How the numbers work: retirement planning

The Future Value of Annuity Formula

1. FV = PMT × [((1 + r)^n - 1) / r]
2. Where:
3. FV = Future Value
4. PMT = Regular payment amount
5. r = Interest rate per period
6. n = Number of periods

Simple Example

1. Monthly contribution: $500
2. Annual interest rate: 6%
3. Time period: 10 years
4. Number of payments: 120 (10 years × 12 months)
5. Interest per period: 0.5% (6% / 12 months)
6. FV = $500 × [((1.005)^120 - 1) / 0.005]
7. FV = $500 × [(1.8194 - 1) / 0.005]
8. FV = $500 × 163.88

FV = $81,940

Common Applications

1. Age: 25, retiring at 65 (40 years)
2. Contribution: $200/month
3. Employer match: $200/month
4. Total monthly: $400
5. Expected return: 5% annually
6. Future value at 65: ~$610,000

Common Applications

1. Start when child is born
2. Save $300/month for 18 years
3. Conservative return: 4% annually
4. Future value at age 18: ~$82,000
5. Covers university tuition and living costs

Example 1: Monthly Savings

1. Monthly savings: $1,000
2. Annual interest rate: 4.5%
3. Time period: 5 years

Example 1: Monthly Savings

1. Number of periods (n) = 5 years × 12 months = 60
2. Interest per period (r) = 4.5% / 12 = 0.375% = 0.00375

Example 1: Monthly Savings

1. FV = PMT × [((1 + r)^n - 1) / r]
2. FV = $1,000 × [((1.00375)^60 - 1) / 0.00375]
3. FV = $1,000 × [(1.2516 - 1) / 0.00375]
4. FV = $1,000 × [0.2516 / 0.00375]
5. FV = $1,000 × 67.09

FV = $67,090

Example 1: Monthly Savings

1. Total contributions: $1,000 × 60 = $60,000
2. Interest earned: $67,090 - $60,000 = $7,090
3. Interest as % of contributions: 11.8%

Example 2: Quarterly Investment

1. Quarterly investment: $2,500
2. Annual return: 7%
3. Number of quarters: 15 × 4 = 60
4. Quarterly rate: 7% / 4 = 1.75% = 0.0175
5. FV = $2,500 × [((1.0175)^60 - 1) / 0.0175]
6. FV = $2,500 × 95.54

FV = $238,850

Reverse Calculation: Finding Required Payment

1. PMT = FV / [((1 + r)^n - 1) / r]
2. PMT = FV × [r / ((1 + r)^n - 1)]

Reverse Calculation: Finding Required Payment

1. n = 8 × 12 = 96 months
2. r = 5% / 12 = 0.4167% = 0.004167
3. PMT = $100,000 × [0.004167 / ((1.004167)^96 - 1)]
4. PMT = $100,000 × [0.004167 / 0.4856]
5. PMT = $100,000 / 116.52

PMT = $858 per month

Comparing Lump Sum vs Regular Contributions

1. Initial: $10,000
2. Rate: 6% annual
3. Time: 20 years
4. FV = $10,000 × (1.06)^20 = $32,071

Comparing Lump Sum vs Regular Contributions

1. Annual payment: $500
2. Same 6% return
3. Total invested: $10,000 (over 20 years)
4. FV = $500 × [((1.06)^20 - 1) / 0.06] = $18,393

🌍 Real-World Future Value Applications

1. Salary: $60,000
2. Employee contribution: 3.5% = $2,100/year ($175/month)
3. Employer match: 3.5% = $2,100/year ($175/month)
4. Government contribution: $260.72/year ($21.73/month)
5. Total monthly: $372
6. Expected return: 6% annually
7. Years until retirement (65): 40 years

Who this happens to, and what it meant for them

🌍 Real-World Future Value Applications

1. n = 40 × 12 = 480 months
2. r = 6% / 12 = 0.5%
3. FV = $372 x [((1.005)^480 - 1) / 0.005]

