Financial mathematics: annuities, loans and bond valuation

Actuarial Science (B.Sc) · Financial mathematics · free study notes for exams, placements and interviews

In short

Annuities are regular, level or systematically varying cash flows; loans and bonds are the two instruments whose entire valuation reduces to pricing an annuity plus a lump sum. This topic supplies the mechanics — accumulated and present values, the EMI and amortisation schedule, bond price given yield and yield given price, and the duration measures that quantify interest-rate risk. It is the core of the Institute of Actuaries of India's CM1 (formerly CT1) and the IFoA CM1 syllabus, and of B.Sc Actuarial Science, B.Com and MBA finance papers; CFA and FRM Level I test exactly the same formulae. Indian employers test it hard: life insurers (LIC, HDFC Life, SBI Life, ICICI Prudential), consultancies (Milliman, WTW, Aon, Mercer), banks and NBFCs for retail-loan pricing, and PSU bond desks and mutual funds for fixed-income valuation.

Explained

Annuity-certain: the four basic values

An annuity-certain pays 1 per period for n periods, with certainty (no mortality). Effective rate per period i; discount factor v = 1/(1+i).
Annuity-immediate (in arrears, payments at times 1, 2, …, n): PV written aₙ, accumulated value at time n written sₙ.
Annuity-due (in advance, payments at times 0, 1, …, n−1): PV written äₙ, AV at time n written s̈ₙ. Each payment is one period earlier, so äₙ = (1+i)·aₙ and s̈ₙ = (1+i)·sₙ.
Derivation outline (examined): aₙ = v + v² + … + vⁿ is a GP with first term v and ratio v; sum = v(1−vⁿ)/(1−v) = (1−vⁿ)/i since (1−v)/v = i. Multiply by (1+i)ⁿ for sₙ = ((1+i)ⁿ−1)/i.
Useful identities: 1 = i·aₙ + vⁿ (invest 1, draw interest i each year, get capital back) → aₙ = (1−vⁿ)/i. Also 1/aₙ = 1/sₙ + i (the loan-repayment = sinking-fund + interest identity), and sₙ = (1+i)ⁿ·aₙ.
Perpetuity: payments forever. a_∞ = 1/i, ä_∞ = 1/i + 1 = (1+i)/i. Growing perpetuity at rate g (g < i): 1/(i−g).
Deferred annuity: m|aₙ = vᵐ·aₙ = a_{m+n} − a_m — the annuity starts after m periods of deferment.

Payments more or less often than interest is credited

p-thly annuity: total 1 per year paid as 1/p at each p-thly interval. PV = aₙ^p = (1−vⁿ)/i^p, where i^p is the nominal annual rate convertible p-thly: i^p = p[(1+i)^1/p − 1]. Due version: äₙ^p = (1−vⁿ)/d^p.
Continuous annuity: āₙ = (1−vⁿ)/δ, δ = ln(1+i) the force of interest. For small i, aₙ < aₙ^p < āₙ < äₙ.
Annuity less frequent than interest (e.g. yearly payment, monthly compounding): first convert to an effective rate per *payment* period — i_eff = (1+i_m)^k − 1 — then use the standard formula. This conversion step is where most exam marks are lost.
Practical Indian point: banks quote nominal annual rates compounded monthly for EMIs, but RBI's Fair Practices requirements make lenders disclose an annual percentage rate (APR) including fees in the Key Facts Statement for retail and MSME loans. APR with fees exceeds the nominal rate.

Varying annuities

Arithmetically increasing (payments 1, 2, …, n): (Ia)ₙ = (äₙ − n·vⁿ)/i. Decreasing (n, n−1, …, 1): (Da)ₙ = (n − aₙ)/i. Note (Ia)ₙ + (Da)ₙ = (n+1)·aₙ.
Geometrically increasing at rate g: PV = (1/(1+i))·[1 − ((1+g)/(1+i))ⁿ]/[1 − (1+g)/(1+i)]. Equivalently evaluate a level annuity at the net rate j = (1+i)/(1+g) − 1 and divide by (1+g): PV = aₙ(j)/(1+g).
Inflation-linked cash flows: discount money (nominal) flows at the money rate, or real flows at the real rate j where (1+i) = (1+j)(1+e), e = inflation. Never mix a real flow with a money rate.
Increasing p-thly and continuously increasing forms (Iā)ₙ, (Īā)ₙ appear in CM1 for salary-linked pension cash flows.

