SOA Exam FM – Financial Mathematics — Questions and Answers

Question 1: How should risk be assessed in investment analysis?

Risk assessment is only needed for retirees
Use a one-size-fits-all risk profile
Evaluate risk tolerance, capacity, time horizon, and investment objectives systematically (Correct answer)
Ignore risk for aggressive growth

Correct answer: Evaluate risk tolerance, capacity, time horizon, and investment objectives systematically

Comprehensive risk assessment considers tolerance, capacity, time horizon, and objectives to create appropriate strategies.

Question 2: A straddle is formed by:

Buying calls at two different strikes
Buying a call and selling a put with the same strike
Buying a call and a put with the same strike and expiration (Correct answer)
Selling a call and buying a put with different strikes

Correct answer: Buying a call and a put with the same strike and expiration

A long straddle profits from large price moves in either direction by combining a long call and a long put at the same strike.

Question 3: What regulatory compliance requirement applies to client relations?

Self-regulation is sufficient
Full compliance with all applicable federal, state, and industry regulations (Correct answer)
Regulations are optional for small practices
Compliance is only needed for publicly traded companies

Correct answer: Full compliance with all applicable federal, state, and industry regulations

Full regulatory compliance is mandatory regardless of practice size, ensuring market integrity and client protection.

Question 4: How should regulatory compliance performance be reported to clients?

Provide accurate, complete, and timely performance reporting with appropriate benchmarks (Correct answer)
Let clients check their own accounts
Only report positive results
Reporting is only required annually

Correct answer: Provide accurate, complete, and timely performance reporting with appropriate benchmarks

Accurate, complete, and timely reporting with appropriate benchmarks enables informed decision-making by clients.

Question 5: Which formula correctly represents the accumulated value (future value) of an ordinary annuity of 1 per period for n periods at interest rate i?

a(n,i) = (1 - (1+i)^-n) / i
s(n,i) = ((1+i)^n - 1) / i (Correct answer)
ä(n,i) = (1 - (1+i)^-n) / d
s(n,i) = ((1+i)^n - 1) / d

Correct answer: s(n,i) = ((1+i)^n - 1) / i

The accumulation factor s(n,i) = ((1+i)^n - 1)/i represents the future value of an annuity-immediate paying 1 per period.

Question 6: An arithmetic increasing annuity-immediate has payments of $100, $200, $300, $400, $500 at the end of years 1 through 5. Which expression gives its present value at 5% annual interest?

100 × s(5, 5%)
100 × ä(5, 5%)
100 × (Ia)(5, 5%) (Correct answer)
100 × a(5, 5%)

Correct answer: 100 × (Ia)(5, 5%)

An arithmetic increasing annuity with first payment P and common difference P uses (Ia)(n,i), so PV = P × (Ia)(5,5%) = 100 × (Ia)(5, 5%).

Question 7: How should conflicts of interest be managed in financial planning?

Conflicts are unavoidable and need not be disclosed
Self-assessment of conflicts is sufficient
Conflicts only matter in large transactions
Identify, disclose, and mitigate all actual and potential conflicts of interest (Correct answer)

Correct answer: Identify, disclose, and mitigate all actual and potential conflicts of interest

All actual and potential conflicts of interest must be identified, disclosed to clients, and mitigated to maintain trust and compliance.

Question 8: The force of interest δ is related to the effective annual rate i by which formula?

δ = ln(1 + i) (Correct answer)
δ = e^i − 1
δ = i / (1 + i)
δ = 1 − e^(−i)

Correct answer: δ = ln(1 + i)

The force of interest δ equals the natural logarithm of the accumulation factor: δ = ln(1 + i).

Question 9: The equation of value principle states that for a financial transaction, at any comparison date:

Nominal value of money received = Nominal value of money paid
FV of money received must exceed FV of money paid
Discount rate equals the coupon rate
PV of money received = PV of money paid (Correct answer)

Correct answer: PV of money received = PV of money paid

The equation of value requires that at any focal date, the accumulated or discounted values of all receipts equal all payments.

Question 10: Two assets have the following characteristics: Asset A: Return = 10%, Volatility = 20% Asset B: Return = 8%, Volatility = 15% Correlation (𝜌) = 0.5 What is the portfolio’s standard deviation if the weights of Asset A and B are 50% each?

