25  Portfolio and Investment Year Rates

A quality of human behavior that many investors possess is that they will search for the highest return for their initial investment, but will not actively manage their investment by continuing to search for the highest return thereafter. Banks and other financial institutions that compete for the deposits of investors use investment year and portfolio interest rates as a means to try to capitalize on this behavior.

25.1 Investment Year Rates

The investment year rate (or new money rate) is the interest rate applied to this investment during the first \(N\) years; the select rate can vary from year to year. These “new money” may be segregated every year for several years in terms of the interest rate earned before being integrated into a larger pooled fund.

We denote the investment year rate for a deposit at year \(y\) during the \(k\)-th investment year to be \(i^{y}_k\).

So, if one deposits an amount \(1.00\) at time \(y\) and keeps the money sitting on the account until the end of the \(k\)-th year, for any \(k \le N\), the amount in the account at time \(k\) will be: \[(1+i^y_1)(1+i^y_2)...(1+i^y_k)\]

25.2 Portfolio Rates

After the \(N\) years have elapsed from the initial deposit, the portfolio rate starts governing the account’s growth; for every year \(z \ge y+N\), the portfolio rate is denoted by \(i^z\).

So, the \(1.00\) deposited at time y grows to \[(1+i^y_1)(1+i^y_2)...(1+i^y_N)(1+i^{y+N})(1+i^{y+N+1})...(1+i^{y+N+r})\] at the end of year \(N+r\).

Investment year and portfolio rates are usually demonstrated via a table, for example:

Year	\[i^y_1\]	\[i^y_2\]	\[i^{y}_3\]	\[i^{y}_4\]	\[i^{y}_5\]	\[i^{y+5}\]
2015	9.00	9.00	9.10	9.10	9.20	8.85
2016	9.00	9.10	9.20	9.30	9.40	9.10
2017	9.25	9.35	9.50	9.55	9.60	9.35
2018	9.50	9.50	9.60	9.70	9.70
2019	10.00	10.00	9.90	9.80
2020	10.00	9.80	9.70
2021	9.50	9.50
2022	9.00

We read the table across and then down. For example, corresponding to the row for year 2015, reading across and then down, we have: \[i^{2015}_1 = 9.00\%, \quad i^{2015}_2 = 9.00\%, \quad i^{2015}_3 = 9.10\%, \quad i^{2015}_4 = 9.10\%, \quad i^{2015}_5 = 9.20\% \] After that, the portfolio rate is then used and we use: \[i^{2020}=8.85\%, \quad i^{2021}=9.10\%, \quad i^{2022}=9.35\%\]