The “decision of State Council on establishment of a unified basic pension system for enterprise employees” ([1997] No. 26 Document) promulgated in 1997 is a symbol of the establishment of pension system for urban workers. The “decision of State Council on the improvement of basic old-age insurance system for enterprise employees” ([2005] No. 38 Document) promulgated in 2005 partly amended the former document. These two documents are applicable for different people. This paper divides the subjects into four samples: the old, the second old, the second new, and the new. The old refers to those who retired before the implementation of [1997] No. 26 Document, and they receive the basic pension; the second old refers to those who work before the implementation of [1997] No. 26 Document, as well as those who retire in time between the implementation of [1997] No. 26 Document and [2005] No. 38 Document, and they receive the basic pension, individual account pension, and transitional pension. The second new refers to those who work before the implementation of [1997] No. 26 Document and retire after the implementation of [2005] No. 38 Document, and they also receive basic pension, individual account pension, and transitional pension. The new refers to those who work after the implementation of [1997] No. 26 Document, they receive basic pension fund and individual account pension.

As of the end of 2014, there are 13 provinces that have fully funded personal accounts, and the process is slow.^{Footnote 8} For convenience in research, this paper assumes that China has not fully funded personal accounts. Then, the personal account only has the payment records of employees, which provides numerical basis for granting individual account pension, as well as the balance of individual account pension. At the same time, the unified financial accounts include personal account payment income and fund payment income, which are used to pay for the basic pension, transitional pension, personal account pension, and individual account return expenses. The return of personal accounts also refers to the balance of individual account that granted to employees’ heirs in the situation that employees died. Moreover, if individual account is exhausted and the employees still survive,^{Footnote 9} the government would continue to grant individual account fund. To guarantee the reliability of the conclusion, this paper also show results of the situation with personal account funding in the part of sensitivity analysis.

### The pension fund income model

The pension fund income in the year of *t* is equal to the value of insured workers in the year of *t* multiplied by the base of pension payment in the year of *t*, and multiplied by the pension payment rate in the year of *t*, the function is as follows:

$$ {(AI)}_t=\left({\displaystyle \sum_{i=1}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={a}_t^j}^{b_t^j-1}{N}_{t, x}^{i, j}}}}\right)\times {\overline{w}}_t\times {R}_t=\left({\displaystyle \sum_{i=1}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={a}_t^j}^{b_t^j-1}{N}_{t, x}^{i, j}}}}\right)\times {\overline{w}}_{t_0-1}\times {\displaystyle \prod_{s={t}_0}^t\left(1+{k}_s\right)\times {R}_t} $$

(1)

(AI)_{
t
} refers to the pension fund income in the year of *t*, and *i* is equal to 1, 2, 3,4, which refers to the old, the second old, the second new, and the new. *j* = 1, 2, 3, which refers to male, female cadres, and female workers respectively. \( {N}_{t, x}^{i, j} \) refers to the number of insured workers that are included in *i* or *j*, as well as at the age of *x* in the year of *t*. \( {a}_t^i\;\mathrm{and}\;{b}_t^i \) refers to the age that employees participate in old-age insurance, and that employees retire respectively. \( {\displaystyle {\sum}_{i=1}^4{\displaystyle {\sum}_{j=1}^3{\displaystyle {\sum}_{x={a}_t^j}^{b_t^j}{N}_{t, x}^{i, j}}}} \) refers to the number of insured employees, *w*
_{
t
} refers to the base of pension payment in the year of *t*, *t*
_{0} refers to the starting time of actuarial analysis, *k*
_{
t
} refers to the growth rate of payment base in the year of *t*, and *R*
_{
t
} refers to payment rate of pension fund.

### The expenditure model of pension fund

The expenditure of pension fund in the year of *t* (AC)_{
t
} includes the basic pension fund expenditure in the year of *t* (AC)_{
t,b
}, the transitional pension expenditure in the year of *t* (AC)_{
t,g
}, the individual account pension expenditure in the year of *t* (AC)_{
t,j
} ,^{Footnote 10} and the return expenditure of individual account (AC)_{
t,i
}.^{Footnote 11} The basic pension expenditure in the year of *t* is equal to the number of insured employees who retire in the year of *t* multiplied by the annual per basic pension in the year of *t*, and the per basic pension in the year of *t* is equal to planned base multiplied by the planed proportion of basic pension fund, and multiplied by the growth coefficient,^{1} the function is as follows:

$$ {\left(\mathrm{AC}\right)}_{t, b}={\displaystyle \sum_{i=1}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={b}_t^j}^{c_t^j}\left[{N}_{t, x}^{i, j}\times {\overline{B}}_{t, x}^{i, j}\times {s}_{t, x}^{i, j}\times {\displaystyle \prod_{s= t- x+{b}_t^j}^t\left(1+{g}_s\right)}\right]}}} $$

(2)

\( {C}_t^j \) refers to the maximum survival age of insured workers who are *j* in the year of *t*, \( {\overline{B}}_{t, x}^{i, j} \) is the planned base of basic pension fund of the insured workers who are *i* and *j* at the age of *x* in the year of *t*. The annual planned base of old and the second old is the average wage before his retirement while that of the new and the second new is the average of the above value and the indexation of expenditure base. \( {s}_{t, x}^{i, j} \) is the planned proportion of basic pension fund of insured employee who are *i* and *j* in the year of *t*. g_{
t
} is the growth rate of basic pension in the year of *t*. 1 + g_{
t
} is the growth coefficient of basic pension fund.