FV = $740,835

🌍 Real-World Future Value Applications

1. First 5 years: $372/month builds $25,954
2. That $25,954 grows for 35 more years to $210,842
3. Years 6-40: $744/month for 35 years adds $1,059,984, giving $1,270,827

🌍 Real-World Future Value Applications

1. Monthly contribution: $300
2. Years of contributing: 40 (age 25 to 65)
3. Return: 7% annually
4. Total contributed: $300 × 12 × 40 = $144,000
5. Future value at 65: $719,147

Who this happens to, and what it meant for them

🌍 Real-World Future Value Applications

1. Monthly contribution: $300 (same as Emma)
2. Years of contributing: 30 (age 35 to 65)
3. Return: 7% annually
4. Total contributed: $300 × 12 × 30 = $108,000
5. Future value at 65: $340,138

1. Current savings: $15,000
2. Target: $150,000
3. Gap to fill: $135,000
4. Timeframe: 7 years
5. Expected return: 3.5% (conservative savings account)

Who this happens to, and what it meant for them

1. FV = $15,000 × (1.035)^7 = $18,964

1. Still need: $150,000 - $18,964 = $131,036
2. n = 7 × 12 = 84 months
3. r = 3.5% / 12 = 0.2917%
4. PMT = $131,036 × [0.002917 / ((1.002917)^84 - 1)]

PMT = $1,456 per month

1. FV of $15,000: $22,542
2. Still need: $127,458
3. Required monthly: $1,344
4. Savings vs 3.5%: $112/month

1. University per child: $60,000
2. Two children: $120,000 total needed
3. Current ages: newborns
4. Time to save: 18 years

Who this happens to, and what it meant for them

1. Expected return: 5% annually
2. Calculate required monthly savings:
3. PMT = $120,000 × [0.004167 / ((1.004167)^216 - 1)]

PMT = $341 per month

1. Total contributed: $341 × 216 = $73,656
2. Future value: $120,000
3. Interest earned: $46,344
4. Compound interest paid for: 38.6% of education costs

The PVA Formula

1. PV = PMT × [(1 - (1 + r)^-n) / r]
2. Where:
3. PV = Present Value
4. PMT = Payment per period
5. r = Interest rate per period
6. n = Number of periods

Simple Example

1. PMT = $10,000/month
2. r = 6% / 12 = 0.5% = 0.005
3. n = 10 × 12 = 120 months
4. PV = $10,000 × [(1 - (1.005)^-120) / 0.005]
5. PV = $10,000 × [(1 - 0.5496) / 0.005]
6. PV = $10,000 × 90.07

PV = $900,700

Pension Valuation Example

1. Current pension: $3,500/month
2. Expected life span: 20 more years
3. Discount rate: 4% (conservative)
4. PV = $3,500 × [(1 - (1.00333)^-240) / 0.00333]

PV = $577,920

Ordinary Annuity vs Annuity Due

1. Ordinary Annuity: $291,530
2. Annuity Due: $291,530 × 1.005
3. Annuity Due: $292,988
4. Difference: $1,458 more

Example 1: Mortgage Calculation

1. Monthly payment: $2,500
2. Interest rate: 6.5% annual (0.5417% monthly)
3. Loan term: 30 years (360 months)

Example 1: Mortgage Calculation

1. PV = $2,500 × [(1 - (1.005417)^-360) / 0.005417]
2. PV = $2,500 × 158.21

PV = $395,525

Example 2: Inheritance Decision

1. Option A: $250,000 lump sum today
2. Option B: $2,000/month for 15 years
3. Your discount rate: 5% (what you could earn)

Example 2: Inheritance Decision

1. PV = $2,000 × [(1 - (1.004167)^-180) / 0.004167]
2. PV = $2,000 × 129.30

PV = $258,600

Example 3: Pension vs Lump Sum

1. PMT = $4,500/month
2. n = 22 years × 12 = 264 months
3. r = 4.5% / 12 = 0.375%
4. PV = $4,500 × [(1 - (1.00375)^-264) / 0.00375]

