Are you preparing for the Society of Actuaries (SOA) Exam FM? This comprehensive cheat sheet covers all the essential financial mathematics concepts you need to master. Use this as your quick reference guide while studying and as a final review before your exam!

Table of Contents #

1. Key Information About Exam FM
2. Time Value of Money
   • Basic Interest Rates
   • Present and Future Values
   • Annuities
   • Perpetuities
3. Loans and Bonds
   • Loan Amortization
   • Bonds
4. Financial Instruments
   • Stocks
   • Duration and Convexity
5. Yield Curves and Forward Rates
   • Spot Rates
   • Forward Rates
6. Investment Portfolio Fundamentals
7. Advanced Topics
8. Study Tips
9. Exam Day Strategies
10. Calculator Guide
11. Common Formulas Quick Reference

Key Information About Exam FM #

The SOA Exam FM (Financial Mathematics) is a crucial step in your actuarial career path. Here’s what you need to know:

Exam Structure:

• Duration: 3 hours
• Questions: 30 multiple-choice questions
• Passing Score: Typically around 70% (varies by sitting)
• Calculator: Only TI BA-II Plus or BA-II Plus Professional allowed
• Format: Computer-based testing (CBT)
• Frequency: Offered 6 times per year

Key Topics Covered:

• General cash flows and portfolios
• Annuities/cash flows with non-contingent payments
• Loans
• Bonds
• General derivatives
• Options

Time Value of Money #

Basic Interest Rates #

Understanding different types of interest rates is fundamental to financial mathematics.

Simple Interest:

• Formula: A = P(1 + rt)
• Where: P = Principal, r = annual interest rate, t = time in years
• Used for: Short-term investments, basic calculations

Compound Interest:

• Formula: A = P(1 + i)^n
• Where: i = interest rate per period, n = number of periods
• More realistic for most financial applications

Effective Annual Rate (EAR):

• Formula: i_eff = (1 + r/m)^m - 1
• Where: r = nominal annual rate, m = compounding periods per year
• Allows comparison between different compounding frequencies

Continuous Compounding:

• Formula: A = Pe^(rt)
• Where: e = Euler’s number (≈2.71828)
• Used in advanced financial modeling

Nominal vs. Effective Rates:

• Nominal rate: Stated annual rate without considering compounding
• Effective rate: Actual annual rate considering compounding effects
• Key relationship: Higher compounding frequency → Higher effective rate

Present and Future Values #

The cornerstone of financial mathematics - understanding how money’s value changes over time.

Present Value (PV):

• Formula: PV = FV / (1 + i)^n
• Represents today’s value of a future cash flow
• Used for: Investment analysis, loan calculations

Future Value (FV):

• Formula: FV = PV × (1 + i)^n
• Represents future value of today’s investment
• Used for: Retirement planning, investment projections

Net Present Value (NPV):

• Formula: NPV = Σ[CFt/(1 + r)^t] - Initial Investment
• Decision rule: Accept if NPV > 0
• Critical for investment decision-making

Annuities #

Annuities are series of equal payments made at regular intervals.

Ordinary Annuity (End-of-Period Payments):

• Present Value: PV = PMT × [(1 - (1 + i)^(-n))/i]
• Future Value: FV = PMT × [((1 + i)^n - 1)/i]
• Examples: Mortgage payments, bond coupons

Annuity Due (Beginning-of-Period Payments):

• Present Value: PV = PMT × [(1 - (1 + i)^(-n))/i] × (1 + i)
• Future Value: FV = PMT × [((1 + i)^n - 1)/i] × (1 + i)
• Examples: Rent payments, insurance premiums

Growing Annuity:

• Present Value: PV = PMT × [(1 - ((1 + g)/(1 + i))^n)/(i - g)]
• Where: g = growth rate
• Used for: Inflation-adjusted payments

Deferred Annuity:

• Present Value: PV = PMT × [(1 - (1 + i)^(-n))/i] × (1 + i)^(-k)
• Where: k = deferral period
• Used for: Retirement planning

Perpetuities #

Perpetuities are annuities that continue forever.

Standard Perpetuity:

• Present Value: PV = PMT/i
• Simple but powerful concept
• Examples: Preferred stock dividends

Growing Perpetuity:

• Present Value: PV = PMT/(i - g)
• Where: i > g (required for convergence)
• Used in: Stock valuation models

Loans and Bonds #

Loan Amortization #

Understanding how loans are repaid over time is crucial for FM.

