Question

An individual with cash to invest has two investment choices:

Buy a stock fund that every year either earns 40% or -20% with a 50/50 probability.

Buy a bond fund that every year returns either 5% or 0% also with a 50/50 probability.

Assume that the returns on the two funds are independent, and that returns from year to year are also independent. Also assume an initial portfolio value of $1. (The answers, however, will be unaffected if you use a different initial portfolio value.)

In addition, suppose the value function is linear and is specified as:

v(z) = z for z > 0

v(z) = 3z for z < 0

a. Which fund does the investor prefer if he looks at his portfolio i) once a year; or ii) once every two years?

b. How does your answer to part a. help us understand the equity premium puzzle?

Answer 1

A B C D F G 1 a) Option-1 "Stock Option-2 2 "Bond" Return per Probability Net Return Return per year Probability Net Return year 40% 20% 5% 2.50% 5 0.5 0.5 -20% 0.5 -10% 0% 0.00% 6 0.5 Total Net return 7 Total Net return 10% 2.50% 8 Stock Holding Period Investment Return Nature of Compounding Future Value Period Value (in %) return 10 Normal Componding Continuous Compounding Normal Componding Continuous Compounding 10.00% 11 1 1.1 1 0.1 10.52% 12 1 1.105 1 0.105 20.00% 13 2 1.2 1 0.2 22.14% 14 2 1.221 1 0.22 15 Bond 16 Holding Period Return Investment Nature of Compounding Period Future Value Value (in %) return 17 Normal Componding Continuous Compounding Normal Componding Continuous Compounding 2.50% 18 1 1.025 1 0.025 0.025 2.53% 19 1 1.025 1 5.00% 20 2 1.05 1 0.05 5.13% 21 1.051 1 0.05 22 23 Investor will prefer the fund which generates the higher return over the period he expects to hold the 24 security. According to this view, Investor will prefer to invest in "Stock" since it generates higher return 25 that is being generated by "Bond" 26 Co LON CO O

D A C F G 27 28 b) Equity premium puzzle help in understanding the premium of equity return over the returns offered by 29 bonds. This can be calculated as below for above case: Equity Nature of Compounding Period Stock Returns Bond Returns Premium Puzzle 30 Normal Componding Continuous Compounding Normal Componding Continuous Compounding 2.50% 7.50% 31 1 10.00% 10.52% 2.53% 7.99% 32 1 20.00% 5.00% 15.00% 33 2 22.14% 5.13% 17.01% 34 2 35 36 37 Calculations for above solutions are being given as below: 38 39 a) Option-1 "Stock Option-2 40 "Bond" 41 Return per Probability Return per year Probability Net Return Net Return 42 year 43 =A43*B43 -D43*E43 0.4 0.5 0.05 0.5 -A44*B44 =D44*E44 44 -0.2 0.5 C 0.5 -SUM(F43:F44) -SUM(C43:C44) Total Net return 45 Total Net return 46 47 Stock Holding Period Investment Return Nature of Compounding Future Value Period Value (in %) return 48 -F49/E49 =F50/E50 =F51/E51 -F52/E52 Normal Componding Continuous Compounding Normal Componding Continuous Compounding 49 - 1 + 1*10% 1 1 -D49-E49 =1*2.71828^(0.1 1) =D50-E50 50 1 1 -1+ 1*10 % 2 51 2 1 =D51-E51 =1*2.71828^(0.1*2) 52 1 -D52-E52 53

A B C D F 53 Bond 54 Holding Period Investment Return Nature of Compounding Period Future Value Value (in %) return 55 Normal Componding Continuous Compounding Normal Componding Continuous Compounding =F56/E56 =F57/E57 =F58/E58 =F59/E59 -1+ 1* 2.5% 56 1 1 =D56-E56 -1*2.71828^(0.025*1 57 1 1 -D57-E57 -1+ 1*2.5%*2 58 2 1 =D58-E58 -1*2.71828^(0.025*2) 59 1 D59-E59 60 61 Investor will prefer the fund which generates the higher return over the period he expects to hold the 62 security. According to this view, Investor will prefer to invest in "Stock" since it generates higher return 63 that is being generated by "Bond". 64 65 66 b) Equity premium puzzle help in understanding the premium of equity return over the returns offered by 67 bonds. This can be calculated as below for above case: Equity Nature of Compounding Bond Returns Period Stock Returns Premium Puzzle 68 Normal Componding Continuous Compounding Normal Componding Continuous Compounding 69 1 G49 G56 =D69-E69 G57 70 G50 D70-E70 71 2 G51 -G58 =D71-E71 72 2 G52 G59 D72-E72 73 74