Constructing a Covariance-Operator-Using-Matrix-Algebra_ Problem Set

EFIN 401 FALL 2021

PROBLEM SET #2 1. Assume that we have an n x k multivariate sample:

1 2, ....,, kY y y y =   where jy is an n x 1 vector containing the sample values for variable j for j=1,…k. Prove that the kxk sample covariance matrix

2 1 12 1

2 12 2 2

2 1 2

ˆ ˆ ˆ ˆ ˆ ˆˆ

ˆ ˆ ˆ

k

k

k k k

σ σ σ σ σ σ

σ σ σ

     Σ =       





   



can be directly computed as: ( ) ( )1 11ˆ ( [ 1 1 ])nnY Y Y I Y−′ ′ ′Σ = = −

(from the posted handout " Constructing-a-Covariance-Operator-Using-Matrix-Algebra.pdf ")

2. Use the information and procedures from question (4) of the the last problem set i.e. "Suppose that an individual starts with a zero retirement account balance and makes their initial payment one year from today. They wish to save a constant amount (at the end of each year) over the next 35 years to have an amount sufficient to purchase an expected fixed income retirement annuity of $75,000 per year (in today's purchasing power) for 30 years of possible retirement. During their savings years they can invest into one of two index funds. The first fund is a stock index with expected annual returns of 7.0% per year but with an expected annual volatility of 𝜎𝜎 = 0.20 over the next 35 years. The second fund is a bond index with expected annual returns of 4.0% and an expected annual volatility of 𝜎𝜎 = 0.05. Assume both the stock and bond indexes follow the standard log-normal stochastic diffusion process. At retirement, assume that the fixed annuity firm will charge them 4% for a fixed 30-year retirement annuity. …. Assume that the individual feels that having their income fall below $45,000 per year would be a serious hardship and that they are only willing to accept a 10% probability of having their retirement income fall below $45,000. "

We do not know the exact correlation between the returns of the stock and bond indexes. We wish

to examine the effects of potential "S-B log-price shock correlations" upon the minimal required

annual savings amount AS (and the associated 𝑞𝑞𝑆𝑆 = proportion to invest in stocks, and 𝑞𝑞𝐵𝐵 = 1 −

𝑞𝑞𝑆𝑆 = proportion to invest in bonds) required to satisfy both their mean and VaR10 retirement goals.

Find the minimal annual savings amount AS, the associated 𝒒𝒒𝑺𝑺 (the proportion of the annual

savings to invest in stocks), and 𝒒𝒒𝑩𝑩 = 𝟏𝟏 − 𝒒𝒒𝑺𝑺 (the proportion of the annual savings to invest

in bonds) required to satisfy both their mean and VaR10 retirement goals for each of the

potential S-B log-price shock correlations of 𝝆𝝆𝑺𝑺,𝑩𝑩 = -0.25, 𝝆𝝆𝑺𝑺,𝑩𝑩 = 0.00, and 𝝆𝝆𝑺𝑺,𝑩𝑩 = 0.25.