Loans: EMI, amortisation and outstanding balance

A level-repayment loan of L over n periods at rate i per period satisfies L = X·aₙ, so the instalment X = L/aₙ = L·i/(1−(1+i)⁻ⁿ). For monthly EMIs, i = annual nominal/12 and n = months.
Amortisation: each instalment = interest on the opening balance + capital repaid. Interest content falls and capital content rises geometrically with ratio (1+i).
Two ways to find the outstanding balance after t payments (both must give the same answer):
1. Retrospective: B_t = L(1+i)^t − X·s_t.
2. Prospective: B_t = X·a_{n−t} — usually faster, and the one to use in an interview.
Split of the (t+1)-th instalment: interest = i·B_t; capital = X − i·B_t = X·v^{n−t}. Total interest over the life = nX − L.
Flat rate vs reducing balance: a "flat" rate charges interest on the original principal throughout. Approximate effective reducing-balance rate ≈ 2 × flat rate (closer to 1.8–1.9× in practice). Flat-rate quotes persist in some vehicle and consumer-durable finance; always convert before comparing.
Sinking fund method: borrower pays interest i·L each period to the lender and accumulates the capital separately at rate j; instalment = L·i + L/sₙ(j). Equals the amortised instalment only when j = i.
Prepayment and rate reset: on a part-prepayment, reduce B_t then re-solve for either a shorter n or a smaller X. Floating-rate home loans in India are externally benchmarked (repo-linked under RBI's EBLR framework), so a reset changes i and the bank usually holds X fixed and lengthens n.
Credit-linked practice: RBI's August 2023 framework on floating-rate personal loans requires lenders, at each reset, to offer the borrower the option to switch to a fixed rate, to disclose the revised tenor and/or EMI, and to permit prepayment or foreclosure.
APR/effective comparison: processing fee F reduces net proceeds, so solve L − F = X·aₙ for the yield — that yield, not the quoted rate, is the borrower's cost.

Bond valuation

A bond pays coupon D = F·r per year (F = face/nominal value, r = coupon rate), usually in two half-yearly instalments, plus redemption R at maturity. Government of India dated securities (G-secs) pay semi-annual coupons on a face value of ₹100; corporate NCDs may pay annually.
Price (no tax, redeemable at time n): P = D·aₙ² + R·vⁿ at the investor's required yield i. With semi-annual coupons it is standard and simpler to work in half-years: P = (D/2)·a_{2n}(i₂) + R·v^{2n}, i₂ = half-yearly yield.
Price–yield relationships: price falls as yield rises (inverse, convex). If yield = coupon rate → price = par; yield < coupon → premium; yield > coupon → discount. A premium bond's price amortises down to redemption value, a discount bond's accretes up (the "pull to par").
Yield measures:
- Current (running) yield = annual coupon / market price — ignores capital gain/loss and timing.
- Yield to maturity (redemption yield) = the IRR of price against all future cash flows; solved by interpolation or Excel RATE/YIELD. Assumes all coupons are reinvested at the YTM — its main limitation.
- Yield to call for callable bonds; running yield + annualised capital gain is a crude approximation used on dealing desks.
Clean vs dirty price: market convention quotes the clean price; the buyer pays the dirty (settlement) price = clean + accrued interest. Accrued interest = coupon × days since last coupon / days in coupon period. Indian G-secs accrue coupon on a 30/360 day count (FIMMDA/CCIL convention); money-market instruments like T-bills and CDs use actual/365.
Taxation: if income tax at rate t₁ applies to coupons and CGT at t₂ to the gain, P = D(1−t₁)aₙ + R·vⁿ − t₂(R−P)vⁿ, solved for P (note P appears on both sides). Under the Income-tax Act, 2025 (in force 1 April 2026) section numbers were renumbered — name the provision rather than quoting an old number. Zero-coupon bonds and STRIPS (available on G-secs) have no coupon: P = R·vⁿ, and the return is taxed as a gain.
Real-redemption / index-linked bonds (e.g. inflation-indexed securities) require indexing coupons and redemption by the price index before discounting at the money rate.

