Question

Problem: Jonathan and Beth plan on retiring in 15 years. They have made progress on their retirement portfolio (they currently have $100,000 in their 401ks + IRAs), but need to do more. Recognizing their lack of planning they have come to you for help in determining how much they need to save annually to produce an inflation-adjusted equivalent of $50,000 per year paid at the beginning of each your over 20 years of retirement. Moreover, they would like to leave a flat $1 million (not increased for any inflation) to their children. Upon further discussion, you and your clients assume that inflation will average 3% prior to retirement and 4% thereafter. Further, your clients estimate that they will earn 12% per year prior to retirement and 7% per year during retirement. Based on the above data assumptions provide your clients with answers to the following questions. • Question 1.1 – What income do the Havertons want to have for their first year of retirement (i.e., the inflated value of $50,000)? • Question 1.2 – What is the lump sum needed on the first day of retirement to fund 20 years of annual retirement incomes and a $1 million legacy for their children (Remember that there will be 20 years of retirement income that inflates each year, the income is taken at the start of each year, and there needs to be a residual $1 million legacy (not adjusted for inflation) that will be left to their children at the end of the period). • Question 1.3 – How much would they have to contribute at the end of every year (up to the start of retirement) to fund their lump sum need?

Answer 1

1.1 In 15 years at 3% inflation rate, the inflated value of $50000 would be = 50000 * (1+3%)¹⁵ = 77898.37

1.2 We have two requirements here : inflation adjusted $50000 for 20 years (paid beginning of the year) and $1 million at the end of 20 years from date of retirement. The first stream is like a growing annuity due and second one is simple lumpsome - we will calculate their respective PV and that is the amount required at the time of retirement.

PV of 1 million (we use 7% which is the post retirement rate) = 1000000/(1+7%)²⁰ = $258419

PV of growing annuity payed at start of period (annuity due) = ; where r is the applicable interest rate (7%) and t is the time period (20 years) and g is the growth rate (4% - post retirement inflation). The initial CF will be 77898.37. Plugging in the values as solving:

PV = 77898.37 * (1+7%)*[1 - (1+4%)²⁰ * (1+7%)^-20]/(7% - 4%) = 1205181.77

Hence the total corpus required at the beginning of retirement = (1205181.77 + 258419) = 1463600.78

1.3 The current retirement savings at $100,000 and they will grow at 12% in 15 years to :

100000 * (1+12%)¹⁵ = 547356.58

Hence there is a gap of = (1463600.78 - 547356.58) = 916244.20

Now we use the FV of annuity formula to calculate the amount required to fund this gap:

FV annuity = Initial CF * [(1+r)^t - 1]/r

916244.20 = CF * [(1+12%)¹⁵ - 1]/12% ; solving for CF, we get 24577.55 which is the amount they need to contribute end of each year for 15 years to meet their lumpsum need