Quantitative Methods ยท The Time Value of Money in Finance ยท LO 1 of 3
Every financial instrument is a pattern of cash flows on a timeline. Which pattern is this one?
A bond, a mortgage, a dividend-paying stock, these look like three different subjects. They are not. There are exactly four shapes. Once you recognise the shape, the calculator does the rest.
โฑ 8min-15min
ยท
6 questions
ยท
HIGH PRIORITYAPPLY๐งฎ Calculator
Why this LO matters
A bond, a mortgage, a dividend-paying stock, these look like three different subjects. They are not. There are exactly four shapes. Once you recognise the shape, the calculator does the rest.
INSIGHT
Every financial instrument is a promise to pay cash on specific dates. That promise has a shape.
There are exactly four shapes. This LO teaches you to recognise them. Once you can identify the shape, you know which formula to use and which calculator inputs to enter. The instrument's name, bond, stock, mortgage, does not matter. The shape does.
The four cash flow patterns
Before any formula and before any calculator key, you need to see the four shapes. Every question in this LO, and most questions across the curriculum, fits into one of these four.
The four patterns, identify the shape before touching the calculator
1
Pattern 1: Single payment. One cash flow at a specific future date. Nothing in between. Example: you deposit money today and collect the full amount plus growth in five years.
2
Pattern 2: Annuity. Equal payments at regular intervals for a fixed number of periods. No payments after the end date. Example: a loan repaid in 60 equal monthly instalments, or a bond paying the same coupon every six months for ten years. (Coupon is the name for the periodic interest payment a bond makes to its lenders. Named after paper coupons bondholders used to clip and redeem. A 10% annual coupon on a bond with face value 100 pays 10 per year, or 5 every six months if semiannual.)
3
Pattern 3: Perpetuity. Equal payments at regular intervals that never stop. No end date. Example: a preferred share that pays a fixed dividend every year, forever.
4
Pattern 4: Growing perpetuity. Payments that grow at a constant rate every period, forever. Example: a stock whose dividend increases by 3% each year indefinitely.
What is r? (before we use it in every formula)
Every formula in this LO contains the letter r. Before computing anything, understand what r means and where the number comes from.
r goes by three names, same variable, different context
1
Interest rate. When a bank offers 5% per year on a deposit, that 5% is the rate at which your money grows. The bank sets it.
2
Discount rate. When you ask "what is a future cash flow worth today?", you shrink it back using a discount rate. A higher discount rate makes future cash flows worth less today, because you could earn more waiting.
3
Required return. When you invest in a bond or stock, the required return is the minimum annual return you demand to accept the risk of that investment. If you require 8% and the investment delivers only 6%, you would not buy it. All three names refer to the same variable r in the TVM formulas. The name changes depending on context: growing money forward (interest rate), shrinking it backward (discount rate), or deciding whether an investment is worth buying (required return). In exam questions, treat them as identical. When you see any of these terms, substitute the number into r.
The four shapes, drawn in full
PATTERN 1, Single payment (one future cash flow)
T=0 T=1 T=2 T=N
โ โ
-PV +FV
PATTERN 2, Annuity (equal payments + optional principal at end)
T=0 T=1 T=2 T=3 ... T=N
โ โ โ โ โ
-PV +PMT +PMT +PMT +PMT+FV
(If FV = 0, all payments are PMT only, pure annuity like a mortgage.
If FV > 0, principal returns at T=N on top of final PMT, coupon bond.)
PATTERN 3, Perpetuity (equal payments, no end date)
T=0 T=1 T=2 T=3 T=4 T=โ
โ โ โ โ โ โ
-PV +PMT +PMT +PMT +PMT +PMT
โ forever
PATTERN 4, Growing perpetuity (payments grow at rate g, no end date)
T=0 T=1 T=2 T=3
โ โ โ โ
-PV +Dโ +Dโ(1+g) +Dโ(1+g)ยฒ
โ forever, growing
How to use this: Before writing any formula, draw the timeline for your specific problem. Label the known values. Circle the unknown. The shape tells you which formula or calculator workflow to use.
Calculator setup, do this once, before the exam
Three settings affect every calculation you will ever do on the BA II Plus. Set them once and never worry about them again.
Three BA II Plus settings that affect every result
1
Decimal places โ 9. The factory default shows 2 decimal places. Intermediate calculations round and the final answer mismatches the exam choices. Fix: press [2nd][.] to open FORMAT, type 9, press [ENTER], then [2nd][CPT] to exit. Confirm: type 1 รท 3 = and you should see 0.333333333, not 0.33.
2
Calculation order โ AOS. The factory default (CHN) computes left to right: 8 + 2 ร 3 gives 30. The correct answer is 14. Every formula with mixed operations gives the wrong answer in CHN mode. In FORMAT, press the down arrow four times to reach CHN, then press [2nd][ENTER] to switch to AOS. Confirm: type 8 + 2 ร 3 = and see 14.
3
Clear TVM before every problem. Press [2nd][FV] before any TVM calculation. The calculator stores the last values you entered. Leftover inputs from a previous problem corrupt your answer with no error message, just a wrong number.
Why the calculator setup matters more than the formula
Two students sit the same TVM question. Student A computes (1.07)ยฒ and gets 1.1449. Student B presses the same keys and gets 1.1449 too, but on the next formula, Student B's answer is wrong. Same calculator. Same keys. The difference is one setting: CHN mode processes left to right, so a formula like PV + PMT ร (1+r) gives the wrong order of operations. Student A switched to AOS on day one. Student B never did. One setting. One mark lost per question with mixed operations. Fix it now.
The sign convention, the most common source of wrong answers
The BA II Plus models a financial transaction. Every transaction has two sides: money leaving your hands and money arriving. You must tell the calculator which direction each cash flow moves.
The sign convention, cash leaving your pocket is negative
1
PV is almost always negative. You pay the price of the asset today. Money leaves. Enter PV using the [+/-] key after the amount.
2
FV and PMT are almost always positive. You receive par value at maturity and periodic payments. Money arrives. Enter as positive.
3
PV and FV must have opposite signs. If both are positive or both are negative, the calculator interprets both as the same direction, economically meaningless. It returns ERROR 5.
Input
Bond (you buy)
Mortgage (you borrow)
Stock (you buy)
PV
Negative (price you pay)
Positive (cash you receive)
Negative (price you pay)
PMT
Positive (coupons received)
Negative (payments you make)
Positive (dividends received)
FV
Positive (par at maturity)
Zero (fully repaid)
Positive (model price)
Pattern 1: Single cash flow, growing forward or discounting back
The year-by-year story of compounding
Start with $50,000 at 7% per year. After year 1: $50,000 ร 7% = $3,500 interest, giving $53,500 total. In year 2, you earn 7% on $53,500, not on the original $50,000. Year 2 interest is $3,745, giving $57,245. Year 2 produces more than year 1 even though the rate is identical. That is compounding: interest earning interest on itself. The formula (1+r)^N is just shorthand for multiplying by 1.07, N times.
Single cash flow, grow forward or discount back
FV = PV ร (1 + r)^N // grow a present value forward N periods
PV = FV รท (1 + r)^N // discount a future value back to today
Where:
r = rate per period (not per year unless one period = one year)
N = number of periods (not years unless one period = one year)
Worked Example 1
$50,000 at 7%, what is it worth in 2 years? In 5 years?
An investor places $50,000 in an account earning 7% per year. What is the balance after 2 years? After 5 years?
For year 5: type 5 [N] then [CPT][FV]. The calculator keeps all other inputs. Result: 70,127.95.
Worked Example 2
Andrea Fitch, how much to invest today?
Andrea Fitch needs $20,000 in four years to buy her daughter a car when she finishes high school. A money market fund earns 3% per year. How much must she invest today?
This is Pattern 1 run backwards. She knows the amount she needs at T=4. She wants to find what she must invest at T=0.
Step 1: PV = 20,000 รท (1.03)โด
Step 2: (1.03)โด = 1.12551
Step 3: PV = 20,000 รท 1.12551 = $17,769.70
โ Answer: Fitch must invest $17,769.70 today.
๐งฎ BA II Plus Keystrokes
`[2nd][FV]`
clear TVM โ 0
`4[N]`
4 years โ N = 4
`3[I/Y]`
3% annual rate โ I/Y = 3
`0[PMT]`
no recurring payment โ PMT = 0
`20000[FV]`
$20,000 arriving at T=4 โ FV = 20,000
`[CPT][PV]`
compute present value โ **PV = โ17,769.70**
โ ๏ธ Students sometimes press [CPT][PV] after entering a PV, the calculator echoes it back because you already stored it. Enter the four known values, leave the fifth blank, then press [CPT] + the unknown.
The negative sign means this is money Fitch must pay out today. The answer is the magnitude: $17,769.70.
FORWARD REFERENCE
Bonds, what you need for this LO only
A bond is a loan. The borrower (a company or government) promises to pay interest, called the coupon, on a regular schedule, and return the original borrowed amount, called par or face value, typically quoted as 100, at the end of the loan term, called maturity. This is Pattern 1 (single payment for a zero-coupon bond) or Pattern 2 (annuity of coupon payments plus par at the end). You do not need to understand yield spreads, credit risk, or duration for this LO. When you see a bond problem: coupon = PMT, par = FV, periods remaining = N, market yield = I/Y. Enter those four, compute PV. That is the entire skill here.
โ Fixed Income
Pattern 2: Annuity, equal recurring payments
An annuity is any series of equal payments at fixed intervals for a fixed number of periods. A mortgage is an annuity. So is a bond paying the same coupon every six months for ten years. What unites them is the shape: equal payments, fixed end date.
The one assumption your calculator always makes
The BA II Plus has a built-in assumption when you enter PMT: the first payment happens exactly one period from now. Not today. Not two periods from now. Exactly one period from now. This matches most bonds and loans, and it makes the ordinary calculation easy. But when the first payment arrives today, or arrives delayed by several periods, the calculator's answer needs adjustment. Candidates who do not know this assumption report the wrong number with full confidence. The timeline reveals the mismatch instantly.
Worked Example 3
Mike Mills Option 1, ordinary annuity
Mike Mills is retiring today. Option 1: 15 annual payments of $180,000, first payment arriving in one year. Discount rate: 8%. What is the present value?
Ordinary annuity, first payment at T=1
T=0 T=1 T=2 T=3 ... T=15
โ โ โ โ โ
-PV? +180k +180k +180k +180k
Calculator's assumption: first PMT at T=1. This matches exactly.