Interest-rate risk: duration, convexity, immunisation

Macaulay duration (discounted mean term) = Σ t·CFₜ·vᵗ / Σ CFₜ·vᵗ — a PV-weighted average time to payment, in years.
Modified duration (volatility) = Macaulay/(1+i) = −(1/P)·dP/di. Effective duration for small yield moves: ΔP/P ≈ −ModD·Δi.
Price value of a basis point (PVBP) = ModD × P × 0.0001 — the standard desk measure in rupees.
Convexity C = (1/P)·d²P/di² corrects the linear estimate: ΔP/P ≈ −ModD·Δi + ½·C·(Δi)². Because of convexity, duration overstates the loss when yields rise and understates the gain when they fall.
Duration rises with maturity, falls with coupon rate and falls with yield; a zero-coupon bond's Macaulay duration equals its maturity.
Redington immunisation (against small rate changes) needs three conditions: (1) PV of assets = PV of liabilities; (2) duration of assets = duration of liabilities; (3) convexity (spread) of assets > that of liabilities. Full immunisation uses one asset cash flow either side of each liability and works for any size of parallel shift.
Limitations: assumes a flat yield curve and parallel shifts; needs periodic rebalancing; ignores credit and reinvestment risk beyond the horizon.

Project-appraisal links examined with annuities

NPV = Σ CFₜ·vᵗ − initial outlay; accept if NPV > 0 at the required return. IRR sets NPV = 0 — may be multiple or non-existent for sign-changing flows.
Discounted payback period, profitability index (PV of inflows/outlay) and equivalent annual cost = NPV/aₙ, used to compare assets of unequal lives.
Money-weighted (IRR) vs time-weighted vs linked internal rate of return for fund performance — MWRR is distorted by cash-flow timing, which is why SEBI-regulated mutual funds report XIRR/CAGR-style measures for investors and time-weighted returns for manager comparison.

Key points

Annuity-immediate (aₙ) — PV of 1 at the end of each of n periods = (1−vⁿ)/i.
Annuity-due (äₙ) — payments at the start; äₙ = (1+i)aₙ = aₙ₋₁ + 1.
Accumulated value (sₙ) — ((1+i)ⁿ−1)/i; sₙ = (1+i)ⁿ·aₙ.
Perpetuity — a_∞ = 1/i; growing perpetuity = 1/(i−g) requires g < i.
Deferred annuity — m|aₙ = vᵐaₙ = a_{m+n} − a_m.
EMI — X = L·i/(1−(1+i)⁻ⁿ) with i monthly and n in months.
Prospective balance — B_t = X·a_{n−t}; fastest route to outstanding principal.
Instalment split — interest = i·B_t, capital = X·v^{n−t}; capital content grows by factor (1+i) each period.
Total interest — nX − L, independent of how the schedule is tabulated.
Flat vs reducing — flat rate ≈ half the equivalent reducing-balance rate; never compare them directly.
Sinking fund identity — 1/aₙ = 1/sₙ + i.
Bond price — PV of coupons (an annuity) + PV of redemption (a lump sum), at the required yield.
Par/premium/discount — determined by yield vs coupon rate; premium amortises, discount accretes to par.
Clean vs dirty — quoted price excludes accrued interest; settlement price includes it; G-secs accrue on 30/360.
YTM — IRR of the bond; assumes reinvestment of coupons at the YTM, which is its weakness.
Macaulay vs modified duration — mean term in years vs % price change per unit yield; ModD = Mac/(1+i).
Convexity — second-order correction; duration alone overstates losses and understates gains.
Redington immunisation — equal PVs, equal durations, greater asset convexity; valid for small parallel shifts only.

