Actuarial Mathematics for Modelling · Financial instruments and insurance contracts as cashflow models

Fixed Interest Securities and Bond Cashflows

Q: What is the difference between a zero-coupon bond and a coupon bond?

A coupon bond pays regular interest payments and a redemption amount at maturity. A zero-coupon bond pays only one amount at maturity, so all of the return comes from buying it below that amount. Its price is C·vⁿ and it needs no annuity factor.

Q: How do I model index-linked bond cashflows?

Take the real (base-date) payments and multiply each by the index ratio at the payment date. Then either discount the money cashflows at the money yield, or discount the real cashflows at the real yield. Check whether the question gives a time lag for the index.

Q: When does a bond sell at a premium or a discount?

If the coupon rate is higher than the yield required (for a bond redeemed at par), the price is above par, which is a premium. If the coupon rate is lower than the yield, the price is below par, which is a discount. Equal rates give a price equal to par.

Q: Is Makeham's formula needed, or can I always use the annuity method?

The annuity method always works. Makeham's formula is a time saver for level-coupon bonds when you can find K = C·vⁿ easily. It is also useful for checking whether your price is above or below the redemption value.

Updated 11 October 2026 · Fact-checked

A fixed interest security is a loan that pays known coupons and a redemption amount on set dates. To value one, list every cashflow by time and discount each at the yield: Price = coupons × annuity factor + redemption × vⁿ. Zero-coupon bonds have only the redemption. Index-linked bonds scale payments by an index.

Understand Fixed Interest Securities and Bonds

A bond is a loan you make to a government or company. In return, the issuer promises to pay you cash on fixed dates. Your job in CM1 is to turn that promise into a list of cashflows and value it.

A normal coupon bond has three features. The nominal (face) value is the amount the coupons are based on. The coupon rate is the annual coupon as a percentage of nominal, often paid half-yearly. The redemption value is paid at maturity. It is often at par (equal to nominal), but it can be above or below par. A zero-coupon bond pays no coupons. You pay less now and receive one payment at maturity. Its price is just a discounted single payment.

A government bond is usually treated as default-free. A corporate bond carries default risk, so investors demand a higher yield. In basic questions, you assume all promised payments are made unless told otherwise. If default is modelled, the cashflows become expected values, not promises.

An index-linked bond protects against inflation. Coupons and redemption are multiplied by the ratio of an inflation index at the payment date to the index at the base date. Real cashflows stay constant, while money cashflows grow with the index. Many real bonds use a time lag in the index, such as eight months. Use a lag only if the question states it.

The price is the present value of the cashflows at the investor's required yield. The yield to maturity is the rate that makes this present value equal to the price. If the coupon rate is higher than the yield, the bond sells above par (at a premium). If it is lower, the bond sells below par (at a discount).

Key rules to remember

Price of a coupon bond: P = F·r·aₙ + C·vⁿ; F = nominal, r = coupon rate per period, C = redemption amount, n = number of coupon periods, v = 1/(1+i) at the yield per period. The annuity and the power of v must use the same period.
Price when redeemed at par: P = F(r·aₙ + vⁿ); Use when C = F. This is the same formula with C replaced by F.
Makeham's formula: P = K + (g ÷ i)(C − K), where K = C·vⁿ and g = F·r ÷ C; g is the modified coupon rate: annual coupon per unit of redemption. The rate i and the n in K must be per coupon period. It is quick when you already know K.
Zero-coupon bond price: P = C·(1 + i)⁻ⁿ; A single cashflow C at time n. i is the effective annual yield when n is in years.
Index-linked payment: Payment at t = Nominal payment × I(t − lag) ÷ I(base); Without a lag, use I(t) ÷ I(0). Check the question for the base date and lag.
Money and real yields: 1 + i = (1 + r)(1 + e); Exact when inflation e is constant. i = money yield, r = real yield. An index-linked bond can be valued at the real yield using the real (unindexed) cashflows.
Premium or discount test: g > i: premium; g < i: discount; g = i: price = redemption; Use as a reasonableness check on your answer. It holds for a bond redeemed at C with modified coupon rate g.

