You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 2 of Quantitative Methods 6–9% of the exam

Time Value of Money in Finance

6 lessons. About 36 minutes in cram mode, 170 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
Pricing bonds: zero-coupon, coupon and perpetual
calculate 6 min cram, 30 min full
2
Annuities and level-payment loans
calculate 6 min cram, 24 min full
3
Valuing a stock from its dividends
calculate 6 min cram, 30 min full
4
Implied return and implied growth
calculate 6 min cram, 28 min full
5
Cash flow additivity and implied forward rates
explain 6 min cram, 26 min full
6
Forward exchange rates and option values by no arbitrage
explain 6 min cram, 32 min full

What the exam expects you to do

Calculate and interpret the present value (PV) of fixed-income and equity instruments based on expected future cash flows.Pricing bonds: zero-coupon, coupon and perpetual
Calculate and interpret the implied return of fixed-income instruments and required return and implied growth of equity instruments given the present value (PV) and cash flows.Implied return and implied growth
Explain the cash flow additivity principle, its importance for the no-arbitrage condition, and its use in calculating implied forward interest rates, forward exchange rates, and option values.Cash flow additivity and implied forward rates

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

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How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Pricing bonds: zero-coupon, coupon and perpetual

Known

MemorizeCoupon rate against yield

Coupon rate vs yield	Price vs face value	Called
Above	Above	Premium
Equal	Equal	Par
Below	Below	Discount

Memory hook The bond is paid in coupons, the market asks for the yield. Pay more than asked and buyers pay extra.

Know it cold Compare the coupon with the yield, never the yield with zero. A low yield alone does not make a discount.

MemorizeSemiannual bond on the calculator

Memory hook Everything per period: twice as many periods, half the rate, half the coupon.

Know it cold $N = 2 \times$ years, $I/Y =$ yield $\div\, 2$ , $PMT =$ coupon $\div\, 2$ , $FV = 100$ .

Key terms

zero-coupon bond Memorize
discount instrument Use it
coupon instrument Use it
perpetual bond Memorize
perpetuity Memorize
market discount rate Use it
yield to maturity Use it
premium bond Memorize

Three cash-flow patterns, three pricing shortcuts

A bond pricethe present value of every promised payment, at the market yield

Zero-coupon (discount) bond Memorize one payment, the face value at maturity
- PV = FV ÷ (1 + r)ᵗ always below face value unless the yield is negative
Coupon bond Memorize coupons every period, then the face value
- TVM keys: N, I/Y, PMT, FV semiannual: double N, halve I/Y and PMT
Perpetual bond Memorize coupons for ever, no face value to repay
- PV = PMT ÷ r a perpetuity

Coupon rate against yield decides the price

Compare the coupon rate with the market yield

Coupon rate above yield Memorize price above par: a premium
Coupon rate equal to yield Memorize price equals par
Coupon rate below yield Memorize price below par: a discount

Formulas

Price of a zero-coupon bond Memorize

PV = \dfrac{FV}{(1 + r)^{t}}

Quick example

Face value 100, matures in 8 years, yield 4.5%. 100 ÷ 1.045⁸ = 70.32.

On your calculator

2NDCLR TVM with P/Y set to 1
8N4.5I/Y100FV
CPTPV −70.32

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · FV future value · CPT compute · PV present value

fCLEAR FIN
8n4.5i100FV
PV −70.32

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · FV future value · PV present value

Price of a zero-coupon bond (continuous compounding) Know the idea

PV = FV \times e^{-rt}

Quick example

Face value 100, 8 years, continuously compounded yield 4.5%. 100 × e^(−0.36) = 69.77.

On your calculator

0.36+|-2NDe^x×100= 69.77

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.36CHSge^x100× 69.77

CHS change sign · g blue shift key · e^x e to the power x

Price of a coupon bond Memorize

PV = \sum_{k=1}^{N} \dfrac{PMT}{(1 + r)^{k}} + \dfrac{FV}{(1 + r)^{N}}

Quick example

5 years, 3.5% annual coupon, face value 100, yield 5%. N = 5, I/Y = 5, PMT = 3.5, FV = 100 gives 93.51, a discount.

