Case 026Fixed income, credit and LDIWarm up

A client can buy a 10-year government bond at 7.1% or an inflation-indexed bond at a 2.4% real yield. What inflation rate makes them equal, and which does better on Rs 50 lakh if inflation averages 4% or 6%?

The situation

Pratibha Menon, 48, has Rs 50 lakh to set aside for ten years towards her parents' care. Her adviser shows her two government bonds, both held to maturity. A conventional ten-year bond yields 7.1% a year. An inflation-indexed bond yields 2.4% a year above inflation: its principal is uplifted each year by consumer price inflation, so what it pays rises with prices.

For the case, assume coupons are reinvested at each bond's own yield, ignore tax, and treat both as free of default risk. Pratibha asks which one is the safer choice.

Your task

Find the inflation rate at which the two bonds pay the same, then show what Rs 50 lakh is worth in today's rupees after ten years if inflation averages 4% and if it averages 6%.

Quick check

Roughly what average inflation makes the two bonds pay the same?

Worked solution

Try it on paper, then open one step at a time.

30-second answerThe answer to give first

The two bonds pay the same if inflation averages about 4.59%. At 4% inflation the nominal bond wins: Rs 50 lakh grows to Rs 67.1 lakh in today's rupees against Rs 63.4 lakh. At 6% the indexed bond wins by Rs 7.9 lakh. Choosing between them is a bet on inflation against 4.6%; the indexed bond is the one that removes the bet.

Step 1What is the break-even rate actually measuring?

Think of two rent agreements. One fixes the rent at a set rupee amount for ten years; the other starts lower but rises with prices every year. Which is better depends only on how fast prices rise, and there is one inflation rate at which they cost the same. The nominal bond fixes rupees; the indexed bond fixes purchasing power, and the break-even inflationThe inflation rate at which a conventional bond and an inflation-indexed bond of the same maturity give the same return. Roughly the nominal yield minus the real yield. is the rate at which the two agree.

The relationship

1 + \pi^{*} = \frac{1 + y_{\text{nominal}}}{1 + y_{\text{real}}} = \frac{1.071}{1.024} = 1.0459 \qquad \pi^{*} \approx 4.59\%

\pi^{*}	break-even inflation
y_{nominal}	7.1%, the conventional bond's yield
y_{real}	2.4%, the indexed bond's yield above inflation

What it says in wordsInflation of about 4.59% a year makes the fixed 7.1% and the 2.4% plus inflation exactly equal; the quick answer 7.1 minus 2.4 gives 4.7%.

Step 2What does Rs 50 lakh become at 4% and at 6% inflation?

The nominal bond turns Rs 50 lakh into Rs 99.3 lakh whatever happens to prices. The indexed bond pays 2.4% a year after inflation, so in today's rupees it is worth Rs 63.4 lakh in every scenario. At 4% inflation the nominal bond's real return is 2.98% a year and it ends at Rs 67.1 lakh in today's rupees; at 6% its real return falls to 1.04% and it ends at Rs 55.4 lakh. In rupees on the statement, the indexed bond would show Rs 93.8 lakh at 4% inflation and Rs 113.5 lakh at 6%.

Average inflation	Nominal bond, real return	Nominal bond, today's Rs lakh	Indexed bond, today's Rs lakh	Better choice
4%	2.98%	67.1	63.4	Nominal, by 3.7
4.59% (break-even)	2.40%	63.4	63.4	Equal
6%	1.04%	55.4	63.4	Indexed, by 7.9

In today's rupees the indexed bond is always worth Rs 63.4 lakh after ten years, while the nominal bond is worth Rs 67.1 lakh at 4% inflation and Rs 55.4 lakh at 6%.

The indexed bond's real value is flat at Rs 63.4 lakh whatever inflation does, while the nominal bond's real value falls as inflation rises, crossing it at the 4.59% break-even, so the choice between them is a view on inflation.

Step 3So which is the safer choice for Pratibha?

Safety depends on what she is protecting. Her goal is care costs, which rise with prices, so her risk is measured in purchasing power, not rupees. The indexed bond is the one that removes the inflation bet; the nominal bond is a bet that inflation averages below 4.6%. Notice the asymmetry in the table: being wrong by two points on the high side costs her more than being right by half a point on the low side earns. Break-even inflation also carries an inflation risk premiumExtra yield that holders of conventional bonds demand for bearing the risk that inflation turns out higher than expected., so a break-even of 4.6% can sit above what the market actually expects; the nominal buyer is paid a little for carrying that risk.

Say the practical limits too. Indexed bonds trade thinly, so selling before maturity can cost a wide spread; the uplift follows a published price index with a lag, which may not match her parents' actual costs, where medical inflation often runs faster; and the uplift to principal can be taxed each year even though it is paid only at maturity. Tax treatment is a framework to confirm with current rules before comparing post-tax returns.

Where candidates lose it

The common slip is comparing 7.1% with 2.4% and calling the nominal bond better because the number is bigger. The two yields are in different units: one is rupees, the other is rupees after inflation.

The second is answering which bond yields more instead of which risk the client should hold. A client whose spending rises with prices is short inflation already, and the break-even tells you the price of removing that exposure, not which bond wins.

What the interviewer asks next

If the market's break-even were 6% and you expected 4.5% inflation, which bond would you lean towards and why?
How would a two-point rise in real yields next year affect the price of each bond before maturity?
Pratibha's parents' costs are mostly hospital care. What does that do to the case for the indexed bond?

Company names and figures are illustrative.