Deriving a yield to maturity

Chapter 5 defined yield to maturity and gave a figure for the example bond — "roughly 6.7%". This chapter produces that number, and the production turns out to explain the assumption chapter 5 warned about.

The equation

YTM is the single rate $y$ that makes the present value of every promised payment equal the price:

$P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n}$

For chapter 5's bond — a three-year 9% bond on ₹1,000 face, trading at ₹1,060:

$1060 = \frac{90}{(1+y)} + \frac{90}{(1+y)^2} + \frac{1090}{(1+y)^3}$

This is a cubic in $(1+y)$, and for a longer bond it is a polynomial of degree $n$. There is no formula for $y$. Beyond degree four, no general algebraic solution exists for any polynomial — so the absence of a YTM formula is not an oversight in finance but a fact about algebra.

So it is solved numerically, which means guessing and improving.

Solving it

The function is monotonic: a higher yield always gives a lower price, as chapter 3 established. That is what makes guessing reliable — each guess tells you unambiguously which way to move.

Guess	Price it produces	Verdict
50.0000%	₹422.96	far too low → lower the yield
25.0000%	₹687.68	too low → lower the yield
12.5000%	₹916.65	too low → lower the yield
6.2500%	₹1,073.17	too high → raise the yield
9.3750%	₹990.57	too low → lower
7.8125%	₹1,030.71	too low → lower
7.0312%	₹1,051.64	too low → lower
6.6406%	₹1,062.33	too high → raise
6.8359%	₹1,056.96	too low → lower
6.7383%	₹1,059.64	too low → lower
…
6.7252%	₹1,060.00	solved

Each step halves the remaining range. Ten steps take an interval of 100 percentage points down to about a tenth of one, and twenty take it below a thousandth. This is bisection, and it is the method a spreadsheet's bond function is doing behind the answer.

So the YTM is 6.7252% — chapter 5's "roughly 6.7%", derived.

And the ordering chapter 5 noted is now visible as a consequence: coupon 9.00%, current yield 8.49%, YTM 6.73%. For a bond bought above face value, each measure that accounts for more of the truth is lower than the one before.

How precise does this need to be?

Worth knowing, because it tells you when to stop iterating.

Change in the yield guess	Change in the computed price
+0.10% (10 bp)	−₹2.74
+0.01% (1 bp)	−₹0.27

A basis point of yield is worth about 27 paise on this bond. So pinning the yield to two decimal places is pinning the price to the nearest quarter-rupee, which is finer than the market quotes it.

That sensitivity is duration in disguise. Chapter 6's modified duration is exactly the proportional price change per unit of yield, and here it is being read off the other direction — from a yield guess to a price. The two chapters are the same derivative used for opposite purposes.

Where the reinvestment assumption comes from

Chapter 5 stated the assumption: "coupons are reinvested at the same rate... the arithmetic assumes each ₹90 goes back to work at the YTM." Here is why the arithmetic assumes it, which is more interesting than the fact.

Discounting at a single rate $y$ is the same operation as compounding at $y$. A ₹90 coupon at year 1 is being valued as $90/(1+y)$ — which says that ₹90 received then is equivalent to a smaller sum now that would grow at $y$ into ₹90.

Run the equivalence forward instead of backward. If the bond's price grows at $y$ for three years:

$1060 \times (1.067252)^3 = ₹1{,}288.57$

And if the coupons are collected and reinvested at $y$ until maturity:

$90(1.067252)^2 + 90(1.067252) + 1090 = ₹1{,}288.57$

Identical, necessarily. Solving for the rate that discounts the cash flows to the price is the same as solving for the rate at which the price compounds to the reinvested cash flows. The assumption is not added to the definition; it is the definition, seen from the other end.

What you actually earn

So the YTM is only the realised return if the reinvestment rate happens to equal it.

Coupons reinvested at	Terminal wealth	Realised annual return
3%	₹1,278.18	6.4377%
5%	₹1,283.73	6.5914%
6.7252% (the YTM)	₹1,288.57	6.7252%
9%	₹1,295.03	6.9033%
12%	₹1,303.70	7.1413%

Over three years the spread is modest — about 0.7 percentage points across reinvestment rates from 3% to 12%. Chapter 5 said the same thing qualitatively: "for a short bond this barely matters."