On your calculator

2NDCLR TVM
5N5I/Y3.5PMT100FV
CPTPV −93.51

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

fCLEAR FIN
5n5i3.5PMT100FV
PV −93.51

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Price of a perpetual bond Memorize

PV = \dfrac{PMT}{r}

Quick example

Pays 4.25 a year for ever, yield 5%. 4.25 ÷ 0.05 = 85.00.

On your calculator

4.25÷0.05= 85

4.25ENTER0.05÷ 85

ENTER enter (on the HP, puts the number on the stack)

Annuities and level-payment loans

Known

MemorizeAnnuity formulas

Want	Formula
Present value	$A \times [1 - (1 + r)^{-t}] / r$
Payment	$r \times PV \,/\, [1 - (1 + r)^{-t}]$
Interest in a payment	$r \times$ opening balance
Principal in a payment	$A -$ interest

Memory hook The payment formula is the PV formula turned over: PV in, A out.

Know it cold Rate and number of payments must match the payment period: monthly loan, monthly rate, monthly count. With a fully amortizing loan, FV = 0.

Key terms

annuity Memorize
ordinary annuity Memorize
level-payment instrument Use it
amortizing loan Use it
mortgage Know the idea

How a loan can be repaid

Loan cash-flow patternseach is a different stack of payments, priced by discounting

Interest only Know the idea interest every period, the whole principal at the end
Fully amortizing Memorize equal payments that repay interest and principal, balance reaches zero
- Level-payment loan priced as an annuity: mortgage, car loan
Partly amortizing (balloon) Know the idea smaller payments, then a large final payment

Formulas

Present value of an ordinary annuity Memorize

PV = A \times \dfrac{1 - (1 + r)^{-t}}{r}

Quick example

EUR 2,000 at the end of each year for 6 years, discounted at 5%. 2,000 × (1 − 1.05⁻⁶) ÷ 0.05 = EUR 10,151.38.

On your calculator

2NDCLR TVM with P/Y set to 1
6N5I/Y2000PMT
CPTPV −10,151.38

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · CPT compute · PV present value

fCLEAR FIN
6n5i2000PMT
PV −10,151.38

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

Level payment on a loan Memorize

A = \dfrac{r \times PV}{1 - (1 + r)^{-t}}

Quick example

Borrow 400,000 over 30 years at 5.4% a year, paid monthly. r = 0.45% a month, t = 360. A = 2,246.12 a month.

On your calculator

2NDCLR TVM
360N0.45I/Y400000PV
CPTPMT −2,246.12

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · CPT compute · PMT payment

fCLEAR FIN
360n0.45i400000PV
PMT −2,246.12

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · PMT payment

Interest in a payment Use it

\text{Interest} = r \times \text{opening balance}, \quad \text{Principal} = A - \text{Interest}

Quick example

Balance 250,000 at 0.4% a month, payment 1,432.49. Interest 1,000.00, principal 432.49.

On your calculator

250000×0.004= 1,000.00 interest
+|-+1432.49= 432.49 principal

+|- change sign

250000ENTER0.004× 1,000.00 interest
CHS1432.49+ 432.49 principal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Valuing a stock from its dividends

Known

MemorizeWhich dividend and how many years

Model	Numerator	Denominator
No growth	$D$	$r$
Constant growth	$D_1 = D_0(1+g)$	$r - g$
Terminal value at year $n$	$D_{n+1}$	$r - g_L$

Memory hook The model always values what comes next: next dividend over (r − g). The terminal value sits at year n, so it is discounted n years, not n + 1.

Know it cold Gordon needs $r > g$ . $D_0$ is the dividend already paid; grow it once to get $D_1$ .

Key terms

dividend discount model Memorize
preferred stock Use it
constant growth Memorize
Gordon growth model Memorize
two-stage dividend growth Use it
terminal value Memorize

Three dividend patterns, three valuation methods

Value of a sharepresent value of all future dividends

No growth Memorize a fixed dividend for ever, such as preferred stock
- V₀ = D ÷ r a perpetuity
Constant growth Memorize dividends grow at g for ever
- V₀ = D₁ ÷ (r − g) needs r above g; D₁ = D₀ × (1 + g)
Changing growth Use it a fast stage, then a stable stage
- Stage 1 discount each dividend one by one
- Terminal value Gordon value at the end of stage 1, discounted n years

Formulas

Value with no growth Memorize

V_0 = \dfrac{D}{r}

Quick example

A preferred share pays 3.00 a year and investors require 7.5%. 3.00 ÷ 0.075 = 40.00.