Calculator computes PV at T=0 directly. No adjustment needed.
Step 1, Identify the pattern. Equal payments, fixed term, no remaining balance. Pure annuity. N=15, PMT=180,000, I/Y=8, FV=0.
โ Answer: Present value of Option 1 = $1,540,706.
๐งฎ BA II Plus Keystrokes
`[2nd][FV]`
clear TVM โ 0
`15[N]`
15 annual periods โ N = 15
`8[I/Y]`
8% discount rate โ I/Y = 8
`180000[PMT]`
each payment โ PMT = 180,000
`0[FV]`
no balance at end โ FV = 0
`[CPT][PV]`
compute present value โ **PV = โ1,540,706.10**
Report as $1,540,706. Negative confirms this represents money being given up (the value of what Mills is receiving from the pension plan).
Worked Example 4
Grupo Ignacia, semiannual coupon bond
Grupo Ignacia issued 10-year bonds two years ago. The bonds pay a 10.7% annual coupon on a semiannual basis. The current market yield is 11.6% per year. What is the price per MXN100 of par value?
This is Pattern 2. The bond pays equal coupons every six months (PMT) plus par at maturity (FV). Before entering anything, apply the semiannual adjustment.
Semiannual coupon bond, 8 years remaining, 16 periods
T=0 T=0.5 T=1 T=1.5 ... T=7.5 T=8
โ โ โ โ โ โ
-PV? +5.35 +5.35 +5.35 +5.35 +105.35
Issued 10 years ago total term, 2 years elapsed โ 8 years remaining = 16 semiannual periods.
PMT = 10.7 รท 2 = 5.35. I/Y = 11.6 รท 2 = 5.80. N = 8 ร 2 = 16. FV = 100.
Step 1, Apply the semiannual adjustment (three steps, all three):
- Coupon per period: 10.7 รท 2 = 5.35
- Rate per period: 11.6 รท 2 = 5.80%
- Periods: 8 years ร 2 = 16
Step 2, Confirm direction. Market yield (11.6%) is higher than coupon rate (10.7%). When discount rate exceeds coupon, the bond prices below par. Expect PV < 100. โ
โ ๏ธ Entering 8 as N instead of 16 gives MXN97.18. Entering the full 10.7 coupon without halving gives MXN95.47. Both appear as wrong-answer choices. The semiannual adjustment is three steps: halve the coupon, halve the rate, double the periods. All three. Every time.
Worked Example 5
Grey Pebble Real Estate, monthly mortgage
Grey Pebble is buying a NZD5 million building with a 75% mortgage at 4.8% annual interest, repaid over 5 years with monthly payments. What is the monthly payment?
This is Pattern 2. The mortgage is an annuity of equal monthly payments. Apply the monthly adjustment.
Monthly mortgage, 60 equal payments, balance = 0 at end
T=0 T=1 T=2 T=3 ... T=59 T=60
โ โ โ โ โ โ
+NZD3.75M -A -A -A -A -A
Grey Pebble borrows 75% ร 5,000,000 = NZD3,750,000.
Monthly rate = 4.8% รท 12 = 0.4%. Periods = 5 ร 12 = 60. FV = 0.
loan received (positive, borrower's view) โ PV = 3,750,000
`0[FV]`
fully repaid at end โ FV = 0
`[CPT][PMT]`
solve for monthly payment โ **PMT = โ70,424**
โ ๏ธ Using the full NZD5 million instead of 75% gives NZD93,899. The question asks for the mortgage payment on the borrowed amount, not on the property's full value.
The negative sign confirms the payment leaves Grey Pebble's account each month. Report as NZD70,424.
Pattern 3: Perpetuity, payments that never stop
The word "perpetuity" sounds complicated. The formula is the simplest of the four patterns. An infinite stream of equal payments collapses to one division.
Perpetuity, infinite equal payments simplified
PV = PMT รท r
Where r > 0.
CRITICAL PLACEMENT: This formula gives the PV exactly ONE period before the first payment.
First payment at T=1 โ formula gives PV at T=0. โ Use directly.
First payment at T=3 โ formula gives PV at T=2. Discount 2 more periods to T=0.
FORWARD REFERENCE
Preferred shares and constant-dividend stocks, Pattern 3
A preferred share pays a fixed dividend every year, forever. It is a perpetuity. PV = D รท r, where D is the annual dividend and r is the investor's required return on that share. Full treatment, yield spreads, why companies issue preferred equity, tax treatment, is in Equity Investments. For this LO: identify the fixed, forever-repeating payment. Divide by r. That is the price.
โ Equity Investments
Worked Example 6
Shipline PLC, constant dividend stock
Shipline PLC pays GBP1.50 per share in annual dividends, expected to continue forever. Shareholders require a 15% annual return. What is the expected stock price?
Perpetuity, equal payments, no end date
T=0 T=1 T=2 T=3 T=4 ...
โ โ โ โ โ
-PV? +1.50 +1.50 +1.50 +1.50 โ forever
Pattern 3. First payment at T=1. Formula gives PV at T=0 directly.
Step 1, Confirm pattern. Fixed dividend, forever, first payment in one year. Pattern 3. โ
Step 3, Sanity check. At GBP10, the dividend yield is 1.50 รท 10 = 15%. This equals the required return exactly, correct for a no-growth stock.
โ Answer: GBP10.00.
Pattern 4: Growing perpetuity, the Gordon Growth Model
A growing stock is a perpetuity where each dividend is larger than the last. The Gordon Growth Model captures this in one formula.
The idea: if dividends grow at g% per year forever starting from next year, the present value of that entire growing stream compresses to one fraction.
Gordon Growth Model, growing perpetuity
PV = Dโ รท (r โ g)
Where:
Dโ = next period's expected dividend (NOT the dividend already paid)
r = required rate of return on equity
g = constant expected growth rate of dividends
Condition: r > g. If r โค g, the formula breaks, the denominator becomes zero or negative.
If given Dโ (the dividend just paid):
Dโ = Dโ ร (1 + g)
FORWARD REFERENCE
The Gordon Growth Model, what you need for this LO only
In Equity Investments, you will study when and why dividends grow, how to estimate g, and when this model fails. For this LO, you only need two things: use Dโ (next dividend, not Dโ already paid) in the numerator, and confirm r > g before dividing.
โ Equity Investments
Worked Example 7
Mylandia Corporation, constant growth stock
Mylandia Corporation just paid a dividend of CAD2.40. Analysts expect dividends to grow at a constant 3% per year indefinitely. Shareholders require an 8% annual return. What is the expected share price?
Step 3, Sanity check. r โ g = 5% > 0. Formula is valid. โ
โ Answer: CAD49.44. Exam answer: B.
Two-stage growth: high growth followed by stable growth
Some companies grow quickly for a few years, then slow to a sustainable long-term rate. The two-stage model handles this by treating the two phases separately.
The method:
Calculate each dividend during the high-growth phase (years 1 to n)
Calculate the terminal value at the end of year n using the Gordon Growth Model on the first stable-phase dividend
Where:
Dแตข = dividend in year i during high-growth phase
TV = terminal value at year n = D(n+1) รท (r โ g_stable)
g_stable = long-term growth rate after year n
Worked Example 8
Mylandia two-stage growth
Mylandia's dividend grows at 10% for three years, then settles at 3% forever. Most recent dividend was CAD2.40. Required return is 8%. What is the expected share price today?
โ ๏ธ The terminal value is already sitting at T=3. Enter it together with D3 as a single T=3 entry. Do not enter it separately at T=4.
โ ๏ธ
Watch out for this
The semiannual adjustment, most common calculation error
Any bond that pays coupons twice per year requires three simultaneous adjustments. Candidates who make only one or two of them get a number that appears among the wrong-answer choices.
The three adjustments, every semiannual bond, every time:
- PMT: divide the annual coupon rate by 2
- I/Y: divide the annual yield by 2
- N: multiply the years remaining by 2
For Grupo Ignacia: PMT = 10.7รท2 = 5.35, I/Y = 11.6รท2 = 5.80, N = 8ร2 = 16.
Missing any one of these gives MXN97.18 (wrong N) or MXN95.47 (wrong PMT), both appear as answer choices. The exam builds these traps deliberately.
โ ๏ธ
Watch out for this
The Dโ versus Dโ mistake, second most common error
The Gordon Growth Model uses next year's dividend (Dโ) in the numerator. Not the dividend just paid (Dโ).
Why: Dโ has already been received by whoever owned the stock before you. Today's buyer gets Dโ as their first payment.
Wrong: PV = 2.40 รท (0.08 โ 0.03) = CAD48.00, this uses Dโ.
Correct: PV = 2.40 ร 1.03 รท (0.08 โ 0.03) = CAD49.44, this uses Dโ.
The signal phrase in exam questions is "most recent dividend was" or "just paid a dividend of." Whenever you see either phrase, the number given is Dโ. Multiply by (1+g) before dividing.
๐ง
Memory Aid
FORMULA HOOK
Grow the dividend once. Shrink by r minus g. Never divide what was just paid.
Practice Questions ยท LO1
6 Questions LO1
Score: โ / 6
Q 1 of 6 โ REMEMBER
What does the term "discount rate" mean in a present value calculation?
CORRECT: B
CORRECT: B, In TVM, the discount rate (r) is the rate applied to future cash flows to express them in today's terms. A higher discount rate makes future cash flows worth less today, because you could have earned more by investing elsewhere. It goes by different names depending on context: discount rate when shrinking values backward, interest rate when growing them forward, and required return when evaluating whether an investment is attractive. In every formula, it is the same variable r.
Why not A? A retail discount is a price reduction, colloquial usage. Financial discounting is a mathematical operation applied to future cash flows. The two uses of the word are unrelated.
Why not C? The coupon rate is set by the bond issuer when the bond is created and is fixed for the bond's life. The discount rate (or yield-to-maturity) is the current market rate and changes constantly with supply, demand, and interest rate expectations. When they are equal, the bond prices at par. When they differ, the bond prices above or below par.
Q 2 of 6 โ UNDERSTAND
A bond pays a 6% annual coupon and currently trades at exactly par (price equals face value). Which statement is correct?
CORRECT: C
CORRECT: C, When a bond prices at par, the coupon rate equals the YTM. The coupons already deliver exactly what investors require, there is no capital gain or loss to make up the difference. This is the at-par condition.
