Spot rate
Also written Spot rate (zero rate) · Zero rate · Zero-coupon rate · Zero coupon yield
The true return on money invested today for one stated term with no interim cash flow — read straight off a zero-coupon instrument, and the only rate a cash flow should be discounted at.
In plain language
Yield to maturity applies one rate to every cash flow of a bond, whether it arrives in six months or in ten years. That is convenient and it is wrong, because money for six months and money for ten years are not priced the same.
A spot rate — the workbook also calls it the zero rate — is the rate for one specific term, with no interim cash flow to reinvest and therefore no assumption to make. A zero-coupon instrument has exactly this shape, which is why the spot rate can be read directly off its price.
The consequence is a reversal of how most people think bond prices are set. The bond price is not determined by demand and supply for the bond. It is determined by the term structure of zero rates — which is set by demand and supply for money — applied cash flow by cash flow.
How it works
For a coupon bond there is no single return. A three-year 8% bond has three cash flows, each earning its own zero rate: the first year's Rs 8 earns the 1-year rate, the second year's Rs 8 earns the 2-year rate, and the Rs 108 at maturity earns the 3-year rate. Three cash flows, three returns. A ten-year semi-annual bond has twenty.
Yield to maturity is then simply the single rate that averages those twenty into one number and reproduces the same price. That makes YTM "not a return measure but another way of quoting bond price", in the workbook's phrase — derived from the price, not the cause of it.
Where there are no interim cash flows at all, the spot rate is directly computable and is the genuine realised return. Where there are, the true realised return is the holding period return, and it can only be known ex post, once every reinvestment rate has actually happened. Coupon, current yield and YTM are the approximations invented to fill the gap.
A 5-year zero rate of 7.5% carries a specific meaning: 7.5% in year one, automatically reinvested at 7.5% for four more years; 7.5% in year two, reinvested for three more; and so on.
The formula
⎡ ⎛ F ⎞ 1/(N×C) ⎤
Spot rate = ⎢ ⎜ ─ ⎟ − 1 ⎥ × C
⎣ ⎝ P ⎠ ⎦
F = final amount received P = amount invested
N = years C = compounding frequency
Price of a coupon bond, discounted properly:
C₁ C₂ Cₙ + M
Price = ──────── + ────────── + ... + ────────────
(1 + Z₁) (1 + Z₂)² (1 + Zₙ)ⁿ
Zᵢ = the zero rate for year i
A worked example
The same money, four different answers. Rs 100 grows into Rs 150 over three years. The realised return depends on the compounding convention alone:
| Compounding | Spot rate |
|---|---|
| Annual | 14.4714% |
| Semi-annual | 13.9826% |
| Quarterly | 13.7464% |
| Monthly | 13.5919% |
One cash flow, one holding period, and a 88 basis point spread across conventions.
Pricing a G-Sec off the zero curve. A 3-year 8% annual coupon bond, with the term structure of zero rates at 7.75%, 8.00% and 8.25%:
| Year | Zero rate | Cash flow | Discounted value |
|---|---|---|---|
| 1 | 7.75% | Rs 8 | 8 ÷ 1.0775 = 7.4246 |
| 2 | 8.00% | Rs 8 | 8 ÷ 1.08² = 6.8587 |
| 3 | 8.25% | Rs 108 | 108 ÷ 1.0825³ = 85.1413 |
| Rs 99.4246 |
Solve for the single rate that reproduces Rs 99.4246 and you get a YTM of 8.2242%. That YTM is an output of the price, not an input to it.
Why this kills the cheapness argument. The same issuer, the same 3-year maturity, one bond with an 8% coupon and one with 9%, both priced off the identical zero curve:
| Bond A (8%) | Bond B (9%) | |
|---|---|---|
| Price | Rs 99.4246 | Rs 101.9983 |
| YTM | 8.2242% | 8.2215% |
Bond A looks cheap at a discount and Bond B dear at a premium. Neither is mispriced. Both are exactly right on the same zero curve; the prices differ only because the coupons do. On Rs 10 crore of face value a dealer who bought A and sold B expecting to capture Rs 2.57 per 100 would have captured nothing at all, minus costs.
Why NISM asks about it
Chapter 1 builds this across two sections. Section 1.9 (Spot Rate and Holding Period Return) derives the zero rate and explains why a true return is computable ex ante only for zero-coupon instruments; section 1.11.4 (Spot Rate Bond Price and YTM) prices a coupon bond off the zero curve and shows YTM falling out of it. Section 1.6 supplies the term structure the rates come from, and Chapter 3, section 3.7.1.1, uses spot rates to derive forward rates for futures pricing.
Questions ask you to compute a zero rate from a growth factor at a stated compounding frequency, to price a bond off a given set of zero rates, and — conceptually — why YTM is not a true return measure.
Common exam traps
- A coupon bond has no single true return. It has one per cash flow. YTM is their average and is derived from the price, not the other way round.
- Each cash flow takes its own zero rate. Discounting all of them at the 3-year rate is the YTM shortcut, not the spot-rate method.
- Price and YTM cannot identify mispricing. Two bonds on the same zero curve show different prices and near-identical YTMs; that is correct pricing, not an opportunity.
- Spot rate here means a zero-coupon interest rate, not the cash-market price of a bond. Chapter 3 uses "spot price" for the latter, and the two are unrelated.
- Holding period return is known only afterwards. Every reinvestment rate has to have happened; that is why the approximations exist at all.
- State the compounding frequency. The same growth from Rs 100 to Rs 150 is 14.47% annually compounded and 13.59% monthly — the number is meaningless without it.
Check yourself
1.An investor can put INR 1,000 into a 2-year bond at 6%, or into a 1-year bond at 5% and reinvest for a further year at the forward rate F. Under no-arbitrage, what is F?
- a)7.0095%
- b)5.5000%
- c)6.0000%
- d)11.0000%
Show the answer
Answer: (a) 7.0095%
The no-arbitrage principle: the same return must be received whether the money is invested over one long term or over multiple shorter terms with reinvestment.
- Option 1 after 2 years: 1000 × (1.06)² = INR 1,123.60
- Option 2 after 2 years: {1000 × (1.05)} × (1 + F) = 1050 × (1 + F)
Setting them equal: 1050 × (1 + F) = 1123.60 → F = 7.0095%
By formula, with S1 = 6%, S2 = 5%, n1 = 2, n2 = 1:
$$F = \left[\frac{(1.06)^2}{(1.05)^1}\right]^{1/(2-1)} - 1 = 7.0095%$$
Sense check: to average 6% over two years when the first year earns only 5%, the second year must earn more than 6% — and it does. Option (b), the simple average of the two spot rates, is the intuitive but wrong answer.
Where this is taught
Free preparation for NISM Series V-DRelated terms
- Current yieldA bond's annual coupon in rupees divided by its current market price — the cash income the bond throws off this year, ignoring any gain or loss at redemption.
- Yield to MaturityThe single discount rate at which a bond's future coupons and redemption amount add up to exactly its market price today — the return you actually earn if you hold it to maturity.
- Effective interest rateThe rate actually earned over a year once compounding within the year is counted — always at or above the quoted nominal rate, and equal to it only when interest is paid once a year.
- Forward rateThe interest rate for a period that starts in the future, implied today by two spot rates — because rolling a short investment must return the same as locking in a long one, or arbitrage follows.
- Risk-free rateThe rate on a sovereign borrowing in its own currency, where credit risk is absent because the government can print the money — the benchmark every other valuation is measured against.
- Term structure of interest ratesInterest rate plotted against term — one curve per credit quality, with the risk-free curve as the base and every other borrower quoted as a spread over it.