On your calculator

3÷0.075= 40

3ENTER0.075÷ 40

ENTER enter (on the HP, puts the number on the stack)

Gordon growth model Memorize

V_0 = \dfrac{D_1}{r - g}, \quad D_1 = D_0\,(1 + g)

Quick example

Last dividend 1.60, growth 3% for ever, required return 8%. D₁ = 1.648, so 1.648 ÷ 0.05 = 32.96.

On your calculator

1.6×1.03÷(0.08−0.03)= 32.96

1.6ENTER1.03× D₁
0.08ENTER0.03− r − g
÷ 32.96

ENTER enter (on the HP, puts the number on the stack)

Two-stage dividend discount model Use it

V_0 = \sum_{t=1}^{n} \dfrac{D_t}{(1 + r)^{t}} + \dfrac{V_n}{(1 + r)^{n}}, \quad V_n = \dfrac{D_{n+1}}{r - g_L}

Quick example

D₀ 2.00, growth 10% for 2 years then 4% for ever, r = 9%. D₁ 2.20, D₂ 2.42, V₂ = 2.42 × 1.04 ÷ 0.05 = 50.336. Value = 2.20 ÷ 1.09 + (2.42 + 50.336) ÷ 1.09² = 46.42.

On your calculator

CF2NDCLR WORK
0ENTER↓2.2ENTER↓↓52.756ENTER
NPV9ENTER↓CPT 46.42

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · NPV net present value · CPT compute

fCLEAR REG
0gCF02.2gCFj52.756gCFj9i
fNPV 46.42

f gold shift key · CLEAR REG clear all registers · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · i interest rate per period · NPV net present value

Implied return and implied growth

Known

MemorizeSolve for the unknown

Known	Unknown	Formula
Zero price, face, years	Yield	$(FV/PV)^{1/t} - 1$
Price, $D_1$ , growth	Required return	$D_1/P_0 + g$
Price, $D_0$ , return	Growth	$(rP_0 - D_0)/(P_0 + D_0)$
Payout, $r$ , $g$	Forward P/E	$(D_1/E_1)/(r - g)$

Memory hook Required return is the dividend yield plus growth. Growth is what is left of the return after the yield: g = r − D₁/P₀.

Know it cold The yield to maturity is earned only if you hold to maturity, nothing defaults and coupons are reinvested at that yield.

Key terms

implied return Memorize
yield to maturity Use it
reinvestment assumption Use it
required return Memorize
implied growth rate Memorize
dividend payout ratio Use it
price-to-earnings ratio Use it
forward price-to-earnings ratio Memorize

What a price lets you solve for

A market price and the expected cash flowsfind the rate that makes the present value equal the price

Bond: implied return Memorize the yield to maturity
- Zero (FV ÷ PV)^(1/t) − 1
- Coupon bond solve I/Y on the time value of money (TVM) keys
- Earned only if held to maturity, no default, coupons reinvested at the yield
Share: required return Memorize dividend yield plus growth
- r = D₁ ÷ P₀ + g growth known, solve for r
Share: implied growth Memorize the growth the price needs, given r
- g = (rP₀ − D₀) ÷ (P₀ + D₀) return known, solve for g
Forward P/E Use it ties the multiple to payout, r and g
- P₀ ÷ E₁ = payout ÷ (r − g) payout ratio = D₁ ÷ E₁

Formulas

Implied return of a zero-coupon bond Memorize

r = \left(\dfrac{FV}{PV}\right)^{1/t} - 1

Quick example

A 7-year zero is priced at 76.00 per 100. (100 ÷ 76)^(1/7) − 1 = 4.00% a year.

On your calculator

2NDCLR TVM with P/Y set to 1
7N76+|-PV100FV
CPTI/Y 4.00

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

fCLEAR FIN
7n76CHSPV100FV
i 4.00

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Required return on a share Memorize

r = \dfrac{D_1}{P_0} + g

Quick example

Price 75, last dividend 2.40, growth 4%. D₁ = 2.496, so r = 2.496 ÷ 75 + 0.04 = 7.33%.

On your calculator

2.4×1.04÷75+0.04= 0.0733

2.4ENTER1.04×75÷0.04+ 0.0733

ENTER enter (on the HP, puts the number on the stack)

Implied growth rate Memorize

g = \dfrac{r\,P_0 - D_0}{P_0 + D_0}

Quick example

Price 36, last dividend 1.20, required return 9%. (0.09 × 36 − 1.20) ÷ (36 + 1.20) = 5.48%.

