Please show all work 3. The following table gives actual data on the U.S. CPI an

ID: 2710784 • Letter: P

Question

Please show all work

3. The following table gives actual data on the U.S. CPI and the S&P; 500 total return index (SPTRI). A. (20) Use this data to compute the average annual nominal return and real return for two consecutive 1-year holding period returns of a stock portfolio that matches the S&P; 500 total return index. (for example you could compute 201 1-12 and 2012-201 3) Please use continuous compounding. Express your final answer in percentage. B. (10) Now compute the average annual real return to a 2-year holding -period portfolio of your two consecutive years. C. (10) You have a friend who has been studying the stock market for the last few years and is about to invest $10,000 in a highly diversified portfolio. After 10 years, your friend expects to have over $66,000 in real dollars. You recall from your favorite report in Econ 135 that the average 10 year holding period return is about 7% with a standard deviation of 6%. Do you think your friend?s expectations are realistic? Why or why not?

Explanation / Answer

Answer A

2012-13

Nominal rate of return = 28.09%

Real rate of return = 26.79%

2013-14

Nominal rate of return = 12.81%

Real rate of return = 11.95%

Answer B

Annual average nominal rate of return for two year period = 20.45%

Annual average real rate of return for two year period = 19.37%

Answer C

Expectation of a return of 7% over a 10 year holding period is realistic.

CPI

SPTRI

12/31/2011

226

2159

12/31/2012

230

2504

12/31/2013

233

3316

12/31/2014

235

3769

For 2012-13

Let r be the rate of return i be the inflation rate

Return for 1 year period 2012-13

e^r*n = (SPTRI as on 12-31-2013)/(SPTRI as on 12-31-2012)

e^r*1 = 3316/2504 ==> e^r = 1.32428115

Taking logarithms on both sides

==> log e^r = log 1.32428115

e=2.71828

r log (2.71828) = log (1.32428115)

r * 0.4342942 = 0.1219801973

r = 0.1219801973/0.4342942 = 0.2808699 or 28.08699%

nominal rate of return on SPTRI in 2012-13 = 28.09% (rounded off)

Similarly inflation rate in 2012-13 is

e^i = 233/230 ==> e^i = 1.0130434782 ==> 2.71828^i = 1.0130434782

inflation rate in 2012-13 = 1.30% (rounded off)

real rate of return on SPTRI in 2012-13 = 28.09 – 1.30 = 26.79%

For 2013-14

e^r * n = (SPTRI on 12-31-2014)/(SPTRI on 12-31-2013)

e^r*1 = 3769/3316 ==> e^r = 1.136610374 ==> 2.71828^ r = 1.136610374

Taking logarithms

log 2.71828^r = log 1.136610374 ==> r * log 2.71828 = log 1.136610374

nominal rate of return on SPTRI in 2013-14 = 12.81% (rounded off)

Let i be the inflation rate

e^i = CPI 2014 / CPI 2013 ==> e^i = 235/233 ==> e^i = 1.00858369

Taking Logarithms

log e^i = log (1.00858369) ==> log 2.71828^i = log (1.00858369) ==> i * 0.4342942 = 0.0037119412

i = 0.0037119412/0.4342942 = 0.008547066 or 0.8547% (rounded off)

Inflation rate in 2013-14 = 0.86% (rounded off)

Real rate of return on SPTRI in 2013-2014 = 12.81% - 0.86% = 11.95%

Average annual nominal rate of return for 2012-14

Nominal rate of return for two year period 2012-14 can be calculated as follows

Let r be the nominal rate of return on a 2 year period

e^r*2 = (SPTRI on 12-31-2014)/(SPTRI on 12-31-2012) ==> e^2r = 3769/2504

log (2.71828)^2r = log (1.50519169329) ==> 2*r * 0.4342942 = 0.17759181

r = 0.17759181/2*0.4342942 = 0.20446 or 20.45% (rounded off)

Similarly inflation rate can be calculated as

e^2*i = CPI 2014/CPI 2012 ==> e^2*i = 235/230 ==> 2*i*log 2.71828 = log 1.02173913

2 * i * 0.4342942 = 0.0093400263 ==> i = 0.0093400263 /2*0.4342942 = 0.010753 or 1.08%

Annual average real rate of return for 2012-2014 = 20.45% - 1.08% = 19.37%

Investment = 10000

Period of investment = 10 years

Value in real dollars after 10 years = 66000

Let r be the real rate of return. Then

10000 * (1+r)^10 = 66000

(1+r)^10 = 66000/10000 ==> (1+r)^10 = 6.6 ==> 1+r = (6.6)^(1/10) ==> 1+r = 1.207687

r = 1.207687 – 1 =0.207687 or 20.77%

The real rate of return for the amounts mentioned is 20.77%

Hence the expectations of a return of 7% is realistic.

CPI