Real rate of return

A deposit paying 8 percent while prices rise 6 percent has not made you 8 percent better off. Almost all of that return has gone to standing still.

The nominal rate is the number on the product brochure. The real rate is what is left once inflation has taken its share, and it is the only one that tells you whether your purchasing power has grown. The workbook is blunt about the consequence: the nominal rate is always positive, because nobody invests for a negative headline — but the real rate can be negative, and a negative real return means rising prices have wiped out everything the investment earned.

There are two ways to compute it and the workbook teaches both, calling them Method 1 and Method 2.

Method 2 (subtraction) simply nets inflation off the nominal rate. It is quick, it is what candidates reach for under time pressure, and it is an approximation.

Method 1 (discounting) is the accurate one. It recognises that the investment compounds at the nominal rate while the value of those cash flows is being discounted by inflation, so the two rates divide rather than subtract.

The gap between the two is small in percentage-point terms and large in rupee terms, because a retirement corpus discounts an income stream over twenty or more years at that rate. At 8 percent against 6 percent inflation, Method 1 gives 1.89 percent and Method 2 gives 2 percent — eleven basis points of difference that moves a three-crore corpus by lakhs.

Method 2 (approximate):   RR = NR − IR

Method 1 (accurate):      (1 + NR) = (1 + RR) × (1 + IR)

                          RR = ((1 + NR) ÷ (1 + IR)) − 1

where NR is the nominal rate and IR the inflation rate.

The workbook's own illustration: a bond paying 10 percent with 5 percent inflation gives 5 percent by subtraction and (1.10 ÷ 1.05) − 1 = 4.76 percent by discounting.

Take the workbook's own case. Rani needs Rs 35,000 a month in today's money, retires in 25 years at 60, expects to live to 80, will invest the corpus at 8 percent and assumes 6 percent inflation.

Step 1 — inflate the income to the retirement date.

35,000 × (1.06)^25 = Rs 1,50,215 a month

Step 2 — discount 20 years of that income at the real rate.

RR = (1.08 ÷ 1.06) − 1 = 1.89% a year  →  0.1575% a month
Nper = 20 × 12 = 240,  PMT = 1,50,215,  Type = 1

Corpus required = Rs 3,00,48,832

Now redo step 2 with the subtraction answer of 2 percent instead of 1.89 percent. A higher discount rate produces a smaller present value:

Eleven basis points of sloppiness leaves Rani roughly Rs 3.1 lakh short of the corpus her own plan says she needs. The approximation is fine for explaining the idea to a client and wrong for sizing the goal.

Note what the corpus calculation is really doing: because the income stream is discounted at the real rate, the corpus is automatically sized to fund an income that rises with inflation every year of retirement. Rani's monthly draw does not stay at Rs 1,50,215 — it grows, and the corpus is exhausted at 80.

Chapter 1, section 1.2.2 sets out both formulas explicitly, and Chapter 3 uses the real rate as the discount rate in section 3.2.4 and in the deterministic corpus calculation of section 3.2.7. This is among the most reliably examined pieces of arithmetic in the paper: expect to be handed a nominal rate and an inflation rate and asked for the real rate by the accurate method, and to be asked which of the two methods the workbook calls approximate.

• Subtraction is the approximation, division is the answer. The workbook labels Method 2 approximate in as many words. If a question offers both 5 percent and 4.76 percent as options for a 10 percent bond with 5 percent inflation, it is testing exactly this.
• The accurate real rate is always below the subtracted one when both rates are positive, because you are dividing by a number greater than one.
• A negative real return is not a computation error. When inflation exceeds the nominal rate the real rate goes negative, and the workbook says so directly for fixed-return products in high-inflation periods.
• Do not apply the real rate to an amount that has already been inflated. Inflate the expense to the retirement date or discount at the real rate — never mix an inflated income with a nominal discount rate, or the inflation gets counted twice.
• Convert before you divide. The monthly real rate is the annual real rate divided by 12 in the workbook's Excel method; it is not the real rate recomputed from monthly nominal and monthly inflation figures.
• Real return is before tax. Tax comes off the nominal return first; a 7 percent deposit taxed at slab rate may have a negative real return long before inflation is quoted at anything alarming.