SOA Exam FM – Financial Mathematics — Questions and Answers

Question 1: Which statement about the amortization method for a level-payment loan is CORRECT?

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

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 2: A straddle is formed by:

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

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: The yield rate (internal rate of return) of an investment is defined as the interest rate i at which:

The present value of all cash outflows equals the present value of all cash inflows (Correct answer)
The future value of all payments equals the original investment amount
The net present value of the investment is maximized
The investment generates the highest possible nominal return

Correct answer: The present value of all cash outflows equals the present value of all cash inflows

The yield rate satisfies NPV = 0, meaning the PV of inflows equals the PV of outflows — it is the break-even discount rate.

Question 4: What is the present value of $8,000 to be received 6 years from now, with a discount rate of 7% annually?

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

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 5: 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 + i × T
F₀ = S₀ × e^(−iT)
F₀ = S₀ × (1 + i)^T (Correct answer)
F₀ = S₀ / (1 + 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 6: An interest rate swap in which Party A pays a fixed rate and receives a floating rate results in Party A having:

Floating-to-fixed exposure, benefiting if floating rates fall
A long bond position with no interest rate risk
Zero net interest cost regardless of rate movements
Fixed-to-floating exposure, benefiting if floating rates rise (Correct answer)

Correct answer: Fixed-to-floating exposure, benefiting if floating rates rise

Paying fixed and receiving floating means Party A benefits when floating rates rise above the fixed rate.

Question 7: 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?

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

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

Bear put spread
Bull call spread (Correct answer)
Collar
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 9: What is the present value annuity factor a(5, 6%) for an ordinary annuity at an annual effective interest rate of 6% for 5 years?

3.9927
4.2124 (Correct answer)
5.6371
4.4651

Correct answer: 4.2124

a(n,i) = (1 - v^n)/i = (1 - 1.06^-5)/0.06 = (1 - 0.74726)/0.06 = 4.2124.

Question 10: A forward contract on the FM exam obligates the buyer to:

Exchange interest payments with a counterparty
Purchase an asset at a predetermined price on a specified future date (Correct answer)
Pay a premium for the right to buy at market price
Sell an asset at any time before expiration

Correct answer: Purchase an asset at a predetermined price on a specified future date

A forward contract locks in a purchase price (the forward price) for delivery of the asset at a future date.

Question 11: Which of the following correctly describes a varying force of interest δ(t)?

The accumulated value is ∏(1 + δ(t))
The accumulated value is 1 + ∫₀ᵗ δ(s) ds
The accumulated value is e^(δ·t)
The accumulated value is exp(∫₀ᵗ δ(s) ds) (Correct answer)

Correct answer: The accumulated value is exp(∫₀ᵗ δ(s) ds)

With a time-varying force of interest, the accumulation factor is the exponential of the integral of δ(s) over [0, t].

Question 12: A loan of $20,000 is repaid with equal annual payments over 10 years at an annual effective interest rate of 5%. What is the annual payment?

$2,800.00
$2,443.67
$2,590.09 (Correct answer)
$2,312.20

Correct answer: $2,590.09

Payment = L/a(10, 5%) = 20000/7.7217 = $2,590.09, where a(10, 5%) = (1 - 1.05^-10)/0.05.

Question 13: 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)
S_T − K
max(S_T − K, 0) (Correct answer)
K − S_T

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 14: How should tax strategies performance be reported to clients?

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

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 15: What is the relationship between the nominal rate compounded monthly (i^(12)) and the equivalent effective annual rate i?

(1 + i) = (1 + i^(12)/12)^12 (Correct answer)
(1 + i) = (1 + 12 × i^(12))^(1/12)
(1 + i) = (1 + i^(12))^12
(1 + i) = 12 × i^(12)

Correct answer: (1 + i) = (1 + i^(12)/12)^12

The effective annual rate i satisfies (1 + i) = (1 + i^(12)/12)^12, converting monthly compounding to an annual equivalent.

Question 16: If the force of interest is δ = 0.05, what is the accumulation factor over 3 years under continuous compounding?

(1.05)^3
e^(0.15) (Correct answer)
e^(0.05)
1 + 0.05 × 3

Correct answer: e^(0.15)

Under continuous compounding the accumulation factor is e^(δt) = e^(0.05 × 3) = e^(0.15).

Question 17: Using the prospective method, the outstanding loan balance at any point in time equals:

The original loan amount minus cumulative principal paid
The original loan amount plus total accrued interest
The present value of all remaining future payments (Correct answer)
The future value of all payments already made

Correct answer: The present value of all remaining future payments

The prospective method defines outstanding balance as the present value of all future scheduled payments discounted at the loan interest rate.

Question 18: Put-call parity for European options states:

C − P = K·v^T − S₀
C × P = S₀ × K·v^T
C + P = S₀ + K·v^T
C − P = S₀ − K·v^T (Correct answer)

Correct answer: C − P = S₀ − K·v^T

Put-call parity: C − P = S₀ − K·v^T, where C and P are call and put prices, S₀ is spot, K is strike, and v = 1/(1+i).

Question 19: A bank has a loan portfolio with the following characteristics: Probability of Default (PD): 3% Loss Given Default (LGD): 40% Exposure: $10,000,000 What is the expected loss?

$300,000
$1,200,000
$400,000
$120,000 (Correct answer)

Correct answer: $120,000

Expected Loss (EL) in credit risk quantifies the average loss anticipated from a loan or portfolio over a specific period. It is calculated by multiplying the Probability of Default (PD), the Loss Given Default (LGD), and the Exposure at Default (EAD). For a PD of 3% (0.03), LGD of 40% (0.40), and an exposure of $10,000,000, the expected loss is 0.03 * 0.40 * $10,000,000 = $120,000.

Question 20: For a continuously varying interest rate, the present value factor v(t) is:

exp(∫₀ᵗ δ(s) ds)
e^(−i·t)
1 / (1 + i·t)
exp(−∫₀ᵗ δ(s) ds) (Correct answer)

Correct answer: exp(−∫₀ᵗ δ(s) ds)

The present value factor is the reciprocal of the accumulation factor: v(t) = exp(−∫₀ᵗ δ(s) ds).

Question 21: 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)
5.00%
4.95%
4.80%

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 22: How should risk be assessed in tax strategies?

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

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 23: A collar strategy involves:

Selling both a call and a put at the same strike
Buying a put at a lower strike and selling a call at a higher strike to hedge an asset (Correct answer)
Buying a call and a put at the same strike
Buying two calls and one 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 24: What regulatory compliance requirement applies to financial planning?

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

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 25: If you receive $200 annually forever, and the discount rate is 5%, what is the present value?

$5,000
$4,000 (Correct answer)
$4,500
$4,200

Correct answer: $4,000

The present value of a perpetuity, which is a constant stream of payments received indefinitely, is calculated by dividing the annual payment by the discount rate. In this scenario, an annual payment of $200 divided by a 5% (0.05) discount rate yields a present value of $200 / 0.05 = $4,000. This formula is a simplified way to value an infinite series of cash flows.

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

35 min

Time Limit

70%

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