How to solve Fixed Interest Securities and Bonds questions

Use this routine for any bond pricing or cashflow question. Most lost marks come from wrong timing or mismatched rates.

1Write down the nominal, coupon rate, coupon frequency, redemption value and term. Note the settlement date if it is not a coupon date.
2Convert everything to the coupon period. A half-yearly bond has 2n periods, a coupon of F·r ÷ 2 and a half-yearly yield.
3List the cashflows on a timeline. Show each coupon and the redemption separately. For an index-linked bond, scale each one by the index ratio.
4Choose the discount rate. Convert a nominal yield (for example i⁽²⁾) to the per-period yield. If asked for an effective annual yield, convert properly.
5Calculate the price as the present value: coupons × aₙ plus redemption × vⁿ. Or use Makeham's formula if K is easy.
6Check against the rule: coupon rate above yield means price above redemption. Fix any result that breaks it.
7If solving for yield, set up the equation of value, then solve by trial and interpolation. State the answer in the form asked (per period, nominal or effective).
8State the answer with units and the basis, such as 'per ₹100 nominal' or the total for the holding.

Quickest way: Makeham shortcut and real-yield shortcut

When to use it: Use Makeham when you need price for a bond with level coupons and the redemption term vⁿ is easy to compute. Use the real-yield shortcut for an index-linked bond with constant inflation, as it avoids growing every payment.

Compute K = C·vⁿ at the yield per period.
Compute g = F·r ÷ C and compare it with i. Predict whether the price is above or below C.
Apply P = K + (g ÷ i)(C − K) with i per period.
For an index-linked bond with constant inflation, discard the index. Value the real cashflows at the real yield. The result is the price per unit of nominal at the base date (with no lag).
Do a sense check: premium or discount direction matches g versus i.

Common mistakes in Fixed Interest Securities and Bonds

Using the annual coupon with a half-yearly discount rate
The coupon rate is quoted per year, and students forget to split it when the bond pays twice a year.
Fix: Convert coupon, term and yield to the same period before you start. Write 'per half-year' next to each figure.
Applying the coupon rate to the redemption value instead of the nominal value
Students see C and F as the same, which is true only for redemption at par.
Fix: The coupon is F × r. In Makeham's formula use g = F·r ÷ C, not r.
Forgetting the redemption payment on the final date, or missing the final coupon
The last date carries two payments and students value only one of them.
Fix: On the timeline, write the last coupon and the redemption separately at time n and add both.
Treating a nominal yield as an effective rate
A yield 'of 10% p.a. convertible half-yearly' is read as 10% effective per year.
Fix: Divide a nominal rate i⁽ᵐ⁾ by m to get the per-period rate. Convert to effective annual only if asked.
Discounting index-linked cashflows at the real yield while still using indexed cashflows (or the reverse)
Money and real bases are mixed in one calculation.
Fix: Either money cashflows at the money yield, or real cashflows at the real yield. Never one of each.
Thinking a zero-coupon bond has no yield
With no coupons, students think there is nothing to solve.
Fix: Solve C·(1 + i)⁻ⁿ = P. The yield is (C ÷ P)^(1/n) − 1.

Worked examples

Example 1

A bond has nominal ₹100, pays coupons of 8% p.a. half-yearly, and is redeemed at par in exactly 3 years. Find the price per ₹100 nominal for an investor who wants a yield of 10% p.a. convertible half-yearly. Then give the price of ₹10,00,000 nominal.

