Troy and Kristy Reynolds (both currently 56) have determined that they will require retirement income equal to $93,000 in today's dollars, based on their current income. They plan to retire in 8 years and wish to assume an after-tax return on their investments, prior to retirement, of 8%. They plan to readjust their assets after retirement and believe that their net return will drop to 6%. Troy's parents are both in their late eighties, and Kristy's parents are in their seventies. Troy and Kristy assume that retirement will last for 30 years, and that inflation will average 2%.

 

3. For purposes of Social Security retirement benefits, the Reynolds reach full-retirement age at age 67. Their full benefit is expected to be $30,000 in today's dollars. If the Reynolds decided to factor in Social Security and begin taking Social Security benefits when they retire at age 64, how much personal capital would they need to accumulate at retirement?

 

1) Income that Troy and Kristy need in their first year of retirement

Income that Troy and Kristy need in their first year of retirement = Current Income * (1+inflation rate)^n

Income that Troy and Kristy need in their first year of retirement = 93000 * (1+2%)^8

Income that Troy and Kristy need in their first year of retirement = 108964.32

2) The amount of capital necessary at the start of retirement to support their income needs throughout retirement

The amount of capital necessary at the start of retirement = Present Value of annuity of 108964.32

• Real rate = (1+interest rate after retirement) /(1+inflation rate) - 1
• Real rate = (1+6%)/(1+2%) - 1 = 3.9216%

The amount of capital necessary at the start of retirement = Annuity * (1-(1+r)^-n)/r

The amount of capital necessary at the start of retirement = 108964.32 * (1-(1+3.9216%)^-30)/3.9216%

The amount of capital necessary at the start of retirement = 1902281.11

3) Personal capital Required at age of 64

Social Security benefit at age of 67 = Current Value * (1+inflation rate)^n, where, n = 67 years - 56 years = 11 years

Social Security benefit at age of 67 = 30000*(1+2%)^11 = 37301.23

Present value of Social Security benefit at age of 67 = Social Security benefit at age of 67 * (1-(1+r)^-n)/r where, n = (30 - 3) = 27 years,

Present value of Social Security benefit at age of 67 = 37301.23 * (1-(1+3.9216%)^-27)/3.9216% = 614505.17

Present value of Social Security benefit at age of 64 = Present value of Social Security benefit at age of 67 / (1+r)^3

Present value of Social Security benefit at age of 64 = 614505.17/(1+3.9216%)^3 = 547530.19

Personal capital required at the age of 64 = The amount of capital necessary at the start of retirement - Present value of Social Security benefit at age of 64

Personal capital required at the age of 64 = 1902281.11 - 547530.19 = 1354750.92