Home › Library › Financial Mathematics: YTM & Bonds
JAIIB Paper 3 • Accounting & Financial Management for Bankers (AFM) ⭐⭐⭐⭐⭐
Financial Mathematics: YTM & Bonds
How debt securities are priced, and why bond prices move opposite to interest rates. This guide covers Bond Terminology, Types of Bonds, Callable/Puttable Optionality, Bond Valuation, Semi-Annual Interest, Current Yield vs YTM, Bond Value Theorems, and Duration — with a free bond price calculator and 50 exam-style MCQs with hidden answers.
📅 Updated: September 2026 • 29 min read
1 Meaning of Debt & Introduction to Bonds
📜
Definition
A Bond is a long-term debt instrument issued by a government or corporate borrower, representing a loan from the investor to the issuer — repaid with periodic interest (coupon) and return of principal at maturity.
2 Key Terms Associated with Bonds
Face Value (Par Value)
The redemption value paid at maturity — typically ₹100 or ₹1,000.
Coupon Rate
The stated annual interest rate, paid on the face value.
Maturity Date
The date on which the face value is repaid to the bondholder.
Market Price
The price at which the bond actually trades — may differ from face value.
3 Types of Bonds
Government Securities (G-Secs)
Sovereign-issued, considered free of credit risk.
Corporate Bonds/Debentures
Issued by companies, secured or unsecured.
Zero-Coupon Bonds
No periodic interest — issued at a deep discount, redeemed at face value.
Convertible Bonds
May be converted into equity shares at a specified ratio.
Perpetual Bonds
No fixed maturity date — e.g., AT1 (Additional Tier 1) bonds issued by banks.
Floating Rate Bonds
Coupon linked to a benchmark rate, reset periodically.
4 Optionality in Bonds
Callable Bond
The issuer may redeem the bond before maturity — benefits the issuer, so investors demand a higher yield/lower price as compensation.
Puttable Bond
The investor may force early redemption — benefits the investor, generally commanding a lower yield.
5 Valuation of Bonds
Bond Price = Coupon × [1 − (1+y)−n] ÷ y + Face Value ÷ (1+y)n
= PV of the coupon annuity + PV of the face value at maturity, discounted at yield "y"
A bond's price is simply the present value of all its future cash flows — the periodic coupons plus the final face-value redemption — discounted at the investor's required yield.
6 Bond Value with Semi-Annual Interest
Most bonds pay coupons semi-annually. To value such a bond, the same formula is used, but with the number of periods doubled (n × 2) and the rate per period halved (y ÷ 2), and the coupon amount also halved per period.
7 Current Yield vs Yield-to-Maturity
Current Yield = Annual Coupon ÷ Market Price × 100
A simple "running yield" — ignores time value of money and any capital gain/loss on redemption
🎯
Definition — YTM
Yield-to-Maturity (YTM) is the single discount rate that equates the present value of all future cash flows (coupons + redemption) to the bond's current market price — the true total annualised return if held to maturity, assuming coupons are reinvested at the same rate.
8 Bond Value Theorems
Inverse Relationship
Bond prices move inversely to yields — as yields rise, prices fall, and vice versa.
Maturity Effect
Longer-maturity bonds are more price-sensitive to yield changes than shorter-maturity bonds.
Diminishing Sensitivity
Price sensitivity to maturity increases, but at a decreasing rate, as maturity lengthens.
Coupon Effect
Lower-coupon bonds are more price-sensitive to yield changes than higher-coupon bonds of the same maturity.
Convexity
A given fall in yield raises price more than an equal rise in yield lowers it — the price-yield relationship is convex, not linear.
9 Duration and Bond Price Volatility
Macaulay Duration
The weighted-average time to receive a bond's cash flows, weighted by each cash flow's present value.
Modified Duration
Macaulay Duration ÷ (1 + y/n) — directly estimates % price change for a 1% yield change.
Zero-Coupon Bond
Its Duration always equals its maturity, since there is only one cash flow, at maturity.
Duration & Coupon
Duration falls as the coupon rate rises — more cash flow arrives earlier, pulling the weighted average forward.
JAIIB tip:Modified Duration is a bond's single best summary measure of interest rate risk — the higher it is, the more the bond's price swings for a given change in yield.
🧮 Bond Price Calculator
Enter a bond's face value, coupon rate, required yield and years to maturity to estimate its price and current yield.
Bond Details
Results
Annual Coupon (₹)—
Bond Price (Annual Compounding)—
Current Yield—
If Required Yield > Coupon Rate, the bond prices at a discount; if lower, at a premium.
✓ Key Takeaways
A bond's price is the present value of its future coupons plus face value, discounted at the required yield.
Current Yield is a simple running-yield measure; YTM is the true total return if held to maturity.