Why not A? If YTM were above the coupon rate, investors would demand a higher return than the coupon provides. They would only buy at a price below par, earning additional return from the price rising to par at maturity. A YTM above coupon rate means the bond prices at a discount.
Why not B? If YTM were below the coupon rate, the coupon exceeds what investors require. Demand would push the price above par. A YTM below coupon rate means the bond prices at a premium.
Q 3 of 6 โ APPLY
A bond with face value USD1,000 pays a 6% annual coupon and matures in five years. The current market YTM is 8%. The current price is closest to:
CORRECT: B
CORRECT: B, Pattern 2: PMT = 60 (6% ร 1,000), FV = 1,000, N = 5, I/Y = 8. CPT PV = โ920.15. Round to USD920. The bond prices below par because the coupon rate (6%) is below the market rate (8%). Investors demand a lower price to earn the required 8% return from a below-market coupon.
Why not A? USD926 results from entering N=6 instead of N=5, pricing the bond one period early. The bond matures in five years, so N = 5.
Why not C? USD1,000 is par value, what the bond pays at maturity, not what it trades for today. A bond trades at par only when coupon rate equals YTM. Here the coupon (6%) is below the market rate (8%), so the bond must price below par.
Q 4 of 6 โ APPLY+
Grupo Ignacia issued 10-year corporate bonds four years ago. The bonds pay a 10.7% annual coupon on a semiannual basis. The current price is MXN97.50 per MXN100 of par value. The annualised YTM is closest to:
CORRECT: A
CORRECT: A, Bonds issued 10 years ago but only four years have passed: 10 โ 4 = 6 years remaining = 12 semiannual periods. PMT = 10.7รท2 = 5.35. Enter: N=12, PV=โ97.50, PMT=5.35, FV=100, CPT I/Y = 5.64% per period. Annualised: 5.64 ร 2 = 11.28%.
Why not B? 11.50% results from using 8 periods instead of 12, counting the 4 years elapsed rather than the 6 years remaining. The bond matures in year 10; today is year 4; 6 years of coupons remain. Draw the timeline to see this.
Why not C? 11.71% results from using 6 periods (treating the remaining life as 6 annual periods) instead of applying the semiannual adjustment to get 12. Both N and I/Y must reflect the same frequency, semiannual throughout.
Q 5 of 6 โ ANALYZE
A UK company pays a constant annual dividend of GBP1.50 per share indefinitely. At a required return of 12%, the share price is GBP12.50. An analyst now forecasts that dividends will grow at 3% per year forever starting from the next dividend. Holding the required return constant at 12%, the new share price is closest to:
CORRECT: B
CORRECT: B, The Gordon Growth Model: PV = Dโ รท (r โ g). Next dividend: Dโ = 1.50 ร 1.03 = 1.545. PV = 1.545 รท (0.12 โ 0.03) = 1.545 รท 0.09 = GBP17.17. Adding growth raises the price because investors pay more for a growing income stream and the denominator (r โ g) shrinks from 0.12 to 0.09.
Why not A? GBP16.67 results from using Dโ = 1.50 directly: 1.50 รท 0.09 = 16.67. The Gordon Growth Model requires Dโ, the next expected dividend. Dโ has already been paid and is not a future cash flow.
Why not C? GBP12.50 is the original no-growth price: 1.50 รท 0.12. Adding positive growth raises both the numerator (Dโ > Dโ) and reduces the denominator (r โ g < r), making the new price unambiguously higher than 12.50.
Q 6 of 6 โ TRAP
Mylandia Corporation paid a dividend of CAD2.40 yesterday. Analysts forecast 3% constant annual dividend growth. Required return is 8%. A candidate computes the share price as CAD2.40 รท (0.08 โ 0.03) = CAD48.00. This result is:
CORRECT: B
CORRECT: B, The Gordon Growth Model is P = Dโ รท (r โ g), where Dโ is next year's dividend, not Dโ paid yesterday. Dโ = CAD2.40 has already been received by previous shareholders, it is no longer in the stock price. The buyer today receives Dโ = 2.40 ร 1.03 = 2.472 as their first payment. Correct price = 2.472 รท 0.05 = CAD49.44. The candidate's error understates the price by exactly one year of growth (CAD1.44).
Why not A? The recently paid dividend is a historical event, not a future cash flow. A buyer today does not receive Dโ, it was paid before their purchase. The stock price reflects only future dividends starting from Dโ. Using Dโ treats a past payment as if it were still upcoming, systematically understating the price whenever g > 0.
Why not C? The formula is consistent whether you use decimals (0.08, 0.03) or percentages as long as you are consistent throughout: 0.08 โ 0.03 = 0.05 is identical to 8% โ 3% = 5%. The candidate's error is Dโ versus Dโ, not unit convention.
---
Glossary
coupon
The periodic interest payment made by a bond to its lenders. Usually quoted as an annual percentage of face value; if semiannual, the stated rate is divided by 2 for each actual payment. Named after paper coupons that bondholders used to clip and redeem before electronic records.
par value
The face value of a bond, the amount returned to the lender at maturity. Exam questions typically quote bond prices per 100 of par (so par = 100, a price of 95.39 means MXN95.39 per MXN100 of face value). Also called face value or principal.
yield-to-maturity
The single discount rate that, when applied to all of a bond's future cash flows, produces the current market price. Think of it as the internal rate of return for holding the bond to maturity. It is the I/Y input in the BA II Plus for bond pricing.
annuity
A series of equal cash flows at regular intervals over a fixed number of periods. A mortgage, a coupon bond, a fixed-term pension, all are annuities.
perpetuity
A stream of equal payments that continues forever with no end date. Formula: PV = PMT รท r. The formula gives the value one period before the first payment.
Gordon Growth Model
A formula for the present value of a stock whose dividends grow at constant rate g forever: PV = Dโ รท (r โ g). Requires r > g. Dโ is next year's expected dividend, not the dividend already paid.
terminal value
In a two-stage dividend model, the value at the end of the high-growth phase, calculated using the Gordon Growth Model on the first stable-phase dividend. Captures all cash flows after the forecast horizon in a single number, then discounted back to today.
required return
The minimum annual return an investor demands to accept the risk of a particular investment. Used as r in the Gordon Growth Model and in perpetuity pricing. Also called required rate of return.
discount rate
The rate applied to future cash flows to express them in present-value terms. Higher discount rates make future cash flows worth less today. Same variable as r in TVM formulas.
semiannual adjustment
The three-step conversion required when a bond pays coupons twice per year: (1) divide the annual coupon by 2 to get PMT, (2) divide the annual YTM by 2 to get I/Y, (3) multiply years remaining by 2 to get N.
LO 1 Done โ
Ready for the next learning objective.
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Real-world applications and interview questions from top firms.
Quantitative Methods ยท The Time Value of Money in Finance ยท LO 2 of 3
You know the price and you know the cash flows. What return does this instrument actually promise?
LO 1 solved for price given the rate. This LO runs the same calculation in reverse: given the price and the cash flows, find the rate. One formula family. One new unknown.
โฑ 8min-15min
ยท
6 questions
ยท
HIGH PRIORITYANALYZE๐งฎ Calculator
Why this LO matters
LO 1 solved for price given the rate. This LO runs the same calculation in reverse: given the price and the cash flows, find the rate. One formula family. One new unknown.
INSIGHT
LO 1 solved for PV. Given the rate and the cash flows, find the price.
This LO solves for r. Given the price and the cash flows, find the rate.
The formula does not change. The unknown does. Every technique from LO 1 applies here, run in reverse.
The one reframe that makes this LO easy
Before any formula. Before any calculator key. Hold this idea:
A return is just a price in disguise.
The second-hand car offer
Imagine a friend offers you their car for USD8,000 today, saying you can sell it in three years for USD9,500. They have just told you the price (USD8,000), the future value (USD9,500), and the holding period (three years). They have not told you the annual return. But you can calculate it: what annual rate takes USD8,000 to USD9,500 over three years? The answer, about 5.9% per year, is embedded in the numbers they gave you. The return was always there. You just need to pull it out.
This is the only insight in LO 2. The rest is technique.
What the calculator actually solves when you press CPT I/Y
The BA II Plus solves for the one rate that makes all your inputs consistent.
You provide PV, FV, PMT, and N. The calculator finds the I/Y that satisfies all four simultaneously. There is no formula you need to memorise for this. The calculator does the algebraic work.
Four versions of the implied return problem
1
Single payment (discount bond or holding-period return). You know PV today, FV at a future date, N periods, and PMT = 0. CPT I/Y gives the annualised return. Formula: r = (FV/PV)^(1/N) โ 1.
2
Annuity with principal (coupon bond YTM). You know the current price (PV), all coupon payments (PMT), par value at maturity (FV), and periods remaining (N). CPT I/Y gives the yield-to-maturity.
3
Perpetuity (constant dividend stock). You know price (PV) and annual payment (PMT), and no end date. Rearrange PV = PMT/r to get: r = PMT/PV. Divide annual payment by current price.
4
Growing perpetuity (Gordon Growth Model, rearranged). You know price (PV), next dividend (Dโ), and growth rate (g). Solve for required return: r = Dโ/PV + g. Or solve for growth rate: g = r โ Dโ/PV.
When you are not given PV and FV, make them up
Some exam questions only give you a percentage change, not actual cash flow amounts.
"An investment must grow by 50% over nine years. What annual return does this require?"
No PV is given. No FV is given. Only the ratio matters.
The percentage growth shortcut
The teacher in these sessions demonstrates this directly: pick any convenient numbers that preserve the ratio. If the investment grows by 50%, choose PV = 100 and FV = 150. Or PV = 200 and FV = 300. The 50% relationship is maintained. The calculator gives the same I/Y either way, because I/Y only depends on the ratio FV/PV, not the absolute values. Under exam conditions, choosing simple round numbers saves calculation time.
Worked Example 1
How much must an investment grow per year to rise 50% in nine years?
An investment must grow by exactly 50% over a nine-year period. What annual return is required?
Single cash flow, solve for rate
T=0 T=1 ... T=9
โ โ
-100 +150
Pattern 1. PV and FV not given, use PV=100, FV=150 (50% growth).
Known: PV=100, FV=150, N=9, PMT=0. Unknown: r.
Step 1, Choose convenient numbers.
Growth of 50% means FV is 1.5 ร PV. Use PV = 100 and FV = 150.
โ ๏ธ Candidates who try to enter percentage signs or convert to decimals before pressing I/Y. The BA II Plus I/Y input and output is always in percent (4.61, not 0.0461). No conversion needed.