Level Payment Formula:

• Payment: PMT = PV × [i(1 + i)^n]/[(1 + i)^n - 1]
• Creates equal periodic payments
• Most common loan structure

Outstanding Balance After k Payments:

• Balance: OB = PMT × [(1 - (1 + i)^(-(n-k)))/i]
• Alternative: OB = PV(1 + i)^k - PMT × [((1 + i)^k - 1)/i]

Principal and Interest Components:

• Interest Payment: IP_k = OB_(k-1) × i
• Principal Payment: PP_k = PMT - IP_k
• Total Principal: Σ PP_k = Original Loan Amount

Amortization Schedule Components:

• Payment number
• Beginning balance
• Interest payment
• Principal payment
• Ending balance

Bonds #

Bonds are debt securities with specific characteristics that affect their valuation.

Basic Bond Price Formula:

• Price: P = Σ[C/(1 + i)^t] + F/(1 + i)^n
• Where: C = coupon payment, F = face value, i = yield rate, n = maturity

Bond Classifications:

• Premium Bond: Price > Face Value (coupon rate > yield rate)
• Discount Bond: Price < Face Value (coupon rate < yield rate)
• Par Bond: Price = Face Value (coupon rate = yield rate)

Zero-Coupon Bonds:

• Price: P = F/(1 + i)^n
• No periodic interest payments
• All return comes from price appreciation

Bond Yield Measures:

• Current Yield: Annual coupon / Current price
• Yield to Maturity (YTM): IRR of bond’s cash flows
• Yield to Call (YTC): IRR assuming early redemption

Callable Bonds:

• Can be redeemed before maturity
• Price: P = min(Regular Bond Price, Call Price)
• Typically called when interest rates fall

Financial Instruments #

Stocks #

Equity valuation using dividend discount models.

Constant Growth Model (Gordon Growth Model):

• Price: P = D₁/(r - g)
• Where: D₁ = next year’s dividend, r = required return, g = growth rate
• Assumes constant dividend growth rate

Multi-Stage Growth Models:

• Combines different growth phases
• More realistic for many companies
• Requires careful forecasting

Price-to-Earnings Ratios:

• P/E = Price per share / Earnings per share
• Useful for comparative valuation
• Industry and growth dependent

Duration and Convexity #

These measures help understand bond price sensitivity to interest rate changes.

Macaulay Duration:

• Formula: D_Mac = Σ[t × PV(CFt)]/Bond Price
• Measures weighted average time to receive cash flows
• Higher duration = greater interest rate risk

Modified Duration:

• Formula: D_Mod = D_Mac/(1 + i)
• Measures price sensitivity to yield changes
• Price change ≈ -D_Mod × Δi

Dollar Duration:

• Formula: DD = D_Mod × Bond Price
• Measures dollar price change for 1% yield change
• Useful for portfolio management

Convexity:

• Formula: C = Σ[t(t+1) × PV(CFt)]/[Bond Price × (1 + i)²]
• Measures curvature of price-yield relationship
• Improves duration-based price estimates

Price Change Approximation:

• ΔP/P ≈ -D_Mod × Δi + 0.5 × C × (Δi)²
• More accurate than duration alone
• Essential for risk management

Yield Curves and Forward Rates #

Understanding the relationship between interest rates of different maturities.

Spot Rates #

Spot rates are yields on zero-coupon bonds.

Spot Rate Relationship:

• (1 + s_n)^n = (1 + s₁) × (1 + f₁,₂) × ... × (1 + f_{n-1,n})
• Links spot rates to forward rates
• Foundation of yield curve analysis

Bootstrapping Process:

• Method to extract spot rates from bond prices
• Start with shortest maturity
• Work forward to longer maturities

Forward Rates #

Forward rates are future interest rates implied by current spot rates.

