Question

Question 8 Suppose that you have $10,000 to invest, and you can invest it in stocks or bonds. Each month, bonds yield a certain return of 1.1%. Each month, stocks yield a risky return of 2% with probability 0.8 and - 1.2% with probability 0.2. Assume returns are independent across months. You choose your portfolio as suggested by Benartzi & Thaler. Let x be the change in your portfolio's value between now and the next time you evaluate your portfolio. Of course x will be risky — that is, your choice will generate a lottery over possible outcomes for x. For any lottery (x1, p1;...;XN, Pn), you evaluate this lottery according to prospective utility where if x;> 0 v(x) = { Xi 2.2x; if x; <0. (a) Suppose you plan to evaluate your portfolio after one month. If you invest in all bonds, what is the lottery over x? If you invest in all stocks, what is the lottery over x? Which do you prefer, all bonds or all stocks? (b) Suppose you plan to evaluate your portfolio after two months. If you invest in all bonds, what is the lottery over x? If you invest in all stocks, what is the lottery over x? Which do you prefer, all bonds or all stocks?

Answer 1

Part (a0

All in bonds:

Return is certain and that is 1.1%. Hence, x = change = 1.1% x 10,000 = 110; p = 1 (as it's a certain return); v(x) = x = 110

Hence, lottery over x = p x v(x) = 1 x 110 = 110

All in stocks:

p = 0.8, x = 2% x 10,000 = 200; v(x) = x = 200 as x > 0

p = 0.2, x = -1.2% x 10,000 = -120; v(x) = 2.2x = 2.2 x (-120) = - 264 as x < 0

Hence, lottery over x = Sum of (p x v(x)) = 0.8 x 200 + 0.2 x (-264) = 107.2

Since lottery over x is higher in case you invest all in bonds, I will prefer all in bonds.

Part (b)

Note that investment horizon is now two months.

All in bonds:

x = change in value = 10,000 x (1 + 1.1%)² - 10,000 = 221.21; v(x) = x = 221.21 as x > 0; p = 1 as the returns are certain in case of all bonds. Hence, lottery over x = p x v(x) = 1 x 221.21 = 221.21

All in stocks:

There are four states possible now:

Month 1	Month 2	Portfolio value		x = change	v(x)	probability	p	p x v(x)
Up	Up	10,000 x (1 + 2%) x (1 + 2%) =	10,404.00	404.00	404	0.8 x 0.8 =	0.64	258.56
Up	down	10,000 x (1 + 2%) x (1 - 1.2%) =	10,077.60	77.60	77.6	0.8 x 0.2 =	0.16	12.42
Down	up	10,000 x (1 - 1.2%) x (1 + 2%) =	10,077.60	77.60	77.6	0.2 x 0.8 =	0.16	12.42
Down	down	10,000 x (1 - 1.2%) x (1 - 1.2%) =	9,761.44	(238.56)	(524.83)	0.2 x 0.2 =	0.04	(20.99)
		Total						262.40

The lottery on x = 262.40 > that in case of all bonds.

Hence, I will now prefer all stocks.