Pattern 1: Holding-period return on a discount bond
When you buy a bond today and sell it (or hold to maturity) at a known price, you have a Pattern 1 problem. One payment out, one payment in, N periods between them.
Single cash flow, solve for implied return
r = (FV/PV)^(1/N) โ 1
Where:
FV = what you receive (sale price or par at maturity)
PV = what you paid
N = number of periods you hold the bond
r = annualised return per period
Calculator shortcut: enter PV, FV, N, PMT=0, then CPT I/Y.
A Swiss Confederation zero-coupon bond has a par value of CHF100, a remaining time to maturity of 12 years, and a current price of CHF89. In three years' time, the bond is expected to have a price of CHF95.25. If purchased today, what is the bond's expected annualised return?
๐ง Thinking Flow โ Identify the holding period
The question asks
What annual return does an investor earn by buying at CHF89 today and selling at CHF95.25 after three years?
The key concept
Pattern 1, single payment in, single payment out. N is the holding period, not the total remaining maturity.
Step 1, Find the signal
"Expected to have a price of CHF95.25 in three years." FV = 95.25. PV = 89. N = 3. Not 12.
Step 2, Apply
r = (95.25/89)^(1/3) โ 1.
Step 3, Sanity check
95.25 > 89, so r must be positive. Result should be a small positive percent. โ
Buying and selling a discount bond, only the holding period matters
T=0 T=1 T=2 T=3 T=12
โ โ (bond matures here, irrelevant)
-89 +95.25
Pattern 1. Known: PV=89, FV=95.25, N=3. Unknown: r.
The 12-year maturity is irrelevant. You sell in year 3.
Step 1, Identify what the question is actually asking.
The investor buys today and sells in three years. The holding period is 3, not 12.
โ ๏ธ Using N = 12 gives 0.58% (answer A). Using FV = 100 (par value) instead of 95.25 (expected price in three years) gives 1.64% (answer B). Both wrong numbers appear as answer choices. The question says "expected to have a price of CHF95.25 in three years", that is your FV. Maturity is a distraction here.
Pattern 2: Yield-to-maturity on a coupon bond
Yield-to-maturity (YTM) is the single rate that, when used to discount all of a bond's future cash flows, produces the current market price. It is the implied return of holding the bond to maturity.
The calculator solves for it directly: enter the current price as PV, coupon as PMT, par as FV, periods remaining as N, CPT I/Y.
Worked Example 3
Grupo Ignacia, what YTM does the current price imply?
Grupo Ignacia issued 10-year corporate bonds four years ago. The bonds pay an annualised coupon of 10.7% on a semiannual basis. The current price is MXN97.50 per MXN100 of par value. What is the YTM?
Semiannual coupon bond, 6 years remaining = 12 periods
Issued 10 years ago, four years have elapsed: 10 โ 4 = 6 years remaining = 12 semiannual periods.
Step 2, Apply the semiannual adjustment.
PMT = 10.7 รท 2 = 5.35. N = 12. I/Y is unknown.
Step 3, CPT I/Y.
N=12, PV=โ97.50, PMT=5.35, FV=100, CPT I/Y = 5.64% per period.
Step 4, Annualise.
Annualised YTM = 5.64 ร 2 = 11.28%.
โ Answer: 11.28%. Exam answer: A.
๐งฎ BA II Plus Keystrokes
`[2nd][FV]`
clear TVM โ 0
`12[N]`
6 years ร 2 โ N = 12
`97.50[+/-][PV]`
current market price โ PV = โ97.50
`5.35[PMT]`
10.7% รท 2 per period โ PMT = 5.35
`100[FV]`
par at maturity โ FV = 100
`[CPT][I/Y]`
semiannual rate โ I/Y = 5.64
`ร2=`
annualise โ **YTM = 11.28%**
โ ๏ธ Using N = 8 (four years ร 2, counting elapsed time instead of remaining) gives 11.50%. Using N = 6 (treating remaining years as annual periods) gives 11.71%. Both appear as answer choices. Count years remaining to maturity: 10 โ 4 = 6 years = 12 semiannual periods.
Pattern 4: Implied return and implied growth for equity
The Gordon Growth Model has three variables: price, next dividend, required return, and growth rate. You already solved for price in LO 1. Now you solve for the remaining unknowns.
The model: PV = Dโ รท (r โ g). Rearrange for either unknown.
Gordon Growth Model, solving for r and g
To solve for required return r:
r = Dโ/PV + g
(dividend yield on next dividend + constant growth rate)
To solve for implied growth rate g:
g = r โ Dโ/PV
(required return minus dividend yield on next dividend)
Where:
Dโ = next expected dividend (NOT the dividend just paid)
PV = current share price
r = required return
g = constant growth rate
Condition: r > g
Worked Example 4
Mylandia, what required return does this price imply?
Mylandia Corporation stock trades at CAD60.00. Its most recent annual dividend (Dโ) was CAD2.40, paid yesterday. Analysts expect dividends to grow at a constant 3% per year forever. What is Mylandia's implied required return?
๐ง Thinking Flow โ Find the return embedded in the current price
The question asks
At a price of CAD60 and expected 3% dividend growth, what return must investors require?
The key concept
Gordon Growth Model rearranged for r. r = Dโ/P + g.
Step 1, Find Dโ
The question says "most recent payment was CAD2.40." That is Dโ. Dโ = 2.40 ร 1.03 = 2.472.
Step 2, Apply
r = 2.472/60 + 0.03 = 0.0412 + 0.03 = 7.12%.
Step 3, Sanity check
r > g? 7.12% > 3%. โ
โ Answer: 7.12%. Exam answer: C.
โ ๏ธ Watch for: Using Dโ = 2.40 instead of Dโ = 2.472 gives r = 2.40/60 + 0.03 = 7.00% (answer B). The signal phrase "most recent payment" identifies Dโ. Always grow it by (1 + g) before dividing.
Worked Example 5
NIFTY 50 index, implied growth vs analyst expectation
The Indian NIFTY 50 stock index trades at a forward price-to-earnings ratio (P/E) of 15. The index's expected dividend payout ratio in the next year is 50%. The required return is 7.50%. The analyst expects dividends to grow at 4.50% constantly. Should the analyst view the NIFTY 50 as overpriced or underpriced?
This uses a form of the Gordon Growth Model applied to an index.
๐ง Thinking Flow โ Compare implied P/E vs observed P/E
The question asks
Is the current index price consistent with the analyst's growth expectation?
The key concept
Use the Gordon Growth Model to find the P/E ratio justified by fundamentals. Compare to the observed P/E.
Step 1, Express the Gordon Growth Model in P/E form
Observed P/E = 15. Justified P/E = 16.67. The stock is worth more than its current price implies.
Step 4, Conclusion
Justified P/E (16.67) > Observed P/E (15). The index is underpriced.
โ Answer: The analyst should view the NIFTY 50 as underpriced. Exam answer: B.
Comparing implied return vs required return
This LO introduces a distinction that appears repeatedly in Equity Investments.
Implied return
Required return
Source
Calculated from current market price
Set by investor's risk tolerance and opportunity cost
What it represents
What the market expects this asset to earn
What you personally demand to accept this risk
How to use it
If implied return > required return โ buy. If implied < required โ sell or avoid.
In Gordon Growth Model
Solve for r given P, Dโ, g
Estimate based on risk model; use it to solve for P or g
โ ๏ธ
Watch out for this
The N-counting error, most common error on bond yield questions
Candidates confuse the bond's original term, the time since issue, and the remaining life. The calculator needs only one of these: remaining periods to maturity.
Wrong approach on Grupo Ignacia: the bond was issued 4 years ago and has a 10-year term. Candidate enters N = 8 (using 4 years ร 2 as if timing from now back to issue). This gives 11.50%.
Correct approach: 10-year bond, 4 years elapsed, so 6 years remain = 12 semiannual periods. N = 12. YTM = 11.28%.
The cognitive error: candidates track "when the bond was issued" rather than "when it matures." The timeline prevents this. Mark today on your timeline, mark maturity, count only what is between those two points.
๐ง
Memory Aid
CONTRAST ANCHOR
Implied return: what the price says the market is pricing in. Required return: what you personally demand. When they match, the price is fair. When the market implies more than you require, buy. When the market implies less, walk away.
Practice Questions ยท LO2
6 Questions LO2
Score: โ / 6
Q 1 of 6 โ REMEMBER
Which formula correctly gives the implied annual return on a zero-coupon bond bought today at PV and held for N years until it pays FV?
CORRECT: A
CORRECT: A, This is the Pattern 1 formula rearranged for r. Start from FV = PV ร (1 + r)^N. Divide both sides by PV: FV/PV = (1 + r)^N. Take the N-th root of both sides: (FV/PV)^(1/N) = 1 + r. Subtract 1: r = (FV/PV)^(1/N) โ 1. The exponent 1/N converts the total return over N years into an annual figure by finding the geometric average growth rate.
Why not B? r = FV/PV โ 1 gives the total return over the entire holding period, not the annualised return. If you buy at 89 and sell at 95.25, the total return is 95.25/89 โ 1 = 7.02% over three years. The annualised return is much smaller: 2.29%. Dividing the total return by N would give a simple average, not a compound annual rate. You want the compound rate.
Why not C? PMT/PV is the yield of a perpetuity, a stream of identical, infinite payments. A zero-coupon bond makes no intermediate payments; its entire return comes from one lump sum at maturity. Dividing a periodic payment by price does not apply here.
Q 2 of 6 โ UNDERSTAND
A bond is currently trading below its par value. Which statement correctly describes the relationship between the bond's coupon rate and its current yield-to-maturity?
CORRECT: B
CORRECT: B, When a bond trades below par, it offers investors a return above what the coupon alone provides. The below-par price creates a built-in capital gain: the bond pays par at maturity, but you paid less than par today. That capital gain supplements the coupon, making the total return (YTM) higher than the coupon rate alone. Investors drive the price below par precisely because the market demands a higher return than the coupon rate provides.
Why not A? If the coupon rate exceeded the YTM, the bond would trade above par, at a premium. A higher-than-market coupon is attractive, so investors bid the price up. A below-par price signals the opposite: the coupon is insufficient for current market conditions, forcing the price down to compensate.
Why not C? At-par pricing happens when coupon rate = YTM. The coupon delivers exactly the required return with no price adjustment needed. A below-par price means the coupon rate and YTM are not equal, the YTM is higher.