On your calculator

(0.09×36−1.2)÷(36+1.2)= 0.0548

0.09ENTER36×1.2−36ENTER1.2+÷ 0.0548

ENTER enter (on the HP, puts the number on the stack)

Forward P/E from the Gordon model Use it

\dfrac{P_0}{E_1} = \dfrac{D_1 / E_1}{r - g}

Quick example

Payout ratio 40%, required return 9%, growth 5%. 0.40 ÷ 0.04 = 10.0×.

On your calculator

0.4÷(0.09−0.05)= 10

0.4ENTER0.09ENTER0.05−÷ 10

ENTER enter (on the HP, puts the number on the stack)

Cash flow additivity and implied forward rates

Known

MemorizeReading the forward rate

Notation	Meaning	From spot rates
$F_{1,1}$	1-year rate, starting in 1 year	$(1+S_2)^2/(1+S_1) - 1$
$F_{2,1}$	1-year rate, starting in 2 years	$(1+S_3)^3/(1+S_2)^2 - 1$
$F_{1,2}$	2-year rate, starting in 1 year	$[(1+S_3)^3/(1+S_1)]^{1/2} - 1$

Memory hook The first subscript is the wait, the second is the length. Long way round on top, short way round underneath.

Know it cold Compound the spot rates, never average them. Same cash flows, same price: otherwise arbitrage.

Key terms

cash flow additivity principle Memorize
no-arbitrage condition Memorize
spot rate Use it
implied forward rate Memorize

The no-arbitrage test

Two strategies with identical cash flows on the same dates

Same price today Memorize the no-arbitrage condition holds
Different prices today Memorize an arbitrage exists
- Buy the cheap one, sell the dear one cash flows cancel, the price gap is a riskless profit
- Trading pushes the prices together until the profit disappears

Spot and forward rates

Rates on the yield curve

Spot rate S Memorize the rate today for money lent from now until a date
Forward rate F(A, B − A) Memorize the rate agreed today for a loan that starts in A years and lasts B − A years
- Implied by spot rates the rate that makes rolling over equal to investing long

Formulas

Cash flow additivity Use it

PV(A + B) = PV(A) + PV(B)

Quick example

A 2-year 5% coupon bond equals a 1-year zero paying 5 plus a 2-year zero paying 105. At spot rates 3.0% and 3.8%: 5 ÷ 1.03 + 105 ÷ 1.038² = 102.31.

On your calculator

5÷1.03=STO1 4.8544
105÷(1.038y^x2)=+RCL1= 102.31

STO store · y^x power · RCL recall a stored number

5ENTER1.03÷105ENTER1.038ENTER2y^x÷+ 102.31

ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied one-year forward rate Memorize

(1 + S_2)^2 = (1 + S_1)(1 + F_{1,1}) \;\Rightarrow\; F_{1,1} = \dfrac{(1 + S_2)^2}{1 + S_1} - 1

Quick example

S₁ = 3.0%, S₂ = 3.8%. 1.038² ÷ 1.03 − 1 = 4.61%.

On your calculator

1.038y^x2=÷1.03−1= 0.0461

y^x power

1.038ENTER2y^x1.03÷1− 0.0461

ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied forward rate, general form Memorize

(1 + S_B)^{B} = (1 + S_A)^{A}\,(1 + F_{A,\,B-A})^{B-A}

Quick example

S₁ = 3.0%, S₃ = 4.2%. The 2-year rate starting in a year: (1.042³ ÷ 1.03)^(1/2) − 1 = 4.81% a year.

On your calculator

1.042y^x3=÷1.03=y^x0.5=−1= 0.0481

y^x power

1.042ENTER3y^x1.03÷0.5y^x1− 0.0481

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward rate from zero-coupon prices Use it

F_{1,1} = \dfrac{P_1}{P_2} - 1

Quick example

1-year zero 96.15, 2-year zero 91.80 per 100. 96.15 ÷ 91.80 − 1 = 4.74%.