Formulas and examples

aₙ = (1 − vⁿ)/i, v = 1/(1+i) — PV of 1 per period in arrears; i = effective rate per period (decimal), n = number of periods
sₙ = ((1+i)ⁿ − 1)/i — accumulated value at time n of 1 per period in arrears
äₙ = (1+i)·aₙ ; s̈ₙ = (1+i)·sₙ — annuity-due (payments in advance)
a_∞ = 1/i ; growing perpetuity PV = C/(i − g), g < i
aₙ^p = (1 − vⁿ)/i^p, i^p = p[(1+i)^1/p − 1] ; āₙ = (1 − vⁿ)/δ, δ = ln(1+i)
(Ia)ₙ = (äₙ − n·vⁿ)/i ; (Da)ₙ = (n − aₙ)/i
EMI: X = L·i / (1 − (1+i)⁻ⁿ) — L = principal (₹), i = rate per month, n = months
B_t = X·a_{n−t} (prospective) = L(1+i)^t − X·s_t (retrospective) — outstanding balance (₹)
Interest in payment t+1 = i·B_t ; capital repaid = X·v^{n−t} ; total interest = nX − L
Bond price P = (D/2)·a_{2n}(i₂) + R·v^{2n} — D = annual coupon (₹), R = redemption (₹), i₂ = half-yearly yield
Current yield = annual coupon / market price ; Accrued interest = coupon × (days elapsed / days in period)
Macaulay duration = Σ t·CFₜ·vᵗ / Σ CFₜ·vᵗ ; Modified duration = Macaulay/(1+i)
ΔP/P ≈ −ModD·Δi + ½·C·(Δi)² ; PVBP = ModD × P × 0.0001 (₹ per basis point)
Real rate: (1 + i_money) = (1 + i_real)(1 + e), e = inflation rate

Example 1 — SIP accumulation and the due-versus-arrears gap. ₹10,000 invested at the start of every month for 10 years, 12% nominal convertible monthly (i = 0.01, n = 120).

s₁₂₀ = (1.01¹²⁰ − 1)/0.01. 1.01¹²⁰ = 3.300387 → s₁₂₀ = 230.0387.
Payments in advance → s̈₁₂₀ = 1.01 × 230.0387 = 232.3391.
Fund = 10,000 × 232.3391 = ₹23,23,391.
If instead paid at month-ends: 10,000 × 230.0387 = ₹23,00,387 — ₹23,004 less, exactly one month's interest on the fund.

Example 2 — Home-loan EMI, balance and instalment split. ₹50,00,000 for 20 years at 9% nominal convertible monthly → i = 0.0075, n = 240.

1.0075²⁴⁰ = 6.009152 → (1+i)⁻ⁿ = 0.166413.
a₂₄₀ = (1 − 0.166413)/0.0075 = 111.1449.
EMI = 50,00,000/111.1449 = ₹44,986 (₹44,986.28).
Total paid = 240 × 44,986.28 = ₹1,07,96,707 → total interest ₹57,96,707, i.e. 116% of the principal.
First month: interest = 0.0075 × 50,00,000 = ₹37,500; capital = 44,986 − 37,500 = ₹7,486 (16.6% of the EMI).
Balance after 5 years (t = 60, n−t = 180): 1.0075¹⁸⁰ = 3.838043 → v¹⁸⁰ = 0.260549; a₁₈₀ = (1 − 0.260549)/0.0075 = 98.5935 → B₆₀ = 44,986.28 × 98.5935 = ₹44,35,350. Only ₹5.65 lakh of principal repaid in a quarter of the term.
Check by the retrospective route: 50,00,000 × 1.0075⁶⁰ − 44,986.28 × s₆₀; 1.0075⁶⁰ = 1.565681, s₆₀ = 75.4241 → 78,28,405 − 33,93,052 = ₹44,35,353 (difference is rounding of the EMI).

Example 3 — Flat rate exposed. A dealer offers a ₹6,00,000 car loan for 5 years at a "7% flat" rate, repaid in 60 equal monthly instalments.

Flat interest = 6,00,000 × 0.07 × 5 = ₹2,10,000 → total ₹8,10,000 → instalment = 8,10,000/60 = ₹13,500.
Equivalent reducing-balance rate solves 6,00,000 = 13,500 × a₆₀(i). a₆₀ required = 44.4444.
At i = 1.05%/month: 1.0105⁶⁰ = 1.871449 → a₆₀ = (1 − 0.534346)/0.0105 = 44.3480 (slightly low).
At i = 1.04%: 1.0104⁶⁰ = 1.860374 → a₆₀ = (1 − 0.537526)/0.0104 = 44.4687. Interpolating, i ≈ 1.0420% per month ≈ 12.5% nominal p.a. (13.25% effective) — nearly 1.8× the quoted flat rate.