Q 3 of 6 โ APPLY
A Swiss Confederation zero-coupon bond has a current price of CHF89 and a remaining maturity of 12 years. An investor plans to hold it for 5 years and expects to sell it for CHF93.50. The expected annualised holding-period return is closest to:
CORRECT: B
CORRECT: B, Pattern 1: PV = 89, FV = 93.50, N = 5, PMT = 0. CPT I/Y = 0.96%. Alternatively: r = (93.50/89)^(1/5) โ 1 = (1.0506)^(0.2) โ 1 = 0.96%. The 5-year holding period is the only N that matters. The 12-year remaining maturity and the par value of CHF100 are both irrelevant to this calculation.
Why not A? 0.73% results from using N = 12 (the remaining maturity) instead of N = 5 (the planned holding period). The investor is not holding the bond to maturity, they plan to sell in five years. N must match the actual holding period between cash flows.
Why not C? 1.64% results from using FV = 100 (par value) and N = 12, the return if held to maturity at the par value. The question specifies an expected sale price of CHF93.50 in five years. The par value and full maturity are both distractors.
Q 4 of 6 โ APPLY+
Mylandia Corporation stock trades at CAD60.00. Its most recent annual dividend was CAD2.40, paid yesterday. Analysts expect dividends to grow at a constant 3% per year. The implied required return is 7.12%. An analyst argues the stock should trade at CAD80.00, using an 8% required return. Assuming the analyst's required return is correct, is Mylandia's stock currently overpriced or underpriced?
CORRECT: A
CORRECT: A, At r = 8% and g = 3%, the Gordon Growth Model gives P = Dโ/(rโg) = 2.472/(0.08โ0.03) = 2.472/0.05 = CAD49.44. The current price is CAD60.00, which is higher than the justified price. The stock is overpriced. Note that A says "underpriced" but then correctly identifies the direction: justified price < current price = overpriced. The analyst's required return implies the stock is worth less than it currently trades for.
Why not B? When the implied return (7.12%) is below the required return (8%), the market is offering less than you require. You would not buy. This means the stock is overpriced for an investor who requires 8%, not underpriced. The comparison goes: implied return < required return โ overpriced.
Why not C? CAD80.00 results from using Dโ = 2.40 (not Dโ = 2.472) and r โ g = 0.03 (not 0.05): 2.40/0.03 = 80. This misapplies the formula in two ways simultaneously. The correct denominator is r โ g = 0.08 โ 0.03 = 0.05, not 0.03. And the numerator requires Dโ, not Dโ.
Q 5 of 6 โ ANALYZE
An analyst observes two bonds with the same par value, maturity, and YTM. Bond A has a 3% coupon. Bond B has a 7% coupon. The analyst notes that Bond A has a lower price than Bond B. Which statement is correct?
CORRECT: C
CORRECT: C, Two bonds with the same YTM offer the same annual return but achieve it differently. Bond A's lower coupon means a lower price (higher discount from par). Its return comes mainly from the price rising toward par at maturity. Bond B's higher coupon means a higher price, and its return comes mainly from the coupon payments. The total return is identical, the source of return differs. This distinction is the foundation of bond duration: lower-coupon bonds are more sensitive to rate changes because more of their value is tied to that distant par payment.
Why not A? If two bonds share the same YTM, they offer the same implied return by definition. A lower price does not mean a higher YTM when the coupon also differs. The lower price of Bond A reflects the lower coupon, both bonds are fairly priced given the same market rate.
Why not B? Total cash flows differ. Bond A pays 3% coupon ร N periods plus par. Bond B pays 7% coupon ร N periods plus par. Bond B's total cash flows are substantially higher. The YTMs are equal because the present values of those different cash flow streams happen to be equal at the same discount rate.
Q 6 of 6 โ TRAP
Grupo Ignacia issued 10-year corporate bonds four years ago. The bonds pay an annualised coupon of 10.7% semiannually. The current price is MXN97.50 per MXN100 of par. A candidate enters N = 8 into the BA II Plus (reasoning: four years have passed and there are bonds with semiannual payments, so 4 ร 2 = 8). The resulting YTM is 11.50%. The correct YTM is:
CORRECT: B
CORRECT: B, The bond was issued 10 years ago with a 10-year term. Four years have elapsed. 6 years remain = 12 semiannual periods. N = 12, not 8. The candidate counted time elapsed (4 years ร 2 = 8) instead of time remaining (6 years ร 2 = 12). Using N = 12 gives the correct I/Y of 5.64% per period, annualised to 11.28%. The timeline makes this clear: mark today at the 4-year point, mark maturity at the 10-year point, count the gap.
Why not A? N = 8 means the candidate treated the bond as having only 4 years left rather than 6. This misprices the bond by assuming fewer coupon payments remain, fewer payments means a higher implied yield from the same price, explaining why 11.50% > 11.28%.
Why not C? N = 6 treats the remaining life as annual periods rather than semiannual periods. The semiannual adjustment requires N = 6 ร 2 = 12, not N = 6. Using 6 annual periods instead of 12 semiannual periods gives the wrong coupon count and the wrong yield. Both N and I/Y must be in semiannual terms throughout.
---
Glossary
yield-to-maturity
The single discount rate that makes the present value of all a bond's future cash flows equal to its current market price. It is the implied annual return of holding the bond to maturity. Enter current price as PV, coupon as PMT, par as FV, periods remaining as N, then CPT I/Y.
holding-period return
The total return earned from buying an asset at one price and selling it at another price, over a specific period. For a discount bond: r = (sale price/purchase price)^(1/N) โ 1. The holding period is the number of years between purchase and sale, not the bond's remaining maturity.
price-to-earnings ratio
The current share price divided by annual earnings per share. In the Gordon Growth Model for an index, P/E = (dividend payout ratio) รท (required return โ growth rate). A "forward P/E" uses next year's expected earnings rather than trailing earnings.
implied return
The annual return embedded in a current market price, calculated by solving the relevant TVM formula for r given price and cash flows. Contrasted with required return, which is what an investor demands from their own risk-return preferences.
required return
The minimum annual return an investor demands to accept the risk of a specific investment. Set by the investor's opportunity cost and risk tolerance. Used as r in the Gordon Growth Model when solving for price (LO 1). Compared to implied return in LO 2 to evaluate whether a security is attractively priced.
dividend yield
The next expected dividend (Dโ) divided by the current share price. In the Gordon Growth Model rearranged for r, dividend yield is the first component of the required return: r = (Dโ/P) + g. It represents the income component of the investor's return.
LO 2 Done โ
Ready for the next learning objective.
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How analysts use this at work
Real-world applications and interview questions from top firms.
Quantitative Methods ยท The Time Value of Money in Finance ยท LO 3 of 3
If two roads lead to the same destination, can one be cheaper? What happens when it is?
Two strategies that produce identical cash flows must sell for identical prices. When they don't, a riskless profit exists, and the act of chasing that profit eliminates it. This is how market prices get set.
โฑ 8min-15min
ยท
6 questions
ยท
HIGH PRIORITYUNDERSTAND๐งฎ Calculator
Why this LO matters
Two strategies that produce identical cash flows must sell for identical prices. When they don't, a riskless profit exists, and the act of chasing that profit eliminates it. This is how market prices get set.
INSIGHT
Two things that deliver identical results must cost the same.
If they do not, you can buy the cheaper one and simultaneously sell the more expensive one. You earn a guaranteed profit. No risk. No capital locked up. Everyone with a calculator will try the same trade, flooding demand for the cheap strategy and supply of the expensive one, until the price difference disappears.
This is no-arbitrage pricing. It is not a theory. It is a description of what markets do automatically.
The one principle behind everything in this LO
Two currency exchange windows
Imagine two currency exchange windows side by side at an airport. Both convert US dollars to euros. Window A offers USD/EUR 1.05. Window B offers USD/EUR 1.10. Every traveller with five minutes to spare would queue at Window B. The extra demand at B and the lack of demand at A would eventually force both windows to converge on the same rate. The gap cannot persist. This is arbitrage at work, not in financial markets, but at an airport. The mechanism is identical.
Cash flow additivity, the building block
1
The cash flow additivity principle states: the present value of any set of cash flows equals the sum of the present values of the individual cash flows. This sounds obvious. Its power is in what it implies. If Strategy A produces cash flows of {10, 5, 8} at T=1, T=2, T=3, and Strategy B produces cash flows of {12, 4, 7} at the same dates, their present values might or might not be equal. But you can find out by comparing the discounted sum of each. More importantly: if their present values are equal, an investor has no rational basis to prefer one over the other. And if their present values differ, the investor should choose the higher one, and the market will price them until they equalise.
Step 1: Testing whether two strategies are equivalent
Before applying any formula, know how to check equivalence.
[PROCESS: Comparing two investment strategies]
Calculate the present value of each strategy individually using the relevant discount rate.
Alternatively: subtract one strategy's cash flows from the other at each time period. If the resulting net cash flows have a present value of zero, the strategies are equivalent.
If PV(A) = PV(B): economically identical. No preference.
If PV(A) > PV(B): prefer A. Market will reprice B upward until equal.
Worked Example 1
Two investment opportunities, are they equivalent?
Two investment opportunities each require USD500,000 today and have a discount rate of 8%.
Cash flows (USD thousands)
T=0
T=1
T=2
T=3
Opportunity 1
โ500
195
195
195
Opportunity 2
โ500
225
195
160.008
Which opportunity should an investor prefer?
Strategy comparison, subtract one from the other
T=0 T=1 T=2 T=3
โ โ โ โ
0 โ30 0 +34.992
(Opportunity 1 minus Opportunity 2)
If PV of this difference = 0, the opportunities are equivalent.
Step 1, Subtract Opportunity 2 from Opportunity 1 at each period.
Period
Opp 1
Opp 2
Difference
T=0
โ500
โ500
0
T=1
195
225
โ30
T=2
195
195
0
T=3
195
160.008
+34.992
Step 2, Find PV of the difference at 8%.
PV = 0 + (โ30)/1.08 + 0/1.08ยฒ + 34.992/1.08ยณ
PV = 0 + (โ27.78) + 0 + 27.78 = 0
Step 3, Conclusion.
PV of the difference = 0. The two opportunities are economically identical.
โ Answer: The investor should be indifferent. Exam answer: C.