Bond prices and yields move inversely; longer maturity and lower coupon both increase price sensitivity to yield changes.
The price-yield relationship is convex — a yield fall boosts price more than an equal yield rise cuts it.
Modified Duration is the key summary measure of a bond's interest rate risk; a zero-coupon bond's duration equals its maturity.
📝 Top 50 JAIIB-Style MCQs on Financial Mathematics: YTM & Bonds
Test your understanding with these 50 practice MCQs, closely modelled on the pattern expected in the JAIIB AFM exam — including combination-answer questions. Each question has 5 options — the correct answer is hidden by default; tap "Show Answer" to reveal it along with a short explanation.
📜 Meaning of Debt & Introduction to Bonds
1 A Bond represents:
A. Ownership equity in the issuer
B. A loan from the investor to the issuer
C. A type of insurance policy
D. A demand deposit account
E. A guarantee issued by a bank
Answer: B. A bond is a loan from investor to issuer.
2 Bonds can be issued by:
A. Governments only
B. Corporates only
C. Both governments and corporates
D. Only individuals
E. Only foreign entities
Answer: C. Both governments and corporates issue bonds.
3 A bondholder receives, over the bond's life:
A. Dividends only
B. Periodic coupon interest and, at maturity, the face value
C. Voting rights only
D. Only capital gains
E. No return of any kind
Answer: B. Bondholders get periodic coupons plus face value at maturity.
4 Compared to equity, a bond represents:
A. An ownership stake
B. A debt claim on the issuer
C. A guarantee of profit share
D. A perpetual dividend right
E. A form of insurance
Answer: B. A bond is a debt claim, unlike equity ownership.
5 Bonds are generally classified as which type of financing instrument?
A. Short-term debt instruments always
B. Long-term debt instruments
C. Equity instruments
D. Derivative instruments
E. Insurance contracts
Answer: B. Bonds are long-term debt instruments.
🔑 Key Terms Associated with Bonds
6 The amount repaid to a bondholder at maturity is the:
A. Coupon Rate
B. Face Value (Par Value)
C. Current Yield
D. Market Price always
E. Duration
Answer: B. Face Value is repaid at maturity.
7 The Coupon Rate is:
A. The stated annual interest rate paid on face value
B. The bond's market price
C. The bond's maturity date
D. Always equal to the YTM
E. A type of duration measure
Answer: A. Coupon Rate is the stated interest rate on face value.
8 A bond's Market Price:
A. Always equals its face value
B. May differ from its face value
C. Is fixed by the issuer forever
D. Is irrelevant to investors
E. Is the same as the coupon rate
Answer: B. Market Price may differ from face value.
9 Typical face values for Indian bonds are:
A. ₹100 or ₹1,000
B. ₹1 crore only
C. ₹1 only
D. There is no standard face value ever used
E. Always USD 100
Answer: A. ₹100 or ₹1,000 are typical face values.
10 The Maturity Date is:
A. When the face value is repaid
B. When the bond is first issued
C. Irrelevant to bond valuation
D. The date of the first coupon only
E. Always the same as the coupon date
Answer: A. Maturity Date is when face value is repaid.
🏷️ Types of Bonds
11 Government Securities (G-Secs) are generally considered:
A. High credit risk
B. Free of credit risk (sovereign-backed)
C. Equity instruments
D. Short-term only, never long-term
E. Issued only by corporates
Answer: B. G-Secs are considered free of credit risk.
12 A Zero-Coupon Bond:
A. Pays periodic interest at a high rate
B. Pays no periodic interest, issued at a deep discount
C. Is identical to an equity share
D. Cannot be redeemed ever
E. Pays interest monthly always
Answer: B. Zero-Coupon Bonds pay no periodic interest, issued at a discount.
13 A Convertible Bond may be converted into:
A. Cash only, at face value
B. Equity shares of the issuer
C. A government security
D. A different currency automatically
E. A fixed deposit
Answer: B. Convertible bonds convert into equity shares.
14 A Perpetual Bond, such as a bank's AT1 bond:
A. Has a fixed short maturity of 1 year
B. Has no fixed maturity date
C. Is identical to a zero-coupon bond
D. Cannot pay any coupon
E. Is a type of equity share
Answer: B. Perpetual bonds have no fixed maturity.
15 A Floating Rate Bond's coupon:
A. Never changes throughout its life
B. Is linked to a benchmark and reset periodically
C. Is always zero
D. Is set only by the investor
E. Equals the Current Yield always
Answer: B. Floating Rate Bond coupons reset per a benchmark.
🔀 Optionality in Bonds
16 A Callable Bond gives the right to redeem early to the:
A. Investor
B. Issuer
C. Government only
D. Rating agency
E. Stock exchange
Answer: B. The Issuer holds the call option.