And the reason it is modest here is that the coupons are a small part of the total. Of the ₹1,288.57 terminal wealth, ₹1,090 is the final payment that is not reinvested at all. Only ₹180 of coupons is exposed to the reinvestment rate, and two years and one year of compounding on ₹180 cannot move the total much.

Scale that up and the conclusion inverts. On a thirty-year bond most of the terminal wealth is reinvested coupons, and chapter 5's warning applies: "a large share of the YTM depends on an assumption about rates two decades out."

Which gives the practical rule. Treat the YTM of a short bond as close to a promise, and the YTM of a long bond as a quoted price containing an embedded forecast. The quantity that decides which you are looking at is how much of the terminal value comes from reinvested coupons — and a zero-coupon bond, having none, has a YTM that is exactly what you will earn if you hold it.

Where this leaves the single-rate idea

YTM discounts every cash flow at the same rate. But the RBI's published yields for the week ended 2 October 2026 show that money has different prices at different horizons:

Instrument	Yield
91-day Treasury bill	5.52%
182-day Treasury bill	5.96%
364-day Treasury bill	6.18%
10-year G-Sec par yield	7.23%

Money for a year costs 6.18% and money for ten years costs 7.23%. So discounting a three-year bond's year-one coupon and its year-three principal at the same rate is a simplification, and a known one.

It is not fatal, because the YTM is defined as the single rate that reproduces the observed price — it is a summary of the price rather than a claim about discounting. But it means two bonds with the same YTM and different coupon patterns are not necessarily equivalent, and that is what chapter 15 fixes.

Working the problem

Three-year 9% bond on ₹1,000 face, at ₹1,060. Reinvestment at 5%.

Part 1 — the YTM. The table above, converging on 6.7252%, with the check that

$\frac{90}{1.067252} + \frac{90}{1.067252^2} + \frac{1090}{1.067252^3} = ₹1{,}060.00$

Part 2 — what is actually earned at 5% reinvestment.

Compound each coupon forward to maturity. The year-1 coupon has two years to grow; the year-2 coupon has one; the year-3 payment arrives at maturity and grows not at all.

$90 \times 1.05^2 + 90 \times 1.05 + 1090 = 99.225 + 94.50 + 1090 = ₹1{,}283.73$

Convert to an annual rate over the three years.

$\left(\frac{1283.73}{1060}\right)^{1/3} - 1 = 6.5914\%$

So the investor earns 6.59%, not the quoted 6.73% — a shortfall of about 13 basis points.

Three readings of that result.

The shortfall is small, and that is the honest finding. A 1.7-point error in the reinvestment assumption cost 13 basis points of realised return. For a three-year bond the YTM is a good number, and someone who treats it as their return will be close to right.

The direction is predictable. Reinvestment below the YTM means a realised return below the YTM, always. And note which world that is: rates fell, which raised the bond's price while lowering what the coupons could earn. That is chapter 5's "quiet risk" — the same movement helps the holding and hurts the income, so the two partly offset.

And the fix, where it matters, is structural rather than analytical. If the reinvestment assumption is the problem, buy an instrument that has no coupons to reinvest. A zero-coupon bond's YTM is its realised return with no assumption at all, which is why they are the natural building block — and chapter 15 uses exactly that property to take the curve apart.

The point

Yield to maturity is the rate that equates a bond's discounted cash flows to its price, which is a polynomial with no closed-form solution, so it is found by iteration — bisection on a monotonic function, converging for chapter 5's bond on 6.7252% and confirming the ordering coupon 9.00% > current yield 8.49% > YTM 6.73% for a bond above par. The reinvestment assumption is not an extra condition but the definition read forward: ₹1,060 compounding at the YTM for three years equals the coupons reinvested at the YTM, identically. So the realised return is the YTM only if reinvestment happens at it — 5% reinvestment yields 6.59% rather than 6.73% here, a 13 basis point shortfall that would be far larger on a long bond, and zero on a zero-coupon bond. And because money costs 6.18% for a year and 7.23% for ten, discounting every cash flow at one rate is a known simplification.