๐งฎ BA II Plus Keystrokes
`[CF]`
open CF worksheet โ CF0 =
`[2nd][CE/C]`
clear worksheet โ CF0 = 0
`0[ENTER]`
no T=0 net flow โ CF0 = 0
`โ(โ30)[+/-][ENTER]`
T=1 difference โ C01 = โ30
`โโ0[ENTER]`
T=2 difference โ C02 = 0
`โโ34.992[ENTER]`
T=3 difference โ C03 = 34.992
`[NPV]8[ENTER]`
discount rate โ I = 8
`โ[CPT]`
compute NPV โ **NPV = 0**
A result of zero confirms the two strategies are equivalent.
Step 2: Using no-arbitrage to find implied forward rates
A forward interest rate is the rate that will apply to a future investment period, as implied by current spot rates.
The two-year savings decision
You have GBP100 to invest for two years. You could lock in a two-year rate now. Or you could invest for one year, then reinvest for the second year at whatever rate exists then. For a risk-neutral investor, neither strategy should offer a free lunch over the other. If the two-year locked rate gives GBP107.12 at the end, the one-year-plus-reinvest strategy must also give GBP107.12 at the end, otherwise everyone would choose the better strategy. The implied reinvestment rate for the second year is called the forward rate. It is not observed directly. It is calculated from the two observable rates.
Implied one-year forward rate from spot rates
Fโ,โ = (1 + rโ)ยฒ / (1 + rโ) โ 1
Where:
rโ = current one-year spot rate
rโ = current two-year spot rate
Fโ,โ = implied one-year rate, starting in one year
Reading Fโ,โ: "the one-year rate, one year from now."
Logic: GBP100 invested for two years at rโ must equal
GBP100 ร (1 + rโ) ร (1 + Fโ,โ).
Solve for Fโ,โ.
Worked Example 2
Italian government bonds, implied forward rate
Italian one-year government bonds have a rate of 0.73%. Italian two-year government bonds have a rate of 1.29%. What is the breakeven one-year reinvestment rate, one year from now?
Two strategies for a two-year investment
T=0 T=1 T=2
โ โ
-GBP100 โโโ two-year rate 1.29% โโโ GBP102.60
-GBP100 โ one-year 0.73% โ GBP100.73 โ reinvest at Fโ,โ โ GBP102.60
Both strategies must produce the same result at T=2 (no-arbitrage).
Step 1, Confirm the two strategies must be equivalent.
Both invest GBP100 today. Both must produce the same amount in two years. Otherwise a riskless profit exists.
Step 2, Calculate using the formula.
Fโ,โ = (1 + 0.0129)ยฒ / (1 + 0.0073) โ 1
Fโ,โ = (1.02596) / (1.0073) โ 1
Fโ,โ = 1.01855 โ 1 = 1.85%
โ Answer: 1.85%. Exam answer: C.
โ ๏ธ Watch for: Candidates who average the two rates: (0.73 + 1.29)/2 = 1.01% (answer A). Or who just use rโ minus rโ = 0.56% and adjust somehow (answer B). The formula is a ratio, not an average. The forward rate must make the two-year return of both strategies equal, averaging does not achieve this.
Step 3: Forward exchange rates
The same no-arbitrage logic applies when you can choose to invest in two different currencies.
The dollar-yen decision
You have USD1,000. You could invest in a US Treasury for six months and collect USD1,010.05. Or you could convert to Japanese yen today, invest in a Japanese Treasury for six months, and convert back to US dollars at a rate you lock in today. If both strategies start with USD1,000 and both end with USD1,010.05, the forward exchange rate that achieves this is the no-arbitrage rate. If the forward rate is set differently, everyone would choose the better strategy, and the resulting flows would force it back to the no-arbitrage rate.
Covered interest rate parity, the no-arbitrage condition for FX
1
A forward exchange rate must be set so that investing domestically and investing abroad (converting at today's spot rate, investing, converting back at the forward rate) produce identical returns. The formula uses continuous compounding in the CFA curriculum: `` Forward rate (f/d) = Spot rate ร e^[(r_f โ r_d) ร T] Where: f/d = foreign currency per domestic currency Spot rate = today's exchange rate r_f = foreign risk-free rate (continuously compounded) r_d = domestic risk-free rate (continuously compounded) T = time in years e = Euler's number โ 2.71828 Note on e: the [2nd][LN] key on the BA II Plus computes e^x. Type [2nd][LN], enter the exponent, press [=]. ``
FORWARD REFERENCE
Covered interest rate parity, what you need for this LO only
The full treatment of forward exchange rates, including uncovered interest rate parity and purchasing power parity, is in Economics. For this LO: identify two equivalent strategies (invest domestically vs invest abroad and convert), apply the no-arbitrage condition that both must produce the same home-currency outcome, and solve for the forward rate that equalises them. The formula above is sufficient.
โ Economics
Worked Example 3
Euro-dollar forward exchange rate
The current exchange rate is USD/EUR 1.025 (USD1.025 per EUR1.00). The one-year euro risk-free rate is 0.75%. The one-year US dollar risk-free rate is 3.25%. Both rates are continuously compounded. What is the no-arbitrage one-year forward exchange rate?
๐ง Thinking Flow โ Two strategies, same starting currency
The question asks
What forward rate makes investing in USD vs EUR produce identical outcomes?
The key concept
Start with EUR1.00. Invest domestically at 0.75% or convert to USD, invest at 3.25%, and convert back at the forward rate.
Step 1, Domestic strategy (EUR)
EUR1.00 invested at 0.75% for one year grows to EUR1.00 ร e^0.0075 = EUR1.00752.
Step 2, Foreign strategy (USD)
EUR1.00 at spot rate USD/EUR 1.025 = USD1.025. Invested at 3.25%: USD1.025 ร e^0.0325 = USD1.025 ร 1.03304 = USD1.05886.
Step 3, Find the forward rate that equates outcomes
US rates (3.25%) > Euro rates (0.75%). USD earns more, so USD should depreciate forward (forward rate per EUR should be higher than spot). 1.051 > 1.025. โ
โ Answer: USD/EUR 1.051. Exam answer: A.
โ ๏ธ Watch for: Candidates who leave the forward rate equal to the spot rate (USD/EUR 1.025, answer B). This is the "no-adjustment" trap. The forward rate must adjust to reflect the interest rate differential. Candidates who invert the formula get USD/EUR 0.975 (answer C), placing the lower-yield currency at a forward premium instead of the higher-yield currency.
Step 4: Option pricing using no-arbitrage
An option gives its buyer the right, but not the obligation, to buy (a call option) or sell (a put option) an asset at a pre-set price on a specific date.
You do not need to understand options deeply for this LO. You need to understand one idea: a portfolio containing the option and a certain number of units of the underlying asset can be constructed so that its value is identical in all future scenarios. Because this replicating portfolio is risk-free, it must earn the risk-free rate.
The insurance analogy
Imagine you sell someone insurance on their car. If the car is damaged, you pay them the repair cost. If it is not damaged, you pay nothing. Now suppose someone offers to buy your insurance contract for a price you have not yet set. You can figure out the fair price by finding the combination of assets that replicates your insurance obligation, a mix of assets whose value is the same whether or not the car is damaged. Price that combination, and you have priced your insurance. This is exactly how option pricing works.
Replicating portfolio logic, three steps
1
Identify the two future outcomes. The underlying asset's price will go up to S_u or down to S_d in one period.
2
Calculate the option value in each scenario. For a call: max(0, S โ X) where X is the exercise price. For a put: max(0, X โ S).
3
Find the hedge ratio and construct the risk-free portfolio. The hedge ratio h tells you how many units of the underlying to hold. With h units of the underlying and one option sold, the portfolio value is identical in both the up and down scenarios. Discount that certain value at the risk-free rate to find the option's price today.
Worked Example 4
Call option, replicating portfolio pricing
A stock has a current price of CNY40. In one period, the price will either rise to CNY56 or fall to CNY32. An investor sells a call option with an exercise price of CNY50. The risk-free rate is 5%. The given hedge ratio is 0.25 units of stock per option sold. What is the fair price of the call option?
Two future scenarios for the replicating portfolio
โ S_up = 56, call pays max(0, 56โ50) = 6
CNY40 today
โ S_down = 32, call pays max(0, 32โ50) = 0
Portfolio: sell one call at price cโ, buy 0.25 units of stock.
Portfolio value up: 0.25 ร 56 โ 6 = 8
Portfolio value down: 0.25 ร 32 โ 0 = 8
Same value in both scenarios โ portfolio is risk-free.
Step 1, Set up the portfolio.
Buy 0.25 units of stock, sell one call at price cโ.
Down scenario: 0.25 ร 32 โ 0 = 8. Identical. The portfolio is risk-free.
Step 3, Discount at the risk-free rate.
Vโ = 8 / 1.05 = 7.619.
Step 4, Solve for cโ.
10 โ cโ = 7.619
cโ = 10 โ 7.619 = CNY2.38.
โ Answer: The fair call option price is CNY2.38.
Worked Example 5
Portfolio value check, Q12 exam question
A stock currently trades at USD25. In one year, it will either rise to USD35 or fall to USD15. An investor sells a call option on the stock with an exercise price of USD25, and simultaneously buys 0.5 units of the stock. What is the investor's portfolio value in each scenario?
Up scenario: Stock rises to USD35. Call buyer exercises. Investor pays USD35 โ USD25 = USD10.
Down scenario: Stock falls to USD15. Call buyer does not exercise. Investor pays USD0.
Step 2, Portfolio value = stock holdings โ option liability.
Up: 0.5 ร 35 โ 10 = USD7.50.
Down: 0.5 ร 15 โ 0 = USD7.50.
Step 3, Conclusion.
The portfolio has a value of USD7.50 regardless of which scenario occurs. It is a risk-free portfolio.
โ Answer: USD7.50 in both scenarios. Exam answer: A.
โ ๏ธ
Watch out for this
The forward rate averaging error
Candidates who understand that a forward rate is "somewhere between" two spot rates often average them. This gives a plausible-looking answer that is wrong.
Wrong approach: The one-year rate is 0.73% and the two-year rate is 1.29%. The forward rate must be around (0.73 + 1.29)/2 = 1.01%. Exam answer option: 1.01%.
Correct approach: Fโ,โ = (1.0129)ยฒ / (1.0073) โ 1 = 1.0260 / 1.0073 โ 1 = 1.85%.
Why averaging fails: the forward rate connects two compound-growth paths, not two arithmetic averages. Investing USD100 for two years at rโ earns compound interest twice. Reinvesting at a simple average ignores the compounding in the first year. The correct formula squares rโ (two years of compounding) and divides by (1 + rโ) (removing the first year of compounding), leaving only the second year's implied rate.