PV = $834,565

Example 3: Pension vs Lump Sum

1. PV of pension: $834,565
2. Lump sum offered: $850,000
3. Lump sum is $15,435 MORE

Example 4: Car Lease vs Buy

1. PV = $450 × [(1 - (1.004167)^-36) / 0.004167]
2. PV = $450 × 33.96

PV = $15,282

Example 4: Car Lease vs Buy

1. PV of lease payments: $15,282
2. Purchase price: $15,000
3. Lease costs $282 more in PV

Using PVA for Loan Affordability

1. Gross monthly income: $8,500
2. Maximum 30% for housing: $2,550/month
3. Available interest rate: 6.8%
4. Standard 30-year term
5. PV = $2,550 × [(1 - (1.005667)^-360) / 0.005667]

Maximum loan: $387,600

🌍 Real-World PVA Examples

1. PV = $200,000 × [(1 - (1.06)^-30) / 0.06]
2. PV = $200,000 × 13.765

PV of Option B = $2,753,000

Who this happens to, and what it meant for them

🌍 Real-World PVA Examples

1. Option A: $3,200,000
2. Option B: $2,753,000 (in PV)
3. Option A is $447,000 better!

🌍 Real-World PVA Examples

1. PV = $3,500 × [(1 - (1.004167)^-240) / 0.004167]

PV = $531,947

Who this happens to, and what it meant for them

1. Planned retirement: age 65
2. Pension at 65: $5,000/month for life
3. Expected lifespan: age 85 (30 years of pension)

Who this happens to, and what it meant for them

1. Retire now at 55
2. Immediate pension: $3,200/month for life
3. Plus one-time: $150,000 bonus

1. Start in 10 years, $5,000/month for 20 years
2. PV at 55 = $5,000 × 149.27 × 0.6756

= $503,920

1. $3,200/month for 30 years
2. PV = $3,200 × 209.46
3. = $670,272
4. Plus bonus: $150,000

Total = $820,272

What it would cost to buy

1. Income required: $30,000.00 a year
2. At a 4 percent withdrawal rate: $30,000.00 / 0.04 = $750,000.00

$750,000.00 of capital, to replicate one person's entitlement.

What it would cost to buy

1. For a couple, at an illustrative $46,000.00 a year
2. At the same rate: $46,000.00 / 0.04 = $1,150,000.00

Over a million dollars of capital, and the real thing is wage-linked and cannot run out.

The practical alternatives most people use

1. A term deposit ladder, which gives certainty of capital but not of income for life.
2. A diversified portfolio drawn down, which can run out and needs monitoring.
3. Keeping some work, which is the most effective and least discussed option.
4. Downsizing the house, which converts an asset most retirees hold into income.
5. Spending flexibly, reducing in poor market years, which is worth more than most product choices.

The last two do more for most households than any guaranteed product available here.

The single most useful thing to know

1. Work out your essential annual spending, the amount below which life becomes difficult.
2. Compare it against NZ Super at the current rate for your situation.
3. If Super covers it, your longevity risk on essentials is already handled.
4. If there is a gap, that gap is the only part that needs guaranteeing.
5. Everything above it can come from savings and can flex.

Most people find the gap is far smaller than they assumed, or absent.

The 4% Starting Point

1. $500,000 balance
2. 4% in the first year is about $20,000
3. Plus NZ Super for your situation
4. Adjust the dollar amount for inflation in later years

Sequence Risk

1. Two retirees with the same average return over 20 years
2. One has poor returns early, one has them late
3. The early-poor retiree can run short despite the same average
4. A cash buffer and flexible spending soften this risk

A Simple Drawdown Plan

1. 1. Count NZ Super as your base income
2. 2. Set a starting withdrawal, around 4%, on top
3. 3. Hold a cash buffer for one to two years of spending
4. 4. Keep some growth assets for the later years
5. 5. Review yearly and flex spending with how markets go

What an ORA Is

1. You pay an entry price for the right to occupy a unit
2. You pay ongoing weekly fees for services and facilities
3. You live there under the village rules and the ORA
4. When you leave, a deferred management fee is deducted from your refund