Contents
Introduction
Financial Mathematics (Exam FM) represents a crucial stepping stone in your actuarial journey. While Exam P focuses on probability theory, FM introduces you to the fundamental concepts of financial mathematics – how money grows over time, how different financial instruments work, and how to value various cash flow streams. These concepts form the foundation of actuarial work in insurance, investments, and pension planning.
Exam Structure
The exam consists of 30 multiple-choice questions to be completed in 3 hours. Each question has five possible answers, and there is no penalty for incorrect answers. A score of approximately 70% is typically required to pass. The exam is computer-based and includes access to an SOA-approved calculator, which will be essential for complex financial calculations.
Core Concepts
Time Value of Money
At its heart, financial mathematics is about understanding that money has different values at different points in time. A dollar today is worth more than a dollar a year from now because of its earning potential. This fundamental concept underlies everything else in the exam.
Consider a simple example: If you invest $1,000 at 5% annual interest, after one year you’ll have $1,050. But what if you want to have $1,050 one year from now – how much do you need to invest today? This introduces the concept of present value: you would need to invest $1,000 (= $1,050/1.05) today to reach your goal.
Interest Rates and Their Measurements
Interest rates can be expressed in various ways, and understanding these different expressions is crucial:
Simple Interest: Interest is calculated only on the initial principal.
Example: $1,000 invested at 5% simple interest for 2 years yields:
$1,000 + $1,000(0.05)(2) = $1,100
Compound Interest: Interest is calculated on both principal and previously accumulated interest.
Example: $1,000 invested at 5% compound interest for 2 years yields:
$1,000(1.05)² = $1,102.50
The small difference ($2.50) in this example becomes much more significant with larger amounts, longer time periods, or higher interest rates.
Effective vs. Nominal Rates
One of the most important distinctions in financial mathematics is between nominal and effective rates. A nominal rate is the stated annual rate, while the effective rate accounts for the frequency of compounding.
For instance, a 12% nominal rate compounded monthly (1% per month) produces an effective annual rate of:
(1 + 0.12/12)¹² – 1 = 12.68%
This means $1,000 invested at “12% compounded monthly” grows to $1,126.80 after one year, not $1,120 as simple interest would suggest.
Annuities and Cash Flow Analysis
An annuity is a series of regular payments. Understanding annuities is crucial because many financial instruments, from car loans to pension payments, can be analyzed as annuities.
Types of Annuities:
- Annuity-immediate: Payments occur at the end of each period
- Annuity-due: Payments occur at the beginning of each period
- Perpetuity: An annuity with infinite payments
- Deferred annuity: Payments start after a waiting period
Loan Amortization
Loan amortization combines the concepts of present value and annuities. When you take out a loan, each payment typically consists of both principal and interest. The proportion of principal to interest changes over time, though the total payment often remains constant.
Bonds and Bond Pricing
A bond is a debt instrument that typically provides regular interest payments (coupons) and returns the face value at maturity. Bond pricing requires understanding present value concepts for both the coupon stream and the face value payment.
Financial Derivatives
While not as extensive as in later exams, FM introduces basic concepts of financial derivatives, particularly forwards and futures. These are contracts whose values derive from underlying assets.
Sample Questions with Detailed Solutions
- An investor deposits $10,000 in an account earning 6% convertible quarterly. What is the effective annual yield?
Solution:
Let’s approach this step-by-step:
- Nominal rate = 6% = 0.06
- Quarterly rate = 0.06/4 = 0.015
- Number of compounding periods per year = 4
- Effective annual rate = (1 + r/n)ⁿ – 1
- = (1 + 0.06/4)⁴ – 1
- = (1.015)⁴ – 1
- = 1.0614 – 1
- = 0.0614 or 6.14%
- You want to accumulate $50,000 in 10 years by making equal monthly deposits into an account earning 8% convertible monthly. Calculate the required monthly deposit.
Solution:
This is an annuity-immediate problem:
- Future Value = PMT × s₍₁₂₀₎ᵢ
- where i = 0.08/12 = 0.00667 per month
- n = 10 × 12 = 120 months
- $50,000 = PMT × [(1.00667)¹²⁰ – 1]/0.00667
- $50,000 = PMT × 165.32
- PMT = $302.44
- A 20-year bond with face value $1,000 pays 5% coupons semi-annually. If the yield rate is 6% convertible semi-annually, what is the bond’s price?
Solution:
We need to:
- Calculate present value of coupons
- Each coupon = $1,000 × 0.05/2 = $25
- Number of payments = 40
- Rate per period = 0.06/2 = 0.03
- PV of coupons = 25 × a₍₄₀₎₀.₀₃
= 25 × 23.11 = $577.75
- Calculate present value of face value
- PV of face = 1000 × (1.03)⁻⁴⁰
= 1000 × 0.3066 = $306.60
- Total price = $577.75 + $306.60 = $884.35
- A loan of $200,000 is to be repaid with level annual payments over 30 years. The interest rate is 4.5% convertible monthly. Calculate the annual payment.
Solution:
First, we need to:
- Find the effective annual rate:
- i₁₂ = 0.045/12 = 0.00375
- Effective annual rate = (1.00375)¹² – 1 = 0.0459 or 4.59%
- Use annuity formula:
- PV = PMT × a₍₃₀₎₀.₀₄₅₉
- 200,000 = PMT × [(1 – (1.0459)⁻³⁰)/0.0459]
- 200,000 = PMT × 17.8328
- PMT = $11,215.67
- You are given two investment options:
A) $10,000 today
B) $3,000 now, $4,000 in 2 years, and $5,000 in 5 years
If the effective annual interest rate is 7%, which option has the greater present value?
