Question

On January 1st, 2019 the 1-year Hungarian Government Bonds are offered at 1% yield. An investor is interested in investing in the Hungarian capital market. He has the following information about the expected annual returns, variances and correlations for two Hungarian cargo airlines (assuming these are the only stocks traded on this market): ASL Airlines and Fleet Air.

	correlates with	correlates with
stock	ASL	Fleet	exp annual return	variance
ASL	1	0.6	5%	0.16
Fleet	0.6	1	7%	0.09

a) i) Do the 1-year Hungarian Government bonds bear any risk? Explain any assumptions you made in arriving at your answer. [2 marks]

ii) Find the annual risk free rate in this economy. [1 mark]

iii) Calculate the annual risk premium of ASL Air. [1 mark]

b) In order to decide his optimum portfolio, the investor needs to know the tangency portfolio in this market. Find the weights of the tangency portfolio. then calculate the expected return and variance of this tangency portfolio [4 marks]

c) Calculate the expected annual return and variance of the tangency portfolio. [3 marks]

Answer 1

a. i) No, 1-year Hungarian Government Bonds do not bear any risk because government debt securities are always considered to be risk free.

ii) Annual Risk Free Rate = 1% [i.e. the yield offered by Government Bonds]

iii) Risk Premium = Rm - Rf

where Rm is the return of the market and Rf is the risk free rate.

Expected Return of ASL Air = 5%

Risk premium of ASL Air = Rm-Rf = 5 - 1 = 4%

Meaning thereby, for additional risk undertaken by investing in ASL, the investor is earning an additional return of 4%.

iv) Calculation of optimum weights for minimizing risk

Let weight of security of ASL be Wa and Weight of security of Fleet be Wb.

Optimum weights are given by the formula

where square of Standard Deviation of B (Fleet) = Variance = 0.16

square of Standard Deviation of A (ASL) = Variance = 0.09

Standard Deviation of B (Fleet) = square root of Variance = square root of 0.16 = 0.4

Standard Deviation of A (ASL) = square root of Variance = square root of 0.09 = 0.3

rA,B = Correlation Coefficient of A & B = 0.6

Putting the values into the equation, we get

= ( 0.16 - 0.6 x 0.3 x 0.4 ) / ( 0.09 + 0.16 - 2 x 0.6 x 0.3 x 0.4)

= 0.088 / 0.106

= 0.8302 or 83%

Weight of A = 1 - Wb = 1 -0.83 = 0.17 or 17%

Wa = 0.17

Wb = 0.83

v) Calculation of Expected Return of Tangency Portfolio

Return of Portfolio is weighted average return given by the equation,

Ra x Wa + Rb x Wb

where, Ra is the return of Security A

Rb is the return of Security B

Wa is the weight of Security A

and Wb is the weight of Security B

= 5 x 0.17 + 7 x 0.83 = 6.66%

vi) Calculation of Standard Deviation of Portfolio

Portfolio Risk is given by

= Square Root of ( 0.16 x 0.17 x 0.17+ 0.09 x 0.83 x 0.83 + 2 x 0.4 x 0.3 x 0.17 x 0.83 x 0.6 )

= Square root of ( 0.0869 )

= Standard Deviation = 0.2947

Variance = Square of Standard Deviation = 0.0869 or 8.69 or 8.7%

For any query or clarification, please leave a comment.