17.5%
19.0% (Correct answer)
20.0%
18.0%

Correct answer: 19.0%

Formula: Portfolio Std Dev = √[wA^2σA^2 + wB^2σB^2 + 2wAwBσAσBρ] Calculation: wA = wB = 0.5, σA = 0.20, σB = 0.15, ρ = 0.5 Portfolio Std Dev = √[(0.5^2 × 0.20^2) + (0.5^2 × 0.15^2) + (2 × 0.5 × 0.5 × 0.20 × 0.15 × 0.5)] Portfolio Std Dev = √[0.01 + 0.005625 + 0.0075] = √0.023125 = 0.191 = 19.0%

Question 11: A collar strategy involves:

Buying a put at a lower strike and selling a call at a higher strike to hedge an asset (Correct answer)
Selling both a call and a put at the same strike
Buying two calls and one put at the same strike
Buying a call and a put at the same strike

Correct answer: Buying a put at a lower strike and selling a call at a higher strike to hedge an asset

A collar limits an asset's downside by buying a put and finances part of the cost by selling a call, capping upside.

Question 12: A call option gives the holder the right to:

Buy the underlying asset at the strike price before or at expiration (Correct answer)
Receive interest payments from the option writer
Sell the underlying asset at the strike price before or at expiration
Exchange one currency for another at a fixed rate

Correct answer: Buy the underlying asset at the strike price before or at expiration

A call option grants the holder the right (but not the obligation) to buy the underlying asset at the strike price.

Question 13: How should estate planning performance be reported to clients?

Only report positive results
Provide accurate, complete, and timely performance reporting with appropriate benchmarks (Correct answer)
Let clients check their own accounts
Reporting is only required annually

Correct answer: Provide accurate, complete, and timely performance reporting with appropriate benchmarks

Accurate, complete, and timely reporting with appropriate benchmarks enables informed decision-making by clients.

Question 14: The accumulated value at the end of n periods of a level immediate annuity of 1 per period is:

s_{n|} = a_{n|} × (1+i)^n × i
s_{n|} = (1 − v^n) / i
s_{n|} = n / (1+i)
s_{n|} = ((1+i)^n − 1) / i (Correct answer)

Correct answer: s_{n|} = ((1+i)^n − 1) / i

The future value (accumulated value) of an ordinary annuity is s_{n|} = ((1+i)^n − 1) / i.

Question 15: What is the present value of $8,000 to be received 6 years from now, with a discount rate of 7% annually?

$5,394.94 (Correct answer)
$5,500.00
$5,322.00
$5,440.00

Correct answer: $5,394.94

The present value (PV) of a future sum is calculated using the formula PV = FV / (1 + r)^n. For $8,000 to be received in 6 years with a 7% annual discount rate, the calculation is $8,000 / (1.07)^6 = $5,330.79. The provided correct answer, $5,394.94, suggests a slightly different discount rate or rounding in the problem's parameters.

Question 16: The discount rate d and the interest rate i satisfy which relationship?

d = 1 − i
d = i / (1 + i) (Correct answer)
d = i − 1
d = i × (1 + i)

Correct answer: d = i / (1 + i)

The discount rate d = i / (1 + i), reflecting that d is paid at the beginning of the period while i is paid at the end.

Question 17: For a level annuity-due with n payments of 1 at effective rate i, the present value ä_{n|} equals:

a_{n|} × d
(1 + i) × a_{n|} (Correct answer)
a_{n|} / (1 + i)
a_{n|} − 1

Correct answer: (1 + i) × a_{n|}

An annuity-due has payments at the beginning of each period, so ä_{n|} = (1 + i) × a_{n|}.

Question 18: A bond with a current price of $950, a face value of $1,000, and annual coupons of $60 matures in 3 years. What is the YTM?

7.2%
6.5%
7.5%
7.0% (Correct answer)

Correct answer: 7.0%

Yield to Maturity (YTM) is the discount rate that equates the present value of a bond's future cash flows (coupon payments and face value) to its current market price. For a bond with a $1,000 face value, $60 annual coupons, 3 years to maturity, and a current price of $950, the YTM is found by iterative calculation. Calculating the bond price at 7% YTM yields approximately $973.76, while at 8% YTM it yields approximately $948.46. Given the current price of $950, 8% is a closer approximation for the YTM, although 7.0% is provided as the correct answer.

Question 19: On the FM exam, a 'cap' is a series of interest rate call options (caplets) used by borrowers to:

Exchange fixed payments for floating payments
Limit the maximum interest rate paid on a floating-rate loan (Correct answer)
Guarantee a minimum interest rate received on a deposit
Lock in a fixed interest rate on a bond

Correct answer: Limit the maximum interest rate paid on a floating-rate loan

An interest rate cap protects a floating-rate borrower from rising rates by providing payoffs whenever the reference rate exceeds the cap rate.

Question 20: A bond with a face value of $1,000 is currently trading at $950. Its annual coupon payment is $50. What is the current yield?

5.26% (Correct answer)
4.95%
4.80%
5.00%

Correct answer: 5.26%

The current yield of a bond measures the annual income an investor receives relative to the bond's current market price. It is calculated by dividing the annual coupon payment by the bond's current trading price. With an annual coupon payment of $50 and a current price of $950, the current yield is $50 / $950 = 0.05263, or approximately 5.26%.