17 A Puttable Bond gives the right to force early redemption to the:
A. Investor
B. Issuer
C. Rating agency
D. Government only
E. Stock exchange
Answer: A. The Investor holds the put option.
18 Because a Callable Bond benefits the issuer, investors typically demand:
A. A lower yield
B. A higher yield/lower price as compensation
C. No compensation at all
D. Conversion into equity
E. A shorter maturity always
Answer: B. Investors demand a higher yield/lower price for call risk.
19 A Puttable Bond, benefiting the investor, generally commands:
A. A higher yield than an equivalent plain bond
B. A lower yield than an equivalent plain bond
C. No difference in yield
D. Automatic conversion into equity
E. Zero coupon always
Answer: B. Puttable bonds generally command a lower yield.
20 Optionality in bonds refers to:
A. Embedded call/put features affecting redemption timing
B. The choice of currency only
C. The choice of stock exchange for listing
D. Voting rights of bondholders
E. Dividend payment options
Answer: A. Optionality means embedded call/put redemption features.
💰 Valuation of Bonds
21 A bond's price equals the present value of:
A. Only the face value
B. Future coupons plus the face value at maturity
C. Only the coupon payments
D. The issuer's total assets
E. The issuer's market capitalisation
Answer: B. Bond price = PV of coupons + PV of face value.
22 The discount rate used in bond valuation is the:
A. Coupon rate always
B. Investor's required yield
C. GST rate
D. Inflation rate only
E. Repo rate always
Answer: B. The required yield is used as the discount rate.
23 If the required yield equals the coupon rate, the bond will price:
A. At a premium
B. At par (equal to face value)
C. At a discount
D. At zero
E. Above face value always
Answer: B. Equal yield and coupon means the bond prices at par.
24 If required yield exceeds the coupon rate, the bond prices:
A. At a premium (above face value)
B. At a discount (below face value)
C. Exactly at face value
D. At zero
E. Cannot be determined
Answer: B. Yield above coupon means the bond prices at a discount.
25 If required yield is below the coupon rate, the bond prices:
A. At a discount
B. At a premium (above face value)
C. Exactly at zero
D. Exactly at face value always
E. Cannot be determined
Answer: B. Yield below coupon means the bond prices at a premium.
🔄 Bond Value with Semi-Annual Interest
26 For semi-annual coupon bonds, the number of periods used in valuation is:
A. Halved
B. Doubled (years × 2)
C. Unchanged
D. Tripled
E. Irrelevant
Answer: B. Number of periods is doubled for semi-annual bonds.
27 For semi-annual coupon bonds, the rate per period used is:
A. The full annual rate
B. Half of the annual rate
C. Double the annual rate
D. Zero
E. The GST rate
Answer: B. The rate per period is halved.
28 Most Indian government and corporate bonds pay coupons:
A. Monthly
B. Semi-annually
C. Never
D. Only at maturity always
E. Weekly
Answer: B. Most bonds pay coupons semi-annually.
29 More frequent coupon payments (e.g., semi-annual vs annual), for the same nominal rate, result in a bond price that is:
A. Unaffected in any scenario
B. Slightly different, reflecting the different effective compounding
C. Always exactly at par
D. Always zero
E. Always at a discount
Answer: B. Payment frequency affects effective compounding and price.
30 Each semi-annual coupon payment amount is:
A. The full annual coupon
B. Half of the annual coupon
C. Double the annual coupon
D. Always zero
E. Unrelated to the annual coupon
Answer: B. Each semi-annual payment is half the annual coupon.
🎯 Current Yield vs Yield-to-Maturity
31 Current Yield is calculated as:
A. Annual Coupon ÷ Market Price × 100
B. Annual Coupon ÷ Face Value × 100
C. Face Value ÷ Market Price
D. Market Price ÷ Coupon
E. Duration ÷ Coupon
Answer: A. Current Yield = Annual Coupon ÷ Market Price × 100.
32 YTM is defined as the discount rate that:
A. Equates PV of all future cash flows to the current market price
B. Equals the coupon rate always
C. Is set by the government arbitrarily
D. Ignores all future cash flows
E. Equals zero for all bonds
Answer: A. YTM equates PV of cash flows to market price.
33 Current Yield, unlike YTM, ignores:
A. The coupon amount
B. The time value of money and any capital gain/loss on redemption
C. The market price
D. All future cash flows entirely
E. The face value entirely
Answer: B. Current Yield ignores time value and capital gain/loss.
34 YTM assumes that coupons received are:
A. Reinvested at the same YTM rate
B. Never reinvested
C. Converted into equity automatically
D. Paid out as tax
E. Held as physical cash only
Answer: A. YTM assumes reinvestment at the same rate.
