Time value of money

Rs 100 today and Rs 100 a year from now are not the same thing, and a retirement plan that treats them as the same is wrong before it starts.

Two separate forces pull in opposite directions. The rupee in your hand can be invested, so it grows. The rupee arriving next year buys less, because prices have risen in the meantime. The workbook puts both in one paragraph: at 8 percent, Rs 100 today is Rs 108 in a year; at 6 percent inflation, what costs Rs 100 today costs Rs 106 in a year.

Every number in a retirement plan — the expense figure, the corpus, the monthly saving — has to be stamped with the date it belongs to.

The workbook identifies three places where time value bites in a retirement calculation:

1. Expenses expected in retirement must be inflated. The same standard of living costs more later, so the corpus must be larger.
2. Savings will earn returns and compound. The final corpus is contributions plus returns, and the returns do most of the work — which is what reduces the monthly saving required.
3. The longer the runway, the larger both effects. Time amplifies compounding and inflation together.

The direction of travel tells you which operation to use. Moving a number forward in time is compounding; moving it backward is discounting. They are the same equation read in opposite directions.

Future value:   FV = PV × (1 + r)^n

Present value:  PV = FV ÷ (1 + r)^n

where r is the rate per period and n the number of periods.

In Excel, the workbook uses PV(Rate, Nper, Pmt, FV, Type) and FV(Rate, Nper, Pmt, Type), with the contribution entered as a negative number to signal an outflow, Type = 1 for payments at the start of a period and Type = 0 at the end. For a monthly contribution, divide the annual rate by 12 and multiply the years by 12.

The workbook runs three cases. All three are worth being able to reproduce.

1. Inflating an expense forward. Mr X needs Rs 20,000 a month today. At 5 percent inflation, five years later the same basket costs

20,000 × (1.05)^5 = Rs 25,526 a month

2. Discounting it back. To have Rs 25,526 in five years, invested in a 5-year bank FD at a post-tax 8 percent, he does not need to set aside Rs 25,526 today:

PV = 25,526 ÷ (1.08)^5 = Rs 17,372

Rs 8,154 of the eventual sum is produced by the investment, not by the saver.

3. The cost of a five-year delay. X invests Rs 5,000 a month at 12 percent from age 30 to 60:

Sixty extra instalments of Rs 5,000 — Rs 3 lakh of the saver's own money — are worth Rs 80.8 lakh at 60. That ratio, roughly 27 to 1, is the entire argument for starting early, and it is arithmetic rather than exhortation.

Chapter 1 (Fundamental Concepts in Retirement Planning), section 1.2.1, introduces this and Chapter 3 (Retirement Planning Process) applies it end to end in the deterministic corpus calculation of section 3.2.7. Expect direct computations — inflate an expense forward n years, discount a required sum back — and questions on which Excel inputs go where, including the sign convention and the Type flag. The conceptual question that recurs is why a delay in starting to save costs far more than the instalments missed.

• Inflate first, then discount. The workbook's two-step corpus method inflates today's expense to the retirement date, and only then discounts the retirement income stream back to a lump sum. Doing it in one step with a net rate skips the first number the paper asks for.
• Match the rate to the period. A monthly contribution needs the annual rate divided by 12 and the years multiplied by 12. Feeding an annual rate into a monthly Nper is the most common arithmetic error in this chapter.
• Type matters. Contributions at the beginning of the period (Type = 1) accumulate one extra period of return against Type = 0. On a 35-year series that is not a rounding difference.
• The negative sign is not decoration. Excel returns the PV in the example as minus Rs 17,372 because it is an outflow. Read the magnitude, not the sign, but know why it is there.
• Time value is not the same as the real rate of return. Time value is the mechanism; the real rate is what you get after netting inflation off the nominal return. Use the second inside the first, not instead of it.