Show the solution

Per half-year: coupon = 100 × 0.08 ÷ 2 = ₹4. Number of periods n = 6. Yield per half-year = 10% ÷ 2 = 5%.
v⁶ = 1.05⁻⁶ = 1 ÷ 1.340096 = 0.746215.
a₆ at 5% = (1 − 0.746215) ÷ 0.05 = 5.075692.
Price = 4 × 5.075692 + 100 × 0.746215 = 20.3028 + 74.6215 = 94.9243.
Check: coupon rate 8% is below the yield 10%, so the price is below par. 94.92 < 100, which agrees.
For ₹10,00,000 nominal, multiply by 10,000: 94.9243 × 10,000 = 9,49,243 (to the nearest rupee).

Answer: ₹94.92 per ₹100 nominal (₹94.9243 before rounding); about ₹9,49,243 for ₹10,00,000 nominal.

Example 2

An index-linked bond of nominal ₹100 pays an annual coupon of 3% of nominal and is redeemed in 2 years at ₹100 indexed. Both coupons and redemption are indexed to an index with base value 100 at issue, with no lag. Assume inflation is 5% p.a. and the investor wants a real yield of 2% p.a. effective. (a) Write the money cashflows. (b) Find the price at issue.

Show the solution

Index at time 1 = 100 × 1.05 = 105. Index at time 2 = 100 × 1.05² = 110.25. Index ratios are 1.05 and 1.1025.
Time 1 money coupon = 3 × 1.05 = ₹3.15.
Time 2 money coupon = 3 × 1.1025 = ₹3.3075. Time 2 redemption = 100 × 1.1025 = ₹110.25. Total at time 2 = ₹113.5575.
Money yield: 1 + i = 1.02 × 1.05 = 1.071, so i = 7.1%.
Method 1, real cashflows at the real yield: v = 1/1.02 = 0.980392, v² = 0.961169. Price = 3 × (0.980392 + 0.961169) + 100 × 0.961169 = 3 × 1.941561 + 96.1169 = 5.8247 + 96.1169 = 101.9416.
Method 2, money cashflows at the money yield: 3.15 ÷ 1.071 = 2.9412, and 113.5575 ÷ 1.071² = 113.5575 ÷ 1.147041 = 99.0003. Sum = 101.9415, which agrees to rounding.

Answer: (a) ₹3.15 at time 1 and ₹113.5575 at time 2 (coupon ₹3.3075 plus redemption ₹110.25). (b) Price ≈ ₹101.94 per ₹100 nominal.

Exam tips

Draw a timeline first. In written questions, a clear cashflow list earns method marks even if your arithmetic slips.
State your assumptions: coupon dates, redemption at par unless told otherwise, no default, no tax, and whether an index lag is ignored.
Always say which rate you are using (per half-year, nominal or effective). Examiners look for the conversion.
In MCQs, use the premium/discount check to remove options. If the coupon rate exceeds the yield, any price below redemption is wrong.
In Paper B, build the cashflows in a column in Excel or a vector in R and discount with a single formula, so you can change the yield and re-run quickly.

Practice questions from Financial instruments and insurance contracts as cashflow models

Fixed Interest Securities and Bonds in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Fixed Interest Securities and Bonds: frequently asked questions

What is the difference between a zero-coupon bond and a coupon bond?

A coupon bond pays regular interest payments and a redemption amount at maturity. A zero-coupon bond pays only one amount at maturity, so all of the return comes from buying it below that amount. Its price is C·vⁿ and it needs no annuity factor.

How do I model index-linked bond cashflows?

Take the real (base-date) payments and multiply each by the index ratio at the payment date. Then either discount the money cashflows at the money yield, or discount the real cashflows at the real yield. Check whether the question gives a time lag for the index.

When does a bond sell at a premium or a discount?

If the coupon rate is higher than the yield required (for a bond redeemed at par), the price is above par, which is a premium. If the coupon rate is lower than the yield, the price is below par, which is a discount. Equal rates give a price equal to par.

Is Makeham's formula needed, or can I always use the annuity method?

The annuity method always works. Makeham's formula is a time saver for level-coupon bonds when you can find K = C·vⁿ easily. It is also useful for checking whether your price is above or below the redemption value.