Question

a) In the first trading day of 1998, Hang Seng Index closed at 10680.60, while in the last trading day of 2017, it closed at 29919.15. What is the average annual growth rate of Hang Seng Index in this 20-year?                              

b) The following yield curve is observed of the U.S. Treasury securities on 28th October 2019:

Maturity (Year)	Yield Rate (%)
1	1.60
2	1.64
3	1.65

Suppose the pure expectation theory is correct. Forecast the expected one-year yield rate of one year later and of two years later respectively.  

Answer 1

Solution:

a) The average annual growth rate of Hang Seng Index in 20 years. is 180.13% (approximately)

Annual growth rate = (29919.15 - 10680.60) / 10680.60

=1.8013173 = 180.13%

b) According to pure expectation theory, the yield for security maturing in 2 years is equal to the yield on 2 securities maturing in 1 year.

=> For one year yield rate of one year later:

Option 1: Buy 2-year security and hold it for 2 years (Rate 1.64%)

Option 2: Buy 1-year security, hold it for 1 year and at the end of year 1 reinvest the proceeds in 1-year security (Rate 1.60%)

=>Assume $100 is to be invested.

Return according to option 1 = 100*(1+0.0164)^2 = $103.306896

Return according to option 2 = 100*(1+0.016)*(1+X) {Here X is the expected return at the end of year 1 for future}

According to the theory both the return needs to be equal:

100*(1+0.016)*(1+X) = 103.306896

X = 0.0168 = 1.68%

Therefore, the one-year yield rate of 1 year later is 1.68%

=> For one year yield rate of two years later:

Option 1: Buy 3-year security and hold it for 3 years (Rate 1.65%)

Option 2: Buy 1-year security, hold it for 1 year and at the end of year 1 and year 2 reinvest the proceeds in 1-year security (Rate 1.60%)

=>Assume $100 is to be invested.

Return according to option 1 = 100*(1+0.0165)^3 = $105.0321242

Return according to option 2 = 100*(1+0.016)*(1+X1)*(1+X2) {Here X1 and X2 are the expected return at the end of year 1 and year 2 respectively}

Accoring to the theory both the return needs to be equal:

100*(1+0.016)*(1+X1)*(1+X2) = 105.0321242

100*(1+0.016)*(1+0.0168)*(1+X2) = 105.0321242

X = 0.0167 = 1.67%

Therefore, one-year yield rate of 2 year later is 1.67%