Plan Comparsion

Goal

The goal is to reproduce the calculation reported in Figure 2 in “Which Teachers Choose a Defined Contribution Pension Plan? Evidence from the Florida Retirement System” (Chingos and West, 2015) [PDF].

All calculation is based on a person starting working at age 32.

DC Plan

Compute the account balance for someone starting working at age 32, enrolled in a DC plan and leaving the system after working for T years where T ranges from 1 to 30.

Parameters

Factors relevant to DC plan

• m0: initial salary
• a: annual salary increasing rate
• contribution: annual contribution rate (including the matching rate from employer)
• ave.r: annual investment return rate

Set some initial parameters.

m0 = 60000
a = 0.03
ave.r = 5 / 100
contribution = 15.6 / 100
T = 30

Code

• xValues: Years of Consecutive Service Prior to Separation (start age = 32). In the Florida paper, this value ranges from 1 to 30.

• yValues: Account balance at age 62.

BalanceDC= function(T, ave.r){
  xValues = c(1:T) + 32
  yValues = rep(0, T)
  
  for (t in 1:T){
    AccountBalance = 0
    mT = m0
    for(j in 1:t){
      anualDeposit = mT * contribution
      monthlyDeposit = anualDeposit/12
    
      M = exp(log(1+ave.r)/12)
      for (m in 1:12){
        AccountBalance = AccountBalance * M + monthlyDeposit
      }
      mT = mT * (1 + a)
    }
    
    AccountBalance = AccountBalance * (1 + ave.r)^(T-t)    
    yValues[t] = AccountBalance
  }
   
 return(data.frame(x=xValues, y=yValues))
}

Plot

test = BalanceDC(T, ave.r)
x = test$x
y = test$y

plot (x, y,
        xlab = "Years of Service",
        ylab = "Pension Wealth",
        main = "Net Present Value of DC Benefits at age 62")

DB Plan

Assume a person starts working at age 32, enrolled in a DC plan and leeaves the system after working for T years where T ranges from 1 to 30.

Parameters

Factors relevant to DB plan

• m0: initial salary
• a: annual salary increasing rate
• pension : annual pension income
• discount.r: discount rate minus the pension annual increase rate. For example, if discount rate is 3% but also pension will increase annually with 3%, then discount.r = 0.

The formula for pension varies from plan to plan. It is usually equal to the product of the following two quantities (determined by the plan):

• ave.m: average salary. For example, the Florida teacher’s DB plan defines ave.m to be the average salary over the last 5 years of the teacher’s career.

• pension.percent: pension percentage. For example, the Florida teacher’s DB plan defines this percentage to be equal to 1.6 times T when T is bigger than 5 and 0 when T is less than or equal to 5.

Some initial values

m0 = 60000
a = 0.03
T = 30
pension.percent = 1.6 / 100

Code

• xValues: Years of Consecutive Service Prior to Separation (start age = 32). In the Florida paper, this value ranges from 1 to 30.

• yValues: Net present value (NPV) of the corresponding pension income

Our calculation involves two steps.

• Step 1: compute the annual pension income, which is the product of ave.m and pension.percent.

• Step 2: compute the (expected) NPV of the stream of pension income from age 63 to age 119 using the Actuarial Life Table from SSA website; we use the death rate averaged over male and female. The NPV formula is given below \[ \sum_{\text{age} = 63}^{119} ... = \sum_{t = 63-62}^{(119 - 62)} \text{annual-pension-income} \times p_t \times v^{t - 0.5} \] where \(v\) \[ v = \frac{1}{1 + \text{discount.rate}} \] is the discount factor and \[ \begin{aligned} p_t = & \mathbb{P}(\text{still alive at age} (62+t) \mid \text{alive at age } 62) \\ = & \quad \mathbb{P}(\text{surve for a year} \mid \text{alive at age } 62) \\ & \times \mathbb{P}(\text{surve for a year} \mid \text{alive at age } 63) \times \\ & \times \cdots \times \mathbb{P}(\text{surve for a year} \mid \text{alive at age } 62+t-1) \\ = & \prod_{j=1}^t \Big [ 1 - \mathbb{P} \big (\text{die within a year } \mid \text{alive at age } (62 + j-1) \big ) \Big ]. \end{aligned} \]

The probabilities in the last row above can be find from the Actuarial Life Table.

First, compute the probabilities.

data = read.table("mtable.txt", sep="\t")
names(data) = c("Age", "m.prob", "m.lives", "m.exp", 
                "f.prob", "f.lives", "f.exp")
newdata = data.frame(Age = data$Age, 
                     prob = data$m.prob + data$f.prob)
probs = newdata$prob[newdata$Age >= 62]
rm("data", "newdata")

Then, compute NPV for DB plan.

BalanceDB= function(T, discount.r){
  xValues = c(1:T) + 32
  yValues = rep(0, T)
  
  if (T < 6) {
    return(data.frame(x=xValues, y=yValues))
    exit
  }
  
  # Compute the annual pension income 
  pension = rep(0, T)
  for (t in 6:T){
    
    # compute ave.m: the average of the last five years' salaries
    tmp = tail(1:t, 5)
    tmp = tmp - 0.5
    ave.m = mean(m0 * (1 + a)^tmp)
    
    # compute pension income
    pension[t]  = ave.m * pension.percent * t
  }
  
  # Compute NPV of pension income
  
  pt = 1 
  v = 1 / (1 + discount.r)
  vt = v^{- 0.5}
  
  for(t in 1:length(probs)){
    
    pt = pt * (1 - probs[t])
    vt = vt * v
    yValues = yValues + pension * pt * vt
  }
   
 return(data.frame(x = xValues, y = yValues))
}

Plot

discount.r = 0.03 - 0.03
# discount.r = 0.07 - 0.03 
test = BalanceDB(T, discount.r)
x = test$x
y = test$y

plot(x, y, type="n", yaxt="n", 
     xlab = "Years of Service",
     ylab = "Pension Wealth (in Dollars)",
     main = "Net Present Value of DC Benefits at age 62")
lines(x, y, lty = 2)

points(x, y, pty = 2)
ytick = (1:11)*100000
axis(2, at = ytick, las = 2, cex.axis = 0.6,
     labels = formatC(ytick, big.mark = ",", format="f", digits = 0))