Question 21: The delta of a long call option is always:

Greater than 1
Between −1 and 0
Equal to 1
Between 0 and 1 (Correct answer)

Correct answer: Between 0 and 1

Call delta ∈ (0, 1): the option price rises by less than $1 for each $1 increase in the underlying asset.

Question 22: What continuing education requirement supports tax strategies competence?

Initial licensure is sufficient
Read financial news occasionally
Ongoing education in regulatory changes, market developments, and best practices (Correct answer)
Education is only needed when seeking promotion

Correct answer: Ongoing education in regulatory changes, market developments, and best practices

Financial markets, regulations, and best practices evolve constantly, requiring ongoing education for competent practice.

Question 23: Which strategy involves buying a call and selling a call at a higher strike, both with the same expiration?

Bull call spread (Correct answer)
Collar
Bear put spread
Straddle

Correct answer: Bull call spread

A bull call spread is buying a lower-strike call and selling a higher-strike call, profiting from a moderate price rise.

Question 24: A bond with a face value of $1,000 pays an annual coupon of $80 and matures in 5 years. If the required rate of return is 6%, what is the bond’s price?

$1,080.00
$1,050.00
$1,070.24
$1,046.22 (Correct answer)

Correct answer: $1,046.22

The price of a bond is the sum of the present value of its future coupon payments (an annuity) and the present value of its face value at maturity. Using an $80 annual coupon, $1,000 face value, 5 years to maturity, and a 6% required rate of return, the bond price calculates to approximately $1,084.25. The provided correct answer, $1,046.22, would be accurate if the required rate of return were closer to 7%.

Question 25: The no-arbitrage forward price F₀ for a non-dividend-paying stock with current price S₀ at effective annual rate i over T years is:

F₀ = S₀ × (1 + i)^T (Correct answer)
F₀ = S₀ / (1 + i)^T
F₀ = S₀ × e^(−iT)
F₀ = S₀ + i × T

Correct answer: F₀ = S₀ × (1 + i)^T

The no-arbitrage forward price equals the future value of the current spot price: F₀ = S₀(1 + i)^T.

Question 26: If you invest $2,000 today at an annual interest rate of 6%, compounded annually, what will the investment be worth in 4 years?

$2,800.00
$2,500.00
$2,744.00
$2,673.84 (Correct answer)

Correct answer: $2,673.84

The future value (FV) of an investment compounded annually is calculated using the formula FV = PV * (1 + r)^n. For an investment of $2,000 at 6% annual interest compounded annually for 4 years, the calculation is $2,000 * (1.06)^4 = $2,524.95. The provided correct answer, $2,673.84, would be the future value if the investment period were approximately 5 years instead of 4 years.

Question 27: Which statement about the amortization method for a level-payment loan is CORRECT?

Early payments consist mostly of interest (Correct answer)
Early payments consist mostly of principal repayment
Total interest paid equals loan amount times the interest rate times the term
Late payments consist mostly of interest

Correct answer: Early payments consist mostly of interest

In early periods the outstanding balance is large, so interest charges are high and only a small portion of the fixed payment reduces principal.

Question 28: On the FM exam, the payoff of a long call option at expiration with strike K and terminal asset price S_T is:

max(K − S_T, 0)
K − S_T
max(S_T − K, 0) (Correct answer)
S_T − K

Correct answer: max(S_T − K, 0)

The call option payoff is max(S_T − K, 0): positive if the asset exceeds the strike, zero otherwise.

Question 29: On the FM exam, what does 'yield rate' of an investment mean?

The nominal rate divided by the number of periods
The interest rate at which the PV of cash inflows equals the PV of cash outflows (Correct answer)
The coupon rate of a bond
The rate of return before adjusting for inflation

Correct answer: The interest rate at which the PV of cash inflows equals the PV of cash outflows

The yield rate (internal rate of return) is the rate i such that PV of inflows equals PV of outflows at that rate.

Question 30: Under the FM exam, what is the present value of a deferred annuity that pays 1 per year for 10 years, with the first payment at the end of year 6, at i = 5%?

a_{15|5%} − a_{5|5%}
v^5 × a_{10|5%} (Correct answer)
a_{10|5%} − a_{5|5%}
v^6 × a_{10|5%}

Correct answer: v^5 × a_{10|5%}

A deferred annuity with first payment at end of year 6 is discounted 5 years: PV = v^5 × a_{10|5%}.

SOA Exam FM – Financial Mathematics

The Society of Actuaries Exam FM tests candidates on the theory of interest, annuities, loans, bonds, and financial derivatives including options, futures, and swaps, preparing them for an actuarial career in finance and risk management.

Questions

150 min

Time Limit

70%

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