๐ง
Memory Aid
ACRONYM
PAIR, the no-arbitrage test for any two strategies:
P
Payoff โ Do both strategies produce the same payoff at the same future date?
A
Arbitrage โ If yes, and prices differ, a riskless profit exists.
I
Implied rate โ The rate (or price) that eliminates the arbitrage is the no-arbitrage rate.
R
Replicating portfolio โ Construct the combination of assets that matches the target payoff, then price it.
When you see a question comparing two strategies, run PAIR in sequence. If payoffs match, prices must match. If prices differ, find the implied rate that closes the gap.
Practice Questions ยท LO3
6 Questions LO3
Score: โ / 6
Q 1 of 6 โ REMEMBER
The cash flow additivity principle states that the present value of a set of future cash flows is equal to:
CORRECT: A
CORRECT: A, Each future cash flow can be discounted to today's value independently. Summing these individual present values produces the same result as discounting the total future amount at once, because discounting is a linear operation. This linearity is what makes the principle useful: you can split complex cash flow streams into parts, value each part, and add them. The whole equals the sum of the parts, but in present-value terms, not in future-value terms.
Why not B? Summing undiscounted cash flows ignores the time value of money. A dollar received today is not equivalent to a dollar received in five years, the future dollar is worth less. Simply adding future amounts treats all time periods as identical, which is precisely what present-value discounting is designed to correct.
Why not C? There is no single "average rate" that produces the correct result when cash flows arrive at different dates. Each cash flow must be discounted by (1+r)^t using its own t. The correct calculation is the sum of PVs, not a present value computed from a single average.
Q 2 of 6 โ UNDERSTAND
An investor can lock in a two-year rate of 3.50% today, or invest for one year at 2.50% and reinvest for the second year. Under the no-arbitrage condition, which statement is correct?
CORRECT: C
CORRECT: C, No-arbitrage requires identical outcomes. GBP100 invested for two years at 3.50% produces GBP107.12. GBP100 invested at 2.50% for one year gives GBP102.50. For the second year, the reinvestment rate must take GBP102.50 to GBP107.12. That rate is Fโ,โ = (1.035)ยฒ / (1.025) โ 1 = 4.51%. Neither the two-year rate (3.50%) nor the average (3.00%) achieves this.
Why not A? If the reinvestment rate were 3.50%, the two-path strategy would give GBP100 ร 1.025 ร 1.035 = GBP106.09. That is less than GBP107.12 from the locked-in two-year rate. The two strategies do not produce the same outcome. An arbitrage would exist.
Why not B? Averaging to 3.00% would give GBP100 ร 1.025 ร 1.030 = GBP105.58. Even further from GBP107.12. The forward rate cannot be a simple average because the first year's compounding creates a base effect that the second year multiplies, not adds to.
Q 3 of 6 โ APPLY
Italian one-year government debt has an interest rate of 0.73%. Italian two-year government debt has an interest rate of 1.29%. The implied one-year reinvestment rate, one year from now, is closest to:
CORRECT: C
CORRECT: C, Apply Fโ,โ = (1 + rโ)ยฒ / (1 + rโ) โ 1 = (1.0129)ยฒ / (1.0073) โ 1. Numerator: (1.0129)ยฒ = 1.02596. Divide by 1.0073: 1.02596 / 1.0073 = 1.01854. Subtract 1: Fโ,โ = 1.85%. This is the breakeven reinvestment rate that makes investing at the two-year rate equivalent to rolling two one-year investments.
Why not A? 1.01% is roughly the average of 0.73% and 1.29%: (0.73 + 1.29)/2 โ 1.01. Averaging does not account for compounding. The forward rate is a compound calculation, not an arithmetic mean.
Why not B? 1.11% results from a common mis-step: computing (rโ โ rโ) ร 2 = (0.0129 โ 0.0073) ร 2 = 1.12%. This subtracts rates and doubles, which is not the correct formula. The no-arbitrage forward rate uses a ratio of compound factors, not a difference of simple rates.
Q 4 of 6 โ APPLY+
The current USD/EUR spot rate is 1.025. The one-year continuously compounded risk-free rate is 3.25% for USD and 0.75% for EUR. The no-arbitrage one-year forward rate is closest to:
CORRECT: A
CORRECT: A, Start with EUR1.00. Domestic path (EUR): EUR1.00 ร e^0.0075 = EUR1.00752 in one year. Foreign path (USD): EUR1.00 ร 1.025 = USD1.025, then USD1.025 ร e^0.0325 = USD1.05886 in one year. Forward rate = USD1.05886 / EUR1.00752 = USD/EUR 1.051. Sanity check: US rates are higher, so USD earns more. The forward rate must be higher than spot (more USD per EUR forward than spot) to prevent arbitrage. 1.051 > 1.025. โ
Why not B? USD/EUR 1.025 is the current spot rate. It ignores the interest rate differential. If the forward rate equalled the spot rate, an investor could borrow euros, convert to dollars, invest at 3.25%, convert back at the same rate, and profit from the rate differential risk-free. The forward rate must adjust to close this opportunity.
Why not C? USD/EUR 0.975 inverts the direction of the adjustment. Because the US rate (3.25%) exceeds the euro rate (0.75%), the dollar earns more, and must depreciate forward relative to the euro (the forward price of a euro in USD must be higher than today's price, not lower). A forward rate below spot implies the dollar appreciates, which would itself create an arbitrage.
Q 5 of 6 โ ANALYZE
A stock trades at USD25. In one year, it will be worth either USD35 (up 40%) or USD15 (down 40%). An investor sells a call option with an exercise price of USD25 and simultaneously buys 0.5 units of the underlying stock. Which statement is correct about this investor's portfolio in one year?
CORRECT: A
CORRECT: A, Up scenario: stock worth USD35, call buyer exercises the USD25 call, costing the investor USD10 (= 35 โ 25). Portfolio = 0.5 ร 35 โ 10 = USD7.50. Down scenario: stock worth USD15, call expires worthless, investor pays nothing. Portfolio = 0.5 ร 15 โ 0 = USD7.50. Identical in both scenarios. This is the definition of a replicating portfolio: a combination of assets engineered to produce the same payoff regardless of which future state occurs. The value is risk-free.
Why not B? The sold call option offsets the stock's movement. When the stock rises, the call obligation rises by an equal and opposite amount (the buyer exercises). The hedge ratio (0.5 units of stock per call) is specifically chosen so that the option obligation exactly cancels the stock gain in the up scenario. This cancellation is the whole point of the replicating portfolio construction.
Why not C? USD17.50 results from ignoring the call option entirely and just computing 0.5 ร 35 = USD17.50 (stock-only up scenario). The sold call imposes a USD10 liability in the up scenario: the investor must deliver the stock at USD25 when it is worth USD35. Netting: 17.50 โ 10 = 7.50. Options have real cash flow consequences. They are not zero.
Q 6 of 6 โ TRAP
An investor observes a stock at CNY40 and constructs a replicating portfolio. The stock rises to CNY56 (call value = CNY6) or falls to CNY32 (call value = CNY0). The hedge ratio is 0.25. The risk-free rate is 5%. A candidate calculates the call option price as CNY2.38. A second candidate calculates it as CNY1.52 by discounting the average call value. Which is correct?
CORRECT: B
CORRECT: B, The replicating portfolio method: portfolio value at T=1 is CNY8 in both scenarios (0.25 ร 56 โ 6 = 8 and 0.25 ร 32 โ 0 = 8). Discount at 5%: present value = 8/1.05 = 7.619. Solve for call price: 0.25 ร 40 โ c = 7.619, so c = 10 โ 7.619 = CNY2.38. The second candidate averages the call payoffs: (6 + 0)/2 = 3, then discounts: 3/1.05 = CNY2.86. This is wrong, it uses an assumed 50/50 probability, which is not given and not required.
Why not A? Averaging payoffs assumes equal probability of the up and down scenarios. The no-arbitrage method does not require any assumption about probabilities. It prices the option by finding what it would cost to replicate its payoff exactly using the stock and a risk-free asset. The probability of each scenario is irrelevant, the replicating portfolio costs what it costs regardless of which scenario occurs.
Why not C? The replicating portfolio price is unique and probability-independent. There is no valid set of probability assumptions under which the second candidate's answer equals the correct no-arbitrage price. The no-arbitrage price is not an expected value, it is a replication cost. These two concepts produce the same number only under specific conditions (risk-neutral pricing) derived from the model, not from assumed probabilities.
---
Glossary
no-arbitrage
The condition that no strategy can generate a guaranteed profit with zero net investment and zero risk. In efficient markets, arbitrage opportunities are eliminated quickly as investors exploit them. Prices are set at levels where no arbitrage exists.
cash flow additivity
The principle that the present value of any combination of future cash flows equals the sum of the present values of the individual cash flows. Follows from the linearity of discounting. Used to compare strategies and find implied prices.
forward interest rate
The interest rate implied by current spot rates for an investment period that starts in the future. For a one-year rate starting in one year: Fโ,โ = (1 + rโ)ยฒ / (1 + rโ) โ 1. It is not observed directly, it is calculated from observable spot rates.
option
A contract giving the buyer the right, but not the obligation, to buy or sell an asset at a specified price on a specific future date. The buyer pays a premium (the option price) for this right. Full treatment in Derivatives.
call option
An option granting the right to buy an asset at the exercise price. The call has value when the market price rises above the exercise price: value = max(0, market price โ exercise price).
put option
An option granting the right to sell an asset at the exercise price. The put has value when the market price falls below the exercise price: value = max(0, exercise price โ market price).
exercise price
The pre-agreed price at which an option buyer can buy (call) or sell (put) the underlying asset. Also called the strike price.
replicating portfolio
A combination of assets whose future cash flows exactly match those of another asset or contract under all future scenarios. By no-arbitrage, the replicating portfolio and the target must have the same price today.
hedge ratio
In option pricing, the number of units of the underlying asset to hold per option written, such that the portfolio value is the same in all future scenarios. The hedge ratio is chosen so that changes in the option's value exactly offset changes in the underlying asset's value. In the worked example, the hedge ratio was 0.25, one unit of stock per four call options sold.
LO 3 Done โ
You have completed all learning objectives for this module.
๐ PRO Feature
How analysts use this at work
Real-world applications and interview questions from top firms.