Example 4 — Bond price, current yield and duration. A G-sec with ₹100 face, 7% coupon paid half-yearly, 5 years to maturity, redeemable at par; required yield 8% p.a. convertible half-yearly (i₂ = 4%, 10 half-years).

1.04¹⁰ = 1.480244 → v¹⁰ = 0.675564; a₁₀(4%) = (1 − 0.675564)/0.04 = 8.1109.
P = 3.50 × 8.1109 + 100 × 0.675564 = 28.388 + 67.556 = ₹95.94 — a discount bond, since yield > coupon.
Current yield = 7/95.94 = 7.30%, well below the 8% redemption yield: the difference is the pull to par.
Macaulay duration (in half-years, using Σt·CFv^t): coupon part = 3.50 × (Ia)₁₀ and redemption part = 10 × 100 × v¹⁰. ä₁₀ = 1.04 × 8.1109 = 8.4353; (Ia)₁₀ = (8.4353 − 10 × 0.675564)/0.04 = (8.4353 − 6.75564)/0.04 = 41.9915 → coupon contribution 146.970; redemption contribution 675.564; total 822.534. Duration = 822.534/95.944 = 8.573 half-years = 4.29 years.
Modified duration = 8.573/1.04 = 8.243 half-years = 4.121 years… in annual terms ModD = 4.29/1.04 = 4.12 years per unit yield. A 50 bp rise in yield → ΔP/P ≈ −4.12 × 0.005 = −2.06%, about ₹1.98, before the convexity correction (which makes the actual fall slightly smaller).
Accrued interest if bought 73 days into a 180-day (30/360) coupon period: 3.50 × 73/180 = ₹1.42, so the dirty price ≈ 95.94 + 1.42 = ₹97.36.

Example 5 — Sinking fund versus amortisation. A firm borrows ₹10,00,000 for 10 years, paying 10% interest annually to the lender and accumulating the capital in a fund earning 8%.

Annual interest = ₹1,00,000. s₁₀(8%) = (1.08¹⁰ − 1)/0.08; 1.08¹⁰ = 2.158925 → s₁₀ = 14.4866 → fund deposit = 10,00,000/14.4866 = ₹69,029.
Total annual outgo = 1,00,000 + 69,029 = ₹1,69,029.
An ordinary amortised loan at 10%: a₁₀(10%) = (1 − 1/1.10¹⁰)/0.10; 1.10¹⁰ = 2.593742 → v¹⁰ = 0.385543 → a₁₀ = 6.14457 → instalment = 10,00,000/6.14457 = ₹1,62,745.
The sinking-fund route costs ₹6,284 more a year, because the fund earns 8% while the debt costs 10% — consistent with 1/aₙ = 1/sₙ + i only when the two rates are equal.

Measure	What it answers	Formula	Blind spot
Current yield	Income return now	Coupon/Price	Ignores capital gain and timing
YTM	Total return if held to maturity	IRR of price vs cash flows	Assumes reinvestment at YTM
Macaulay duration	Average time to payment	Σt·CFv^t/P	Flat curve, parallel shifts
Modified duration	% price change per 1% yield	Mac/(1+i)	Linear — ignores convexity
Convexity	Curvature of price–yield	(1/P)d²P/di²	Still assumes parallel shifts

Feature	Annuity-immediate	Annuity-due
Payment timing	End of period	Start of period
PV of 1 p.a., n years	aₙ	(1+i)aₙ
Typical use	EMIs, bond coupons	Rent, insurance premiums, SIPs

