FM for Actuaries

126 CHAPTER 4
Table 4.2:
An example of investment-year rates and portfolio rates
Calendar year Investment-year rates (%) Portfolio rates (%)
of investment (Y ) i Y 1 i Y 2 i Y 3 i Y +3
2012 5.6 5.6 5.7 5.9
2013 5.6 5.7 5.8 6.2
2014 5.8 5.9 6.0 6.3
2015 6.2 6.3 6.6
2016 6.7 6.4
2017 7.1
In Table 4.2, investments are credited the investment-year rates in the first 3
years. The calendar year of the investment is denoted by Y , the rate of interest
credited for the tth year of the investment made in calendar year Y is i Y t for t =
1, 2, 3, andi Y is the portfolio rate credited for calendar year Y . Thus, using these
notations, i Y t = i Y +t−1 for t>3. For example, in Table 4.2, an investment made
in 2012 earns 5.6%, 5.6% and 5.7% in 2012, 2013 and 2014, respectively, which
are the investment-year rates. Subsequent to that, the investment earns 5.9%, 6.2%
and 6.3% in 2015, 2016 and 2017, respectively, which are the portfolio rates.
Example 4.14: A fund credits interest according to Table 4.2. Find the total
interest credited in the period 2015 through 2017 for (a) an investment in 2010,
(b) an investment in 2014, (c) an investment in 2015, (d) an investment in 2015,
withdrawn every year and reinvested in the fund as new money. All investments,
including reinvestments, are made on January 1.
Solution: The last column of the table gives the portfolio rate of interest in 2015
through 2017. For (a), the investment made in 2010 earns the portfolio rate of
interest in 2015 through 2017. Thus, the total return over the 3-year period is
1.059 × 1.062 × 1.063 − 1=19.55%.
For (b), the investment made in 2014 earns the investment-year rates in 2015 and
2016 of 5.9% and 6.0%, respectively, and the portfolio rate in 2017 of 6.3%. The
total return is
1.059 × 1.06 × 1.063 − 1=19.33%.
For (c), the investment made in 2015 earns the investment-year rates in 2015 through
2017 to give the total return
1.062 × 1.063 × 1.066 − 1=20.34%.