On your calculator

96.15÷91.8−1= 0.0474

96.15ENTER91.8÷1− 0.0474

ENTER enter (on the HP, puts the number on the stack)

Forward exchange rates and option values by no arbitrage

Known

MemorizeFX forward in three forms

Rates	Forward
Annual compounding	$S(1 + r_P)^T / (1 + r_B)^T$
Simple	$S(1 + r_P T) / (1 + r_B T)$
Continuous	$S\,e^{(r_P - r_B)T}$

Memory hook Price-currency rate on top, base-currency rate underneath. The higher-rate currency is worth less forward: more of it per unit of the other.

Know it cold Quote A/B is A per 1 B: A is the price currency. Match each rate to its currency.

MemorizeOption by replication, four steps

Memory hook Hedge, check, discount, solve.

Know it cold 1) $h = (c^+ - c^-)/(S^+ - S^-)$ . 2) Long $h$ shares, short 1 call: same value in both outcomes. 3) Discount that value at $r$ . 4) $c_0 = hS_0 -$ that present value.

Key terms

forward exchange rate Memorize
call option Use it
put option Use it
risk-free portfolio Use it
hedge ratio Memorize
replicating portfolio Use it

Reading an exchange-rate quote

INR/USD 83.00read it as INR 83.00 per USD 1

Price currency (INR) Memorize the currency you pay, the one in the answer
- Its interest rate goes on top F = S × (1 + r_price) ÷ (1 + r_base)
Base currency (USD) Memorize the currency being priced, one unit
- Its interest rate goes underneath

Same cash flows, same price: three uses

No-arbitrage pricing

Forward interest rate Memorize long loan against two short loans
Forward exchange rate Memorize invest at home against convert, invest abroad, convert back
Option value Use it option against a mix of shares and borrowing that pays the same
- Hedge ratio shares per option that makes the portfolio risk-free
- Risk-free portfolio must earn the risk-free rate

Formulas

Forward exchange rate (annual compounding) Memorize

F = S \times \dfrac{(1 + r_P)^{T}}{(1 + r_B)^{T}}

Quick example

Spot 0.9400 CHF per EUR, one-year rates CHF 1.5%, EUR 3.0%. 0.9400 × 1.015 ÷ 1.03 = 0.9263 CHF per EUR.

On your calculator

0.94×1.015÷1.03= 0.9263

0.94ENTER1.015×1.03÷ 0.9263

ENTER enter (on the HP, puts the number on the stack)

Forward exchange rate (simple rates) Use it

F = S \times \dfrac{1 + r_P\,T}{1 + r_B\,T}

Quick example

Spot USD/GBP 1.2700, six-month rates USD 5%, GBP 4% (annual, simple). 1.27 × 1.025 ÷ 1.02 = 1.2762.

On your calculator

1.27×1.025÷1.02= 1.2762

1.27ENTER1.025×1.02÷ 1.2762

ENTER enter (on the HP, puts the number on the stack)

Forward exchange rate (continuous compounding) Use it

F = S\,e^{(r_P - r_B)\,T}

Quick example

Spot INR/USD 83.00, continuous rates INR 6.5%, USD 4.5%, one year. 83 × e^0.02 = 84.68.

On your calculator

0.022NDe^x×83= 84.68

2ND second function (the shifted label above a key) · e^x e to the power x

0.02ge^x83× 84.68

g blue shift key · e^x e to the power x

Hedge ratio Memorize

h = \dfrac{c^{+} - c^{-}}{S^{+} - S^{-}}

Quick example

Share 80, then 100 or 64. Call, exercise price 90, pays 10 or 0. h = 10 ÷ 36 = 0.2778 shares per call.

On your calculator

10÷36= 0.2778

10ENTER36÷ 0.2778

ENTER enter (on the HP, puts the number on the stack)

Call value by replication Use it

c_0 = h\,S_0 - \dfrac{h\,S^{-} - c^{-}}{1 + r}

Quick example

Same tree, risk-free rate 4%. 0.2778 × 80 − (0.2778 × 64 − 0) ÷ 1.04 = 22.22 − 17.09 = 5.13.

On your calculator

10÷36=STO1 0.2778
RCL1×64÷1.04=STO2 17.09
RCL1×80−RCL2= 5.13

STO store · RCL recall a stored number

10ENTER36÷STO1 0.2778
64×1.04÷STO2 17.09
RCL180×RCL2− 5.13

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Cheat sheet · PDF · 5 A4 pages

Time Value of Money in Finance on 5 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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