Forward Rate Formula:

• (1 + f_{t,t+k}) = [(1 + s_{t+k})^{t+k}]/[(1 + s_t)^t]
• Where: f_{t,t+k} = forward rate from time t to t+k
• Represents market’s expectation of future rates

Applications:

• Hedging future borrowing costs
• Investment planning
• Arbitrage opportunities

Investment Portfolio Fundamentals #

Net Present Value (NPV) #

NPV Formula:

• NPV = -C₀ + Σ[CFt/(1 + r)^t]
• Where: C₀ = initial investment, CFt = cash flow at time t

Decision Rules:

• NPV > 0: Accept the project
• NPV < 0: Reject the project
• NPV = 0: Indifferent

NPV Profile:

• Graph of NPV vs. discount rate
• Shows relationship between cost of capital and project value
• Helps in sensitivity analysis

Internal Rate of Return (IRR) #

IRR Definition:

• Discount rate that makes NPV = 0
• Equation: 0 = -C₀ + Σ[CFt/(1 + IRR)^t]
• Requires iterative solution

Decision Rules:

• IRR > Required return: Accept
• IRR < Required return: Reject
• IRR = Required return: Indifferent

IRR Limitations:

• Multiple IRRs possible with non-conventional cash flows
• Scale problem when comparing projects
• Reinvestment assumption issues

Payback Period #

Simple Payback:

• Time required to recover initial investment
• Ignores time value of money
• Easy to calculate but limited usefulness

Discounted Payback:

• Uses present values of cash flows
• More accurate but still ignores cash flows beyond payback

Advanced Topics #

Options Basics #

Call Options:

• Right to buy at strike price
• Payoff: max(S - K, 0)
• Where: S = stock price, K = strike price

Put Options:

• Right to sell at strike price
• Payoff: max(K - S, 0)
• Protective strategy for stock holdings

Put-Call Parity:

• C - P = S - Ke^(-rt)
• Relationship between call and put prices
• Used for arbitrage identification

Interest Rate Swaps #

Basic Structure:

• Exchange of fixed for floating rate payments
• Based on notional principal
• Used for interest rate risk management

Valuation:

• Present value of fixed leg minus floating leg
• Market value depends on current yield curve
• Important for financial institutions

Currency Exchange #

Spot vs. Forward Rates:

• Spot: Current exchange rate
• Forward: Future exchange rate agreed today
• Related by interest rate differential

Purchasing Power Parity:

• Exchange rates adjust for inflation differences
• Long-run equilibrium concept
• Important for international investments

Study Tips #

Mastering Your Calculator #

Essential TI BA-II Plus Functions:

• TVM Keys: N, I/Y, PV, PMT, FV
• Cash Flow Keys: CF, NPV, IRR
• Bond Functions: 2ND, FV (BOND)
• Statistics: 2ND, 7 (DATA)

Calculator Setup:

• Set P/Y = 1 (payments per year)
• Set C/Y = 1 (compounding per year)
• Check BGN/END mode for annuity timing
• Clear TVM registers between problems

Time-Saving Tips:

• Learn keyboard shortcuts
• Use memory functions for complex calculations
• Practice until calculations become automatic
• Always verify your setup before calculating

Conceptual Understanding #

Focus Areas:

1. Time Value Relationships: Understand how PV, FV, rate, and time interact
2. Cash Flow Timing: Beginning vs. end of period makes a significant difference
3. Interest Rate Types: Know when to use nominal vs. effective rates
4. Bond Mechanics: Understand relationship between price, yield, and time
5. Risk and Return: Higher risk requires higher return

Common Misconceptions:

• Confusing nominal and effective rates
• Incorrect annuity timing assumptions
• Misunderstanding yield curve relationships
• Forgetting to adjust for different compounding periods

Practice Strategy #

Problem-Solving Approach:

1. Read Carefully: Identify what’s given and what’s asked
2. Draw Timeline: Visualize cash flows over time
3. Choose Method: Calculator vs. formula approach
4. Check Reasonableness: Does the answer make sense?
5. Verify Setup: Confirm calculator settings and inputs

Study Schedule Recommendations:

• 8 weeks before: Complete all readings
• 6 weeks before: Start practice problems
• 4 weeks before: Take practice exams
• 2 weeks before: Focus on weak areas
• 1 week before: Final review and calculator practice

Exam Day Strategies #

Time Management #

Pacing Guidelines:

• Target: 6 minutes per question maximum
• Easy questions: 3-4 minutes
• Moderate questions: 5-7 minutes
• Difficult questions: 8-10 minutes (if time permits)

Strategy:

1. First Pass: Answer easy questions (aim for 20-25 minutes)
2. Second Pass: Tackle moderate difficulty questions
3. Final Pass: Attempt remaining difficult questions
4. Review: Check flagged questions if time remains

Question Analysis #

Read Questions Carefully:

• Identify key information and requirements
• Watch for specific timing (beginning vs. end)
• Note units (annual vs. semi-annual rates)
• Check for special conditions or assumptions

Common Question Types:

• Direct calculation: Apply formula straightforwardly
• Multi-step problems: Break into components
• Comparison questions: Calculate multiple alternatives
• Conceptual questions: Apply theoretical knowledge

Calculator Efficiency #

Pre-Exam Checklist:

• Clear all memory and registers
• Verify battery level (bring spare)
• Check display and key responsiveness
• Reset to factory settings if needed
• Practice key sequences for common calculations

During Exam:

• Clear TVM registers between bond problems
• Double-check BGN/END setting for annuities
• Use parentheses for complex calculations
• Write down intermediate results for multi-step problems

Backup Strategies:

• Bring second calculator
• Know how to perform calculations manually if needed
• Practice switching between calculators smoothly

Calculator Guide #

TI BA-II Plus Essential Functions #

Time Value of Money (TVM):

• N: Number of periods
• I/Y: Interest rate per year
• PV: Present value (enter as negative for outflows)
• PMT: Payment amount per period
• FV: Future value

Cash Flow Analysis:

• CF: Cash flow entry
• NPV: Net present value calculation
• IRR: Internal rate of return
• 2ND, NPV: Access NPV function
• 2ND, IRR: Access IRR function

Bond Calculations:

• 2ND, FV (BOND): Bond worksheet
• SDT: Settlement date
• CPN: Coupon rate
• RDT: Redemption date
• RV: Redemption value
• 2ND, CLR WORK: Clear bond worksheet

Statistics Functions:

• 2ND, 7 (DATA): Data entry
• 2ND, 8 (STAT): Statistical calculations
• ↓, ↓ (LIN): Linear regression

Common Calculator Sequences #

Ordinary Annuity Present Value:

1. Clear TVM: 2ND, CLR TVM
2. Enter N: [number], N
3. Enter rate: [rate], I/Y
4. Enter payment: [payment], PMT
5. Solve for PV: CPT, PV

Bond Price Calculation:

1. Access bond worksheet: 2ND, FV
2. Enter settlement date: [date], ENTER, ↓
3. Enter coupon rate: [rate], ENTER, ↓
4. Enter redemption date: [date], ENTER, ↓
5. Enter redemption value: [value], ENTER, ↓
6. Enter yield: [yield], ENTER, ↓
7. Calculate price: ↑, ↑, CPT

NPV Calculation:

1. Clear CF: CF, 2ND, CLR WORK
2. Initial outlay: [amount], ENTER, ↓, ↓
3. Cash flows: [CF1], ENTER, ↓, [F1], ENTER, ↓
4. Continue for all cash flows
5. Calculate NPV: NPV, [rate], ENTER, ↓, CPT

Common Formulas Quick Reference #

Interest Rate Conversions #

• Effective Rate: i = (1 + r/m)^m - 1
• Continuous: i = e^r - 1
• Force of Interest: δ = ln(1 + i)

Annuity Formulas #

• PV Ordinary: PV = PMT × [(1-(1+i)^(-n))/i]
• FV Ordinary: FV = PMT × [((1+i)^n-1)/i]
• PV Due: PV × (1+i)
• FV Due: FV × (1+i)

Bond Formulas #

• Price: P = Σ[C/(1+i)^t] + F/(1+i)^n
• Current Yield: CY = Annual Coupon / Price
• Duration: D = Σ[t × PV(CFt)] / Price
• Modified Duration: D_mod = D/(1+i)

Loan Formulas #

• Payment: PMT = PV × [i(1+i)^n]/[(1+i)^n-1]
• Balance: B_k = PMT × [(1-(1+i)^(-(n-k)))/i]
• Interest: I_k = B_(k-1) × i
• Principal: P_k = PMT - I_k

Investment Analysis #

• NPV: NPV = Σ[CFt/(1+r)^t] - C0
• IRR: 0 = Σ[CFt/(1+IRR)^t] - C0
• Payback: PB = C0 / Average Annual CF

This comprehensive cheat sheet should serve as your go-to reference for Exam FM preparation. Remember that understanding the concepts is more important than memorizing formulas. Practice extensively, manage your time effectively, and approach the exam with confidence. The key to success is consistent preparation and thorough understanding of the underlying financial mathematics principles.

Good luck with your exam preparation! Remember to stay calm, trust your preparation, and work through problems systematically. You’ve got this!