Where it is used

Life insurance and pensions: LIC, HDFC Life, SBI Life and ICICI Prudential price immediate and deferred annuity products (Jeevan Akshay-type plans) and NPS annuitisation using annuity functions combined with mortality; IRDAI-regulated valuation of liabilities uses discounted mean term for asset–liability matching.
Retail lending: EMI calculation, amortisation schedules, foreclosure/prepayment quotes and APR disclosure in Key Facts Statements at HDFC Bank, SBI, Bajaj Finance and housing-finance companies; floating-rate resets under RBI's external-benchmark framework.
Fixed-income desks and treasuries: pricing G-secs, SDLs, T-bills and corporate NCDs; CCIL/NDS-OM settlement uses dirty prices; FIMMDA valuation conventions; PVBP and duration limits in bank trading books and in RBI's investment-classification norms (HTM/AFS/FVTPL).
Mutual funds and insurance investment teams: debt-fund NAV computation from mark-to-market bond prices, portfolio duration targeting in gilt and short-duration funds, and immunisation of guaranteed-return products.
Actuarial and consulting roles: gratuity and leave-encashment valuations under AS 15/Ind AS 19 use discount rates from G-sec yields of matching duration; Milliman, WTW, Aon and Mercer test this in their analyst interviews.
Corporate finance and project appraisal: lease-versus-buy, equivalent annual cost, bond issue pricing and NPV/IRR evaluation in infrastructure and PSU capital budgeting.

Interview questions

Q: What is the difference between an annuity-immediate and an annuity-due, and by how much do their values differ? A: An annuity-immediate pays at period ends, an annuity-due at period starts; the due value is exactly (1+i) times the immediate value because every payment is one period earlier.
Q: Derive the EMI formula. A: Set the loan equal to the PV of the instalments: L = X·aₙ = X(1−(1+i)⁻ⁿ)/i, so X = L·i/(1−(1+i)⁻ⁿ), with i the rate per month and n the number of months.
Q: How do you find the outstanding balance on a loan after t payments without building a schedule? A: Prospectively: B_t = X·a_{n−t}, the PV of the remaining instalments; the retrospective form L(1+i)^t − X·s_t gives the same answer.
Q: Why does the early part of a home-loan EMI consist almost entirely of interest? A: Interest is charged on a large outstanding balance; the capital content is X·v^{n−t}, which is smallest at the start and grows by factor (1+i) each month.
Q: A lender quotes 8% flat and another 14% reducing. Which is cheaper? A: Convert first — a flat rate is roughly equivalent to about 1.8–2 times itself on a reducing basis, so 8% flat is around 14–15% reducing; they are close, and you must solve L = X·aₙ for the exact yield.
Q: Why does a bond trade at a discount? A: Because the market's required yield exceeds the coupon rate, so the coupon stream alone is worth less than par and the price must fall to deliver the higher yield.
Q: What is the difference between clean and dirty price? A: The clean price is quoted excluding accrued interest; the dirty price, which the buyer actually pays, adds the coupon accrued since the last coupon date.
Q: What is the main assumption behind YTM, and why does it matter? A: That every coupon is reinvested at the YTM until maturity; if reinvestment rates are lower the realised return falls short, which is why zero-coupon bonds have no reinvestment risk.
Q: What is modified duration and how is it used? A: It is the percentage price change for a unit change in yield, Macaulay/(1+i); desks use it and PVBP to size interest-rate exposure and hedges.
Q: Two bonds have the same maturity but different coupons — which has higher duration? A: The lower-coupon bond, because a greater share of its value sits in the distant redemption payment; a zero-coupon bond's duration equals its maturity.
Q: State Redington's immunisation conditions. A: PV of assets equals PV of liabilities, durations are equal, and asset convexity exceeds liability convexity — it protects only against small, parallel yield-curve shifts and needs rebalancing.
Q: If inflation is 6% and the money yield 9%, what real return do you earn? A: (1.09/1.06) − 1 = 2.83%, not 3% — the real rate is multiplicative, not a subtraction.

Common mistakes

Using the annual rate with monthly payments, or dividing an *effective* annual rate by 12 instead of taking (1+i)^1/12 − 1.
Confusing immediate and due annuities — losing or gaining one period's interest on the whole stream.
Forgetting that the EMI must be recomputed after a prepayment or rate reset; applying the old X to the new balance.
Adding undiscounted cash flows in a duration calculation, or using time in half-years while quoting duration in years (divide by 2).
Quoting a flat rate against a reducing-balance rate without conversion, and comparing a nominal loan rate with an APR that includes fees.
Omitting accrued interest when valuing a bond bought between coupon dates, or discounting coupons at the annual yield instead of the half-yearly yield.
Treating a tax-adjusted bond price equation as explicit when P appears on both sides — it needs rearranging or iteration.
Mixing real cash flows with money discount rates in inflation-linked questions.