Solution:
For Option A:
PV = $10,000
For Option B:
PV = 3000 + 4000v² + 5000v⁵
where v = 1/1.07
= 3000 + 4000(0.8734) + 5000(0.7130)
= 3000 + 3493.60 + 3565
= $10,058.60
Option B has the greater present value by $58.60
- An investor buys a perpetuity-immediate for $100,000 that makes annual payments that increase by 3% each year. The first payment is $5,000 and the annual effective interest rate is 8%. Calculate the present value of payments from year 11 onward.
Solution:
This is a growing perpetuity with a deferral period.
- Present value of the entire perpetuity at time 0:
PV = P/(i-g) = 5000/(0.08-0.03) = $100,000 - Present value of first 10 payments:
Using the growing annuity formula:
PV₁₀ = 5000[1-((1.03)/(1.08))¹⁰]/(0.08-0.03)
= 5000[1-0.6756]/0.05
= $32,440 - Present value of remaining payments:
= 100,000 – 32,440 = $67,560 - A 1-year forward contract on a stock is entered into when the stock price is $50. The stock pays dividends of $1 in 3 months and $1.50 in 9 months. If the risk-free rate is 6% convertible quarterly, what is the forward price?
Solution:
Forward price = S₀(1+r)ᵀ – PV(dividends)
- PV of dividends:
First dividend: 1/(1.015)³ = 0.956
Second dividend: 1.50/(1.015)⁹ = 1.384
Total PV of dividends = 2.340 - Forward price = 50(1.015)⁴ – 2.340
= 50(1.0614) – 2.340
= 50.73 - A callable bond with face value $1,000 paying 6% coupons semi-annually can be called at 102% of face value. The yield rate is 4% convertible semi-annually. If the bond is called in 5 years, what is its price?
Solution:
- Calculate PV of coupons:
- Each coupon = 1000(0.06/2) = $30
- Number of payments = 10
- Rate per period = 0.04/2 = 0.02
- PV coupons = 30 × a₍₁₀₎₀.₀₂
= 30 × 8.9826 = $269.48
- Calculate PV of call price:
- Call price = 1000 × 1.02 = $1,020
- PV = 1020 × (1.02)⁻¹⁰
= 1020 × 0.8203 = $836.71
- Total price = $269.48 + $836.71 = $1,106.19
- An investor can buy a share for $100 or enter into a 6-month forward contract. The continuously compounded risk-free rate is 8%, and the stock will pay a dividend of $2 in 2 months. What is the forward price?
Solution:
F = Se^(rt) – De^[r(T-t)]
where:
- S = 100
- r = 0.08
- T = 0.5
- D = 2
- t = 2/12 = 0.167
F = 100e^(0.08×0.5) – 2e^[0.08(0.5-0.167)]
= 100 × 1.0408 – 2 × 1.0271
= 104.08 – 2.05
= $102.03
- A 25-year mortgage of $300,000 has payments at the end of each month. The nominal annual interest rate convertible monthly is 6%. After 5 years, the interest rate drops to 5%. Calculate the change in monthly payment.
Solution:
- Original payment:
- i = 0.06/12 = 0.005
- n = 300
- PV = 300,000
- PMT = PV/a₍₃₀₀₎₀.₀₀₅
= 300,000/166.7916
= $1,798.65
- Balance after 5 years:
- n = 60 payments made
- Outstanding balance = 1798.65 × a₍₂₄₀₎₀.₀₀₅
= 1798.65 × 157.7547
= $283,759.30
- New payment at 5%:
- i = 0.05/12 = 0.004167
- n = 240
- PV = 283,759.30
- PMT = 283,759.30/163.7307
= $1,733.11
- Decrease in payment = $1,798.65 – $1,733.11 = $65.54
Study Strategies
Understanding vs. Memorization
While FM requires memorizing certain formulas, success comes from understanding the underlying concepts. For example, don’t just memorize the annuity formulas – understand why an annuity-due is worth more than an annuity-immediate (because payments occur sooner).
Calculator Proficiency
Become extremely comfortable with your financial calculator. Know how to:
- Switch between payment modes (BEGIN/END)
- Store and recall values
- Use memory functions
- Handle cash flow analysis
- Calculate bond prices and yields
Practice Time Management
With 30 questions in 180 minutes, you have an average of 6 minutes per question. Some questions might require extensive calculations, while others test conceptual understanding. Practice identifying which is which and budgeting your time accordingly.
Use Multiple Study Resources
Combine textbooks, online resources, and practice problems. Different explanations of the same concept can help deepen understanding. The SOA sample questions and past exam questions are particularly valuable resources.
Conclusion
Success in Exam FM requires both technical proficiency and conceptual understanding. The financial mathematics concepts you learn here will serve as building blocks throughout your actuarial career. Focus on understanding the relationships between different topics – how interest rates affect present values, how present values relate to loan payments, and how all these concepts come together in real-world financial instruments.
Remember that this exam tests your ability to apply concepts, not just perform calculations. When practicing, always ask yourself why a particular approach works and how it relates to other topics you’ve studied. With thorough preparation and understanding of the core concepts, you can successfully pass Exam FM and continue your actuarial journey.
