Question

Future value of a portfolio. Rachel and Richard want to know when their current portfolio will be sufficient for them to retire. They have the following balances in their​ portfolio: Money market account​ (MM): $35,000 Government bond mutual fund​ (GB): $140,000 Large capital mutual fund​ (LC): $105,000 Small capital mutual fund​ (SC):  ​$72,000 Real estate trust fund​ (RE):​$88,000 Rachel and Richard believe they need at least ​$2,300,000 to retire. The money market account grows at 2.5 % ​annually, the government bond mutual fund grows at 6.5 % ​annually, the large capital mutual fund grows at 9.5 % ​annually, the small capital mutual fund grows at 12.5 % ​annually, and the real estate trust fund grows at 4.5 % annually. With the assumption that no more funds will be deposited into any of these​ accounts, how long will it be until they reach the ​$2,300, 000 ​goal? Rachel and Richard will need to invest their accounts for ______ or more years to reach $2,300,000.

Answer 1

Future value of an amount A at an interest rate of R and period N = A x (1 + R)^N

Money Market Account: A_MM = amount in Money Market account = $ 35,000; R_MM = 2.5% annually

Government bond mutual fund​: A_GB = $ 140,000; R_GB = 6.5% annually

Large capital mutual fund​: A_LC = $105,000; R_LC = 9.5% annually

Small capital mutual fund​: A_SC = $ 72,000; R_SC = 12.5% annually

Real estate trust fund​: A_RE = ​$ 88,000; R_RE = 4.5% annually

Let us assume that they have to remain invested for N years to achieve the target future value of $ 2,300,000 = FV_target

Hence, the equation will be:

FV_target = A_MM x (1 + R_MM)^N + A_GB x (1 + R_GB)^N + A_LC x (1 + R_LC)^N + A_SC x (1 + R_SC)^N + A_RE x (1 + R_RE)^N

Hence, 2,300,000 = 35,000 x (1 + 2.5%)^N + 140,000 x (1 + 6.5%)^N + 105,000 x (1 + 9.5%)^N + 72,000 x (1 + 12.5%)^N + 88,000 x (1 + 4.5%)^N

There are two different methods to solve this now:

Method 1: Goal seek in excel. Seek that value of N such that RHS becomes equal to LHS. On doing the goal seek, i got the value of N as =  20.6782 years

Method 2: We have no other method but to adopt hit and trial to solve this.

Let's prepare a table showing the value of right hand side (RHS) of the equation for different values of N.

N	RHS ($)
1	472,910
6	686,431
11	1,016,583
16	1,535,936
21	2,366,227
26	3,713,531

Based on the table above, we can see that the value of N lies between 16 & 21. On a prorated basis, N = 16 + (21 - 16) / (2,366,227 - 1,535,936) x (2,300,000 - 1,535,936) = 20.60 years

So, you final answer should be:

Rachel and Richard will need to invest their accounts for ______ or more years to reach $2,300,000.

Fill in the blank with 21, if only integral value is allowed

Fill in the blank with 20.7, you are allowed to provide final answer up to one place of decimal

Fill in the blank with 20.68, you are allowed to provide final answer up to two places of decimal