35 A bond trading at a discount to face value will have a YTM that is:
A. Lower than the coupon rate
B. Higher than the coupon rate
C. Exactly equal to the coupon rate
D. Always zero
E. Unrelated to the coupon rate
Answer: B. A discount bond has YTM higher than its coupon rate.
📐 Bond Value Theorems
36 Bond prices and yields move:
A. In the same direction
B. Inversely to each other
C. Independently of each other
D. Randomly
E. Only in one direction, never the other
Answer: B. Bond prices and yields move inversely.
37 For the same coupon, a longer-maturity bond is:
A. Less sensitive to yield changes
B. More sensitive to yield changes
C. Equally sensitive as a shorter bond
D. Immune to yield changes
E. Not comparable at all
Answer: B. Longer maturity means higher sensitivity.
38 As maturity lengthens, price sensitivity increases:
A. At an increasing rate
B. At a decreasing rate
C. Not at all
D. Linearly and infinitely
E. Only up to 1 year
Answer: B. Sensitivity rises at a diminishing rate with maturity.
39 For the same maturity, a lower-coupon bond is:
A. Less sensitive to yield changes
B. More sensitive to yield changes
C. Equally sensitive as a higher-coupon bond
D. Immune to yield changes
E. Not comparable at all
Answer: B. Lower coupon means higher price sensitivity.
40 The "convexity" of the price-yield relationship means:
A. A yield fall raises price more than an equal yield rise lowers it
B. Price and yield are perfectly linearly related
C. A yield rise raises price more than a yield fall
D. Price never changes with yield
E. Only applies to zero-coupon bonds
Answer: A. Convexity means yield falls boost price more than equal rises cut it.
⏱️ Duration and Bond Price Volatility
41 Macaulay Duration measures:
A. The weighted-average time to receive a bond's cash flows
B. The bond's coupon rate
C. The bond's face value
D. The issuer's credit rating
E. The GST payable on the bond
Answer: A. Macaulay Duration is the weighted-average time to cash flows.
42 Modified Duration directly estimates:
A. The bond's face value
B. The percentage price change for a 1% yield change
C. The bond's coupon rate
D. The issuer's total debt
E. The bond's credit rating
Answer: B. Modified Duration estimates % price change per 1% yield change.
43 A Zero-Coupon Bond's Duration equals:
A. Zero, always
B. Its maturity
C. Half its maturity
D. Double its maturity
E. Its coupon rate
Answer: B. A Zero-Coupon Bond's Duration equals its maturity.
44 As a bond's coupon rate rises, its Duration:
A. Rises
B. Falls
C. Stays unchanged
D. Becomes zero
E. Becomes infinite
Answer: B. Duration falls as coupon rate rises.
45 A bond with a higher Modified Duration will experience, for a given yield change:
A. A smaller price swing
B. A larger price swing
C. No price change at all
D. An unrelated price change
E. A change only in coupon, not price
Answer: B. Higher Modified Duration means a larger price swing.
🔁 Comprehensive Review
46 When market interest rates rise sharply, a bank's existing bond portfolio (AFS/HFT) will typically show:
A. An increase in market value
B. A decrease in market value (mark-to-market loss)
C. No change whatsoever
D. Automatic conversion to equity
E. Immediate maturity
Answer: B. Rising rates cause a mark-to-market loss on bond holdings.
47 Between two bonds of equal maturity, the one with the lower coupon will show:
A. Lower price volatility for a given yield change
B. Higher price volatility for a given yield change
C. Identical volatility
D. No volatility at all
E. Volatility unrelated to coupon
Answer: B. The lower-coupon bond shows higher price volatility.
48 A callable bond's price, compared to an otherwise identical non-callable bond, is generally:
A. Higher
B. Lower (or its yield higher), reflecting call risk
C. Identical
D. Always zero
E. Unrelated to call risk
Answer: B. Callable bonds price lower/yield higher to compensate for call risk.
49 Which of the following would increase a bond's Duration? (i) Longer maturity (ii) Lower coupon rate (iii) Higher coupon rate
A. (i) only
B. (i) and (ii) only
C. (iii) only
D. All of (i), (ii) and (iii)
E. None of these
Answer: B. Longer maturity and lower coupon both raise Duration.
50 For JAIIB purposes, the single most important practical insight from bond mathematics is:
A. Bond prices are entirely insulated from interest rate movements
B. Interest rate changes directly and predictably affect bond values, via yield, maturity, coupon and Duration
C. Only equity prices react to interest rates
D. YTM and Current Yield are always identical
E. Duration has no relevance to banking
Answer: B. Interest rate changes predictably move bond values via yield/maturity/coupon/Duration.
Disclaimer: This article is prepared for educational and exam-preparation purposes only, as of September 2026. Candidates should cross-check the latest official IIBF syllabus and current notifications before the exam.