Quantitative Methods ยท The Time Value of Money in Finance ยท Job Ready
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Valuation, performance reporting, and derivatives pricing in institutional finance
๐ This is a PRO session. You are previewing it. Unlock full access to get all LO sections, interview questions from named firms, and one-line positioning statements.
Why this session exists
Why this session exists: The exam tests whether you can draw a timeline and press the right calculator keys. Interviews test whether you know which timeline to draw, which direction the cash flows, and when the answer tells you to buy or sell. This session bridges that gap.
Fixed income analysts, equity portfolio managers, performance attribution teams, and derivatives traders all use the same four cash flow patterns every day. This module is the vocabulary they share.
LO 1
Cash flow patterns: pricing bonds, mortgages, and dividend-paying stocks
How analysts use this at work
Fixed income analysts at firms like Vanguard and PIMCO price bonds dozens of times per day as market conditions shift. They receive a bond with a stated coupon, a maturity date, and a current market yield. They must immediately recognise the pattern โ annuity plus principal return โ and enter three adjustments before pressing any key: halve the coupon rate, halve the yield, double the periods remaining. A failure to make any one of these adjustments produces a number that appears in the wrong-answer choices. The decision to buy or sell the bond turns on whether the calculated price is above or below the market quote.
Equity analysts at firms like Fidelity and T. Rowe Price use the perpetuity and Gordon Growth formulas to value dividend-paying stocks. When a preferred share promises a fixed annual dividend, the price is simply that dividend divided by the required return. When a common share's dividend grows at a constant rate, the price is next year's dividend divided by the spread between required return and growth rate. The analyst's output is a buy, hold, or sell recommendation โ and the number on the calculator is the backbone of that recommendation.
Interview questions
BlackRock Investment Analyst "A bond pays a 7% annual coupon and matures in 8 years. It currently trades at USD98.50 per USD100 of par. The market yield is 7.3%. Walk me through what the price tells you and whether the coupon is attractive relative to current market conditions."
Vanguard Portfolio Analyst "A stock pays an annual dividend of USD2.00. Analysts expect that dividend to grow at 4% per year forever. The required return is 9%. What is the stock worth? And if the current market price is USD42, what does that tell you about where the market thinks the required return sits?"
Fidelity Equity Research Associate "Company A is a no-growth preferred share paying USD3.60 per year forever, trading at USD40. Company B is a common share paying USD3.60 this year with dividends expected to grow at 3% per year forever, also trading at USD40. Both are at the same price. Which, if either, would you recommend buying and why?"
One-line to use in your interview
Interviewers listen for industry-specific language. It signals you understand the concept, not just the definition. Use the plain English version to adapt it in your own words.
In practice, I always draw the timeline before entering anything into the calculator, because the sign convention and the timing of the first payment are where most pricing errors happen โ and on a bond, those errors directly affect whether I recommend buying at a discount or avoiding a premium.
In plain English
Before I price anything, I sketch out when cash arrives and who is paying. Getting the direction wrong โ treating money coming in as money going out โ gives the wrong answer with no warning. For bonds, that means checking whether coupons are paid semiannually before I touch the keys.
LO 2
Implied returns: what a bond's price and a stock's valuation actually promise
How analysts use this at work
Bond portfolio managers at firms like PIMCO and Oaktree Capital constantly compare what a bond promises to earn against what they require. They start with the market price, count the periods remaining to maturity, and solve for the yield-to-maturity. That number answers a single question: does this bond deliver more than I need, or less? When YTM exceeds the portfolio's required yield, the bond is attractively priced. When YTM falls below the required yield, the manager passes or recommends selling. The calculation is the same every time, but the comparison is what drives the trade.
Equity analysts at firms like Wellington Management and Morgan Stanley use the Gordon Growth Model to assess whether a stock is fairly valued. Given a current price and an assumed growth rate, they solve for the implied required return. If the implied return exceeds what the firm's risk model demands, the stock is underpriced. If it falls short, the stock is overpriced. This comparison โ implied return against required return โ is the foundation of every buy-side valuation recommendation.
Interview questions
PIMCO Fixed Income Analyst "A 10-year corporate bond was issued 3 years ago with a 9% annual coupon paid semiannually. It now trades at USD97.25 per USD100 of par. Walk me through how you determine what yield the market is demanding on this bond."
Goldman Sachs Investment Analyst "A stock trades at USD75. The most recent annual dividend was USD2.25, paid last month. Analysts expect dividends to grow at 3.5% per year forever. What implied return does the current price represent? And at what price would the implied return equal 8%?"
Wellington Management Portfolio Manager "Two bonds with identical maturity dates and identical yields-to-maturity trade at different prices. Bond A has a 4% coupon. Bond B has an 8% coupon. Explain how two instruments with the same YTM can have different prices, and what this means for how you would position your portfolio if you expected interest rates to fall."
One-line to use in your interview
Interviewers listen for industry-specific language. It signals you understand the concept, not just the definition. Use the plain English version to adapt it in your own words.
When I see a bond trading below par, my instinct is to check whether the market is demanding a yield that reflects genuine credit risk or whether the below-par price is just a function of the coupon being below current market rates โ those are different problems that lead to different investment conclusions.
In plain English
If a bond trades below what it pays at maturity, I need to know why. There are two possible reasons. One is that the company might not repay โ a serious concern. The other is simply that the bond's fixed interest payment is lower than what new bonds are paying today โ not a concern at all, just the market adjusting for rate changes. I have to know which one I am looking at before I can decide whether to buy.
LO 3
No-arbitrage: how forward rates and option prices are derived from observable market data
How analysts use this at work
Interest rate strategists at firms like Deutsche Bank and BNP Paribas use the no-arbitrage condition to calculate implied forward rates every morning as part of their rates commentary. Given the one-year spot rate and the two-year spot rate, they compute the implied one-year rate starting one year from now. That forward rate tells them what the market expects short-term rates to be in the future. If their view diverges from the implied forward, they position accordingly โ long or short rate expectations. The calculation is a direct application of the principle that two equivalent investment strategies must offer the same return.
Derivatives traders at firms like Citadel and Two Sigma use the replicating portfolio approach to price options. They identify the combination of the underlying stock and a risk-free bond that produces the same payoff as the option in every future scenario. Since that portfolio is risk-free, it must earn the risk-free rate. The cost of assembling that portfolio today is the option's fair price. If the market price differs from this fair value, the trader buys the cheap side and sells the expensive side, collecting the difference with no net exposure.
Interview questions
Deutsche Bank Rates Strategist "The 1-year spot rate is 1.2% and the 2-year spot rate is 2.1%. What is the implied 1-year rate starting in one year, and why is it not simply the average of those two rates?"
Citadel Derivatives Trader "A stock trades at USD50. In one year it will either rise to USD70 or fall to USD35. The risk-free rate is 4%. A call option with an exercise price of USD50 is trading at USD8.50 in the market. Is the option fairly priced, overpriced, or underpriced? Walk me through the replication logic."
BNP Paribas FX Analyst "The USD/EUR spot rate is 1.08. The 1-year USD risk-free rate is 5.0% continuously compounded, and the 1-year EUR risk-free rate is 3.0% continuously compounded. What should the 1-year forward rate be, and what does it tell you about how the market expects the dollar to move relative to the euro over the next year?"
One-line to use in your interview
Interviewers listen for industry-specific language. It signals you understand the concept, not just the definition. Use the plain English version to adapt it in your own words.
In derivatives pricing, I do not need to know whether the stock will go up or down. I need to know whether the option price in the market matches the cost of assembling a portfolio that replicates the same payoff regardless of which direction the stock moves โ and if it does not, I know where the arbitrage edge is.
In plain English
An option's price is not a guess about the future. It is the cost of building a combination of the stock and a risk-free investment that performs exactly the same as the option in every possible scenario. If the market charges more than that cost, I sell the option. If it charges less, I buy it. The direction of the stock does not matter โ only the mismatch between the market price and the replication cost matters.
LO 4
Cash flow additivity: comparing strategies and detecting mispriced instruments
How analysts use this at work
Investment consultants at firms like Mercer and Aon help pension fund clients select from competing investment managers. Each manager presents a different cash flow strategy โ some front-load returns, others back-load them. The consultant must determine which strategy delivers the highest present value at the client's required return. They do this by discounting each cash flow stream independently and comparing the totals. The one with the higher present value is the better choice. This comparison is not optional โ it is the quantitative basis of the manager selection recommendation presented to the pension trustee board.
Risk analysts at firms like State Street and Bank of America use cash flow additivity to decompose complex structured products into their component parts. If a structured note promises a return linked to an equity index plus a fixed income floor, the analyst discounts each component separately, then sums the present values. If the structured product's market price differs from that sum, it is either mispriced or the embedded optionality has been misvalued. That gap is the starting point of every trade idea and every risk flag raised with the desk head.
Interview questions
Mercer Investment Consultant "Two private credit strategies offer the same initial investment of USD10 million. Strategy 1 delivers USD3.5 million in year 1, USD4 million in year 2, and USD4.5 million in year 3. Strategy 2 delivers USD5 million in year 1, USD4 million in year 2, and USD2.8 million in year 3. At a 7% discount rate, which strategy offers the higher present value and by how much?"
State Street Risk Analyst "You are given two bonds that have identical YTMs and identical maturities. Bond A has a 5% coupon and trades at USD97. Bond B has a 7% coupon and trades at USD103. A colleague argues these prices are inconsistent because they have the same YTM. How would you explain to them why this is actually consistent with no-arbitrage pricing?"
Aon Senior Investment Advisor "A client is comparing two annuity products. Product 1 pays USD50,000 per year for 20 years starting immediately. Product 2 pays USD60,000 per year for 20 years but the first payment does not begin for 5 years. At a 5% discount rate, which product offers the higher present value, and how does the timing of the first payment affect your conclusion?"
One-line to use in your interview
Interviewers listen for industry-specific language. It signals you understand the concept, not just the definition. Use the plain English version to adapt it in your own words.
When I compare two investment strategies, I always put them on the same timeline and discount every cash flow independently before adding them up โ because mixing undiscounted cash flows with discounted ones, or counting the same time period twice, produces an answer that looks precise but is actually wrong.
In plain English
Money received sooner is worth more than money received later. When I compare two strategies, I shrink all the future cash flows back to today using the same rate before I add them together. If I skip that step and just add up the dollar amounts, I am pretending a dollar in year ten is worth the same as a dollar today โ and that is a mistake a pension fund trustee would notice immediately.