The transitional pension expenditure in the year of *t* is equal to number of second old and second new who retire in the year of *t* multiplied by the per transitional pension fund in the year of *t*. The per transitional pension fund in the year of *t* is equal to the planned base multiplied by payment period, and multiplied by planned granting proportion of transitional pension fund, and multiplied by the growth coefficient. The function is as follows:

$$ {\left(\mathrm{AC}\right)}_{t, g}={\displaystyle \sum_{i=2}^3{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={b}_t^j}^{c_t^j}\left\{{N}_{t, x}^{i, j}\right.\times {\overline{G}}_{t, x}^{i, j}\times}\left[1998-\left( t- x+{a}_t^j\right)\right]\times {v}_{t. x}^{i, j}\times {\displaystyle \prod_{s= t- x+{b}_t^j}^t\left.\left(1+{g}_s\right)\right\}}}} $$

(3)

\( {\overline{G}}_{t. x}^{i, j} \) is the planned granting base of transitional pension of insured employees who are *i* and *j* at the age of *x* in the year of *t*. The planned granting base of the second old and second new are the average wage before retiring and the indexation of average expenditure base respectively. [1998 ‐ (t ‐ x + a_{t}
^{j})] refers to payment term of insured employees who are *j*.^{2}
\( {v}_{t, x}^{i, j} \) is the planned granting proportion of transitional pension fund of insured employees who are *i* and *j* at the age of *x* in the year of *t*.^{1} The growth rate of transitional pension fund in the year of *t* is equal to the growth rate of basic pension in the year of *t*.

The expenditure of individual account pension fund in the year of *t* is equal to the number of second old, second new, and new who retired in the year of *t* multiplied by per individual account pension fund in the year of *t*. The per individual account pension fund in the year of *t.* is equal to the individual account storage divided by the total number of planned granting months, and multiplied by value of 12, and multiplied by growth coefficient. The function is as follows:

$$ {\left(\mathrm{AC}\right)}_{t, i}={\displaystyle \sum_{i=2}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={b}_t^j}^{c_t^j}\left\{\left\{{N}_{t, x}^{i, j}\right.\times 12\times \right.}\left[{\displaystyle \sum_{s={a}_t^j}^{b_t^j}{\overline{w}}_s}\times {R}_s^2\times {\left(1+ r\right)}^{b_t^j- s-1}\right]\left./{m}_t^{i, j}\right\}\times {\displaystyle \prod_{s= t- x+{b}_t^j}^t\left.\left(1+{g}_s\right)\right\}}}} $$

(4)

*r* is the deposit band interest rate of 1 year, \( {R}_t^2 \) is the expenditure rate of individual account in the year of *t*, \( {m}_t^{i, j} \) the number of planed months that grant individual account pension to insured employees who are *i* and *j* in the year of *t.* The return expenditure of individual account is equal to the number of dead employees multiplied by individual account balance. The function is as follows:

$$ {\left(\mathrm{AC}\right)}_{t, i}^2={\displaystyle \sum_{i=2}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={b}_t^i}^{b_1^j+{m}_1^{i, j}/12}\left({D}_{t, x}^{i, j}\times \left({b}_t^j+\frac{m_t^{i, j}}{12}- x\right)\right)}\times \left(12\times \left({\displaystyle \sum_{s={a}_t^j}^{b_t^j-1}{\overline{w}}_s}\times {R}_s^2\times {\left(1+ r\right)}^{b_t^j- s-1}\right)/{m}_t^{i, j}\right)\Big)}+}{\displaystyle \sum_{i=2}^4{\displaystyle \sum_{j=1}^3{\displaystyle \sum_{x={a}_t^j}^{b_t^j-1}\left({D}_{t, x}^{i, j}\times \left({\displaystyle \sum_{s={a}_t^j}^x{\overline{w}}_s\times {R}_s^2\times {\left(1+ r\right)}^{x- s}}\right)\right)}}} $$

(5)

\( {D}_{t, x}^{i, j} \) is the number of dead insured employees who are *i* and *j* at the age of *x* in the year of *t.* Other symbols have the same meaning as the above symbols. The first item at the right of the function is the returned expenditure of individual account of retired insured employees while the second item is that of working insured employees.

### The cumulative balance model of pension fund

The cumulative balance of pension fund in the year of *t* is equal to the total of balance in the last year *t* − 1 (including the interest) and the balance in the year of *t* (including the interest). The balance in the year of *t* is equal to revenue of pension fund minus expenditure of pension fund. The function is as follows:

$$ {F}_t={F}_{t-1}\times \left(1+ r\right)+\left[{\left(\mathrm{AI}\right)}_t-{\left(\mathrm{AC}\right)}_t\right]\times \left(1+ r\right) $$

(6)

*F*
_{
t
} is the cumulative balance of pension fund in the year of *t.* Other symbols are same as the above symbols.