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Financial Risk Manager (GARP) · Part I

FRM P1 Valuation and Risk Models: practice questions with answers

6 free exam-style questions on Valuation and Risk Models for FRM P1 Part I. Try each one, then open the answer to read the explanation. Fincora has 120 original questions on this syllabus area.

Question 1

Halvorsen Capital holds an equity portfolio worth $40 million. Daily returns are assumed to be normally distributed with a mean of zero and a standard deviation of 1.1%, and returns are independent from day to day. What is the 10-day 99% value at risk (VaR) of the portfolio?

  1. A $1,023,593
  2. B $2,288,653
  3. C $3,236,885
  4. D $10,235,931
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Answer: C. With a zero mean, VaR = V x sigma x z.

- 1-day 99% VaR = 40,000,000 x 0.011 x 2.32635 = $1,023,593
- Under independence, volatility scales with the square root of time: 10-day VaR = $1,023,593 x sqrt(10) = $1,023,593 x 3.1623 = $3,236,885.

Question 2

An index trades at 100 and pays a continuous dividend yield of 2% per year. Its volatility is 22% per year and the continuously compounded risk-free rate is 4% per year. Using the Black-Scholes-Merton model, what is the value of a six-month European put option on the index with a strike price of 105?

  1. A 4.49
  2. B 7.86
  3. C 8.41
  4. D 9.72
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Answer: C. - d1 = [ln(100/105) + (0.04 - 0.02 + 0.22^2/2) x 0.5]/(0.22 x sqrt(0.5)) = [-0.04879 + 0.02210]/0.15556 = -0.1716
- d2 = d1 - 0.1556 = -0.3271
- N(-d1) = 0.5681; N(-d2) = 0.6282
- K x e^(-rT) = 105 x e^(-0.02) = 102.9209; S x e^(-qT) = 100 x e^(-0.01) = 99.0050
- p = 102.9209 x 0.6282 - 99.0050 x 0.5681 = 8.41

Check with put-call parity: c - p = S x e^(-qT) - K x e^(-rT) = -3.9159, and 4.4948 - 8.4106 = -3.9159.

Question 3

A portfolio manager holds $20 million face value of Bond A, which has a DV01 of $0.0845 per $100 face value. She wants to hedge against small parallel yield changes using Bond B, which has a DV01 of $0.0632 per $100 face value. Which trade gives a DV01-neutral position?

  1. A Sell $14.96 million face value of Bond B
  2. B Buy $26.74 million face value of Bond B
  3. C Sell $20.00 million face value of Bond B
  4. D Sell $26.74 million face value of Bond B
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Answer: D. - DV01 of position = 20,000,000/100 x 0.0845 = $16,900 per basis point
- DV01 of Bond B per $1 face = 0.0632/100
- Face of Bond B to sell = 16,900/(0.0632/100) = $26,740,506 (about $26.74 million)

Hedge ratio = DV01(A)/DV01(B) = 0.0845/0.0632 = 1.3370. The hedge protects only against small parallel shifts.

Question 4

Larchmont Endowment holds a $15 million equity portfolio. The portfolio's one-year continuously compounded (log) return is normally distributed with a mean of 8% and a standard deviation of 20%, so the portfolio value at the horizon is lognormally distributed. What is the one-year 95% value at risk (VaR) of the portfolio?

  1. A $3,305,958
  2. B $3,734,561
  3. C $4,205,039
  4. D $4,934,561
Show the answer

Answer: A. Under lognormal VaR, the 5% quantile of the log return is mu - z x sigma = 0.08 - 1.6449 x 0.20 = -0.24897.

- Portfolio value at the 5% quantile = 15,000,000 x exp(-0.24897) = 15,000,000 x 0.77960 = $11,694,042
- VaR = 15,000,000 - 11,694,042 = $3,305,958

Because a log-return of -24.90% corresponds to a simple loss of only 22.04%, lognormal VaR is below the normal-return figure of $3,734,561; it also can never exceed the value of the portfolio.

Question 5

A 6-month zero-coupon Treasury trades at 98.30 per 100 of face value, and a one-year Treasury with a 5% coupon paid semiannually trades at 100.45. Using only these two bonds, what is the arbitrage-free price of a one-year Treasury with an 8% coupon paid semiannually?

  1. A 103.36
  2. B 105.85
  3. C 106.16
  4. D 108.40
Show the answer

Answer: A. Bootstrap the discount factors:

- d(0.5) = 98.30/100 = 0.98300
- 5% bond: 2.5 x d(0.5) + 102.5 x d(1) = 100.45, so d(1) = (100.45 - 2.5 x 0.98300)/102.5 = 0.95602

Price of the 8% bond = 4 x 0.98300 + 104 x 0.95602 = 3.9320 + 99.4265 = 103.36

Any other price would allow an arbitrage, because the 8% bond's cash flows can be replicated with a portfolio of the zero-coupon bond and the 5% bond.

Question 6

The GBP/USD spot rate is 1.3000 (USD per GBP). Sterling's volatility is 12% per year, and the continuously compounded risk-free rates are 2% for USD and 6% for GBP. Using a two-step Cox-Ross-Rubinstein binomial tree (u = e^(sigma x sqrt(dt)), d = 1/u), what is the value in USD of a one-year American call option on GBP 1 with a strike of 1.2800?

  1. A $0.0200
  2. B $0.0426
  3. C $0.0530
  4. D $0.0836
Show the answer

Answer: C. - dt = 0.5; u = e^(0.12 x sqrt(0.5)) = 1.08856; d = 0.91865
- p = (e^((0.02 - 0.06) x 0.5) - d)/(u - d) = (0.98020 - 0.91865)/(1.08856 - 0.91865) = 0.3623
- Terminal rates: 1.5404, 1.3000, 1.0971; payoffs 0.2604, 0.0200, 0
- Up node (S = 1.4151): holding value = e^(-0.01) x (0.3623 x 0.2604 + 0.6377 x 0.0200) = 0.1060; exercise value = 0.1351, so exercise early
- Down node: holding value = 0.0072; exercise value = 0
- Today: e^(-0.01) x (0.3623 x 0.1351 + 0.6377 x 0.0072) = $0.0530 (immediate exercise gives only 0.0200)

The European call is worth 0.0426. A high foreign interest rate acts like a high dividend yield, which can make early exercise of an American call optimal.

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