) = 0.05% R(t) = 0.07% T = 5 t = 3

R(T) = (0.05 x (5-3)) + 3 x 0.07 / 5 = 0.062 (6.2%)