Question

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SPY and XIU are ETFs tracking the S&P 500 and S&P/TSX 60 index, which are often...

SPY and XIU are ETFs tracking the S&P 500 and S&P/TSX 60 index, which are often used as proxies for the U.S. and Canadian stock markets, respectively. From a set of their historical data, the annual expected returns and standard deviations of those two ETFs and their covariance are estimated as follows: SPY: E(r)= 0.15 σ=0.28 XIU: E(r)= 0.18 σ=0.32 Covariance between SPY and XIU = 0.0618

Suppose that you have $10 million to invest for one year and you want to invest that money into SPY, XIU, and the Canadian one-year T-bill. Assume that the interest rate of the one-year T-Bill is 3% per annum.

Suppose that you have the following utility function: U=E(r) – ½ Aσ2 and A=3

Answer the following questions using Excel:

Draw the opportunity set offered by these two securities (with an increment of 0.01 in weight).

What is the optimal portfolio of SPY and XIU?

Determine your optimal asset allocation among SPY, XIU, and T-Bill, in percentage and in dollar amounts.

Solutions

Expert Solution

Canadian one-year T-bill is the risk-free asset

Draw the opportunity set offered by these two securities

Weight in SPY	Weight in XIU	Expected return	Expected risk (Standard deviation)
1%	99.00%	0.1797	0.318737698
2%	98.00%	0.1794	0.317488393
3%	97.00%	0.1791	0.316252241
4%	96.00%	0.1788	0.315029395
5%	95.00%	0.1785	0.313820012
6%	94.00%	0.1782	0.312624247
7%	93.00%	0.1779	0.311442258
8%	92.00%	0.1776	0.310274201
9%	91.00%	0.1773	0.309120236
10%	90.00%	0.177	0.307980519
11%	89.00%	0.1767	0.30685521
12%	88.00%	0.1764	0.305744468
13%	87.00%	0.1761	0.304648453
14%	86.00%	0.1758	0.303567324
15%	85.00%	0.1755	0.30250124
16%	84.00%	0.1752	0.301450361
17%	83.00%	0.1749	0.300414847
18%	82.00%	0.1746	0.299394856
19%	81.00%	0.1743	0.298390549
20%	80.00%	0.174	0.297402085
21%	79.00%	0.1737	0.296429621
22%	78.00%	0.1734	0.295473315
23%	77.00%	0.1731	0.294533326
24%	76.00%	0.1728	0.293609809
25%	75.00%	0.1725	0.292702921
26%	74.00%	0.1722	0.291812817
27%	73.00%	0.1719	0.29093965
28%	72.00%	0.1716	0.290083574
29%	71.00%	0.1713	0.289244741
30%	70.00%	0.171	0.2884233
31%	69.00%	0.1707	0.287619401
32%	68.00%	0.1704	0.286833192
33%	67.00%	0.1701	0.286064818
34%	66.00%	0.1698	0.285314423
35%	65.00%	0.1695	0.28458215
36%	64.00%	0.1692	0.283868138
37%	63.00%	0.1689	0.283172527
38%	62.00%	0.1686	0.282495451
39%	61.00%	0.1683	0.281837045
40%	60.00%	0.168	0.28119744
41%	59.00%	0.1677	0.280576763
42%	58.00%	0.1674	0.279975142
43%	57.00%	0.1671	0.279392699
44%	56.00%	0.1668	0.278829554
45%	55.00%	0.1665	0.278285824
46%	54.00%	0.1662	0.277761624
47%	53.00%	0.1659	0.277257065
48%	52.00%	0.1656	0.276772253
49%	51.00%	0.1653	0.276307293
50%	50.00%	0.165	0.275862284
51%	49.00%	0.1647	0.275437325
52%	48.00%	0.1644	0.275032507
53%	47.00%	0.1641	0.27464792
54%	46.00%	0.1638	0.274283649
55%	45.00%	0.1635	0.273939774
56%	44.00%	0.1632	0.273616374
57%	43.00%	0.1629	0.27331352
58%	42.00%	0.1626	0.27303128
59%	41.00%	0.1623	0.27276972
60%	40.00%	0.162	0.272528898
61%	39.00%	0.1617	0.272308869
62%	38.00%	0.1614	0.272109684
63%	37.00%	0.1611	0.271931388
64%	36.00%	0.1608	0.271774024
65%	35.00%	0.1605	0.271637626
66%	34.00%	0.1602	0.271522227
67%	33.00%	0.1599	0.271427854
68%	32.00%	0.1596	0.271354528
69%	31.00%	0.1593	0.271302267
70%	30.00%	0.159	0.271271082
71%	29.00%	0.1587	0.271260981
72%	28.00%	0.1584	0.271271967
73%	27.00%	0.1581	0.271304036
74%	26.00%	0.1578	0.271357182
75%	25.00%	0.1575	0.271431391
76%	24.00%	0.1572	0.271526647
77%	23.00%	0.1569	0.271642927
78%	22.00%	0.1566	0.271780205
79%	21.00%	0.1563	0.271938449
80%	20.00%	0.156	0.272117622
81%	19.00%	0.1557	0.272317682
82%	18.00%	0.1554	0.272538584
83%	17.00%	0.1551	0.272780278
84%	16.00%	0.1548	0.273042707
85%	15.00%	0.1545	0.273325813
86%	14.00%	0.1542	0.273629531
87%	13.00%	0.1539	0.273953792
88%	12.00%	0.1536	0.274298524
89%	11.00%	0.1533	0.274663649
90%	10.00%	0.153	0.275049087
91%	9.00%	0.1527	0.275454751
92%	8.00%	0.1524	0.275880554
93%	7.00%	0.1521	0.276326401
94%	6.00%	0.1518	0.276792196
95%	5.00%	0.1515	0.277277839
96%	4.00%	0.1512	0.277783225
97%	3.00%	0.1509	0.278308246
98%	2.00%	0.1506	0.278852793
99%	1.00%	0.1503	0.27941675
100%	0.00%	0.15	0.28

What is the optimal portfolio of SPY and XIU

We get the optimal portfolio using an excel solver

Initially, enter some random weights ( here 50%-50%)

Solving,

Hence, the weight of SPY in optimal portfolio is 53.50% and of XIU is 46.50%

Determine your optimal asset allocation among SPY, XIU, and T-Bill

We again use an excel solver.

Initially, we input the random weights in SPY, XIU and the Canadian one year T-bill (risk-free -asset) and enter the following constraints in excel solver

Solving we get,

Hence, weight in SPY = 23.90%

weight in XIU = 34.40%

Weight in Canadian T-bill (risk-free asset) = 41.70%

In dollar amounts,

in SPY = 23.90% * 10,000,000 = $2390000

in XIU =34.40%* 10,000,000 = $3440000

in Canadian T-bill (risk-free asset) =41.70%* 10,000,000 = $4170000

jeff jeffy answered 1 year ago

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