Reading 9: Common Probability Distributions LOS n习题精选 between, holding, div, return

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LOS n, (Part 1): Distinguish between discretely and continuously compounded rates of return.

A stock increased in value last year. Which will be greater, its continuously compounded or its holding period return?

A) Neither, they will be equal.

B) Its holding period return.

C) Its continuously compounded return.

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When a stock increases in value, the holding period return is always greater than the continuously compounded return that would be required to generate that holding period return. For example, if a stock increases from $1 to $1.10 in a year, the holding period return is 10%. The continuously compounded rate needed to increase a stock's value by 10% is Ln(1.10) = 9.53%.

 

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Over a period of one year, an investor’s portfolio has declined in value from 127,350 to 108,427. What is the continuously compounded rate of return?

A) -13.84%.

B) -14.86%.

C) -16.09%.

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The continuously compounded rate of return = ln( S₁ / S₀ ) = ln(108,427 / 127,350) = –16.09%.

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If a stock decreases in one period and then increases by an equal dollar amount in the next period, will the respective arithmetic average of the continuously compounded and holding period rates of return be positive, negative, or zero?

A) Zero; positive.

B) Zero; zero.

C) Positive; zero.

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The holding period return will have an upward bias that will give a positive average. For example, a fall from 100 to 90 is 10%, and the rise from 90 to 100 is an increase of 11.1%. The continuously compounded return will have an arithmetic average of zero. Since we can sum continuously compounded rates for multiple periods, the continuously compounded rate for the two periods (0%), means the rates for the two periods must sum to zero, and their average must therefore be zero.

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Assume an investor purchases a stock for $50. One year later, the stock is worth $60. After one more year, the stock price has fallen to the original price of $50. Calculate the continuously compounded return for year 1 and year 2.

Year 1

Year 2

A) 18.23% 16.67%

B) -18.23% -18.23%

C) 18.23% -18.23%

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Given a holding period return of R, the continuously compounded rate of return is: ln(1 + R) = ln(Price₁/Price₀). Here, if the stock price increases to $60, r = ln(60/50) = 0.18232, or 18.23%.

Note: Calculator keystrokes are as follows. First, obtain the result of 60/50, or 1. On the TI BA II Plus, enter 1.20 and then click on LN. On the HP12C, 1.2 [ENTER] g [LN] (the LN appears in blue on the %T key).

The return for year 2 is ln(50/60), or ln(0.833) = negative 18.23%.

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If a stock decreases from $90 to $80, the continuously compounded rate of return for the period is:

A) -0.1250.

B) -0.1000.

C) -0.1178.

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This is given by the natural logarithm of the new price divided by the old price; ln(80 / 90) = -0.1178.

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Given a holding period return of R, the continuously compounded rate of return is:

A) ln(1 + R).

B) eR ? 1.

C) ln(1 ? R) ? 1.

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This is the formula for the continuously compounded rate of return.

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The continuously compounded rate of return that will generate a one-year holding period return of -6.5% is closest to:

A) -6.3%.

B) -5.7%.

C) -6.7%.

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Continuously compounded rate of return = ln(1 ? 0.065) = -6.72%.

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Mei Tekei just celebrated her 22nd birthday. When she is 27, she will receive a $100,000 inheritance. Tekei needs funds for the down payment on a co-op in Manhattan and has found a bank that will give her the present value of her inheritance amount, assuming an 8.0% stated annual interest rate with continuous compounding. Will the proceeds from the bank be sufficient to cover her down payment of $65,000?

A) Yes, Tekei will receive $67,028.

B) Yes, Tekei will receive $68,058.

C) No, Tekei will only receive $61,878.

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Because the rate is 8% compounded continuously, the effective annual rate is e^0.08 - 1 = 8.33%. To find the present value of the inheritance, enter N=5, I/Y=8.33, PMT=0, FV=100,000 CPT PV = 67,028.

Alternatively, 100,000e^-0.08(5) = 67,032.