INTRODUCTIONTO

RISK MANAGEMENT

Defense Resources Management InstituteNaval Postgraduate School

Monterey, California

WHAT IS RISK?

• Arabic - Fortuitous and favorable.

• Greek - Fortuitous and neither favorable nor unfavorable.

• Latin (risicum) - the challenge that a barrier reef presents to a sailor.

• French (risque) - mainly negative connotation, but sometimes positive.

• Oxford Dictionary - “... the chance of hazard, bad consequences, loss, etc....”

DEFINITIONS I

DEFINITIONS II

• Economic risk - the chance of loss due to ….

• Business risk - the chance of loss associated with …

• Market risk - the chance that a portfolio of investments can

lose money because…..

• Inflation risk - the danger that a general increase in prices ...

• Interest-rate risk - market risk due to interest rate fluctuations

• Credit risk - the chance that of a default on a loan ...

• Liquidity risk - the difficulty in selling a fixed asset ...

• Derivative risk - the chance of financial loss due to increased

volatility ….

• Cultural risk - the chance of loss because of product market …..

CHANCERandom Occurrence

BAD CONSEQUENCESense of Loss

Hirschey & Pappas, Fundamentals of Managerial Economics Dryden Press, 1998

ALITTLE BIT

OFPROBABILITY

• It’s a number – it’s JUST A NUMBER!

• It’s a number between 0 and 1 ( 0 ≤ P ≤ 1)

• It quantifies the likelihood of an event

• It’s a function of experience, judgment, subjective assessment, available data

• It’s uses all information you think is relevant to the determination of the likelihood of occurrence of an event

PROBABILITY

Probability Rules

Probability = 0 if “never/impossible”

Probability = 1 if “always/certain”

If we are collectively exhaustive and

mutually exclusive then

the probabilities over the outocmes

SUM to 1.

Probability Rules

• If mutually exclusive then :

P(A or B) = P(A) + P(B)

• If independent :

P(A and B) = P(A) x P(B)

Probability from Data

• Given data we can always derive approximate probabilities using relative frequency.

• Relative frequency can be used as an estimate of the probability of the observed value

• Taken all together, these can represent the underlying PROBABILITY DISTRUBUTION FUNCTION

1.99 3.92 2.16 3.83 2.99 2.91 3.45 2.98 2.60 2.35 1.87 1.50 2.09 2.03 1.52 2.90 2.76 2.50 2.95 1.532.17 1.51 1.64 3.71 2.25 1.53 1.69 2.16 2.11 4.29 1.99 3.12 1.53 2.37 1.87 1.72 5.89 1.84 2.01 5.262.24 6.89 3.70 4.86 7.17 1.56 1.54 1.57 1.51 3.51 4.95 5.06 7.06 2.95 4.63 2.35 1.59 1.84 1.93 1.811.73 6.48 3.63 1.64 2.89 3.20 2.33 5.04 1.69 3.15 1.92 1.94 1.72 2.25 2.02 1.64 3.74 3.95 3.77 4.282.50 1.94 3.75 2.30 3.88 3.57 1.63 5.25 3.33 7.33 1.53 1.81 1.52 1.95 2.55 2.01 2.16 3.04 1.72 1.522.21 1.88 5.71 2.43 3.59 1.88 1.73 5.27 1.99 2.30 2.67 4.13 2.66 4.46 3.18 7.63 5.86 1.83 2.98 1.632.38 1.54 2.73 1.88 2.61 3.39 3.25 2.59 4.32 4.20 3.05 2.52 5.31 5.49 4.05 1.61 3.69 2.86 1.88 6.822.57 2.68 1.69 1.98 1.58 2.78 2.06 4.24 2.44 2.76 1.71 6.16 1.70 1.55 1.93 6.54 6.17 2.33 6.38 2.253.80 2.29 2.33 1.78 6.73 3.69 1.72 3.74 3.10 7.02 2.97 2.30 1.77 6.81 4.46 2.19 1.75 2.97 2.22 5.021.78 2.36 1.71 1.86 4.14 2.09 1.86 3.94 4.17 3.34 1.87 1.54 2.60 2.97 4.30 2.76 3.20 4.72 2.18 1.881.67 2.46 1.59 1.74 1.66 2.02 1.81 3.85 5.31 2.21 4.19 3.48 4.77 2.21 3.62 2.00 1.86 2.46 3.46 3.762.49 2.44 3.70 1.66 4.28 4.75 2.03 1.91 3.84 2.18 3.56 1.66 2.63 2.01 2.93 2.76 3.10 4.98 1.85 5.001.52 1.58 1.79 2.25 5.16 3.65 1.92 4.43 2.54 2.48 1.81 2.24 3.41 2.38 2.57 3.69 1.68 2.78 4.86 3.843.17 4.09 3.94 2.19 7.11 1.65 3.06 2.22 2.32 2.02 2.92 4.12 8.93 2.03 1.69 3.24 1.53 5.52 1.60 1.724.88 3.90 5.13 2.26 2.48 2.84 1.58 3.48 1.55 2.33 1.64 6.69 2.18 1.67 3.27 1.85 3.56 1.81 3.07 1.942.96 2.71 1.92 6.79 3.85 4.07 1.88 2.48 1.51 2.64 2.65 3.37 2.02 2.88 2.04 4.43 1.73 3.50 3.05 3.352.37 1.59 5.02 4.92 2.35 2.20 2.83 2.98 3.35 2.15 2.41 2.09 2.38 2.00 2.01 3.79 1.92 3.97 5.22 3.041.61 1.83 1.72 7.11 2.16 1.60 2.48 5.98 6.21 2.69 3.52 1.68 5.75 2.23 1.56 1.99 3.59 1.84 5.67 1.836.46 2.17 2.85 2.64 2.00 5.43 5.88 2.44 4.73 2.28 3.27 1.95 4.02 2.71 1.80 2.87 1.79 1.62 1.64 2.811.79 3.99 3.17 3.16 3.32 2.15 3.66 2.46 3.65 1.89 3.15 2.23 2.73 1.83 4.33 2.80 2.57 3.38 3.49 2.245.39 1.88 3.95 2.81 2.55 1.56 4.53 1.89 1.67 3.11 4.41 5.83 3.70 2.85 3.59 3.34 4.11 2.48 5.77 2.474.12 1.82 6.37 1.69 5.46 5.89 2.17 3.23 3.75 6.50 5.73 5.71 2.14 1.70 6.14 3.00 3.55 4.95 3.46 1.893.26 4.33 3.30 2.08 2.80 2.35 4.27 2.83 1.90 6.58 5.38 3.10 2.50 2.18 3.79 1.73 1.53 3.46 2.49 1.581.96 3.83 7.10 3.67 4.63 1.79 3.30 3.65 3.05 1.96 1.76 2.72 5.70 1.62 1.50 3.58 3.55 2.51 3.70 1.561.53 2.89 2.10 2.15 1.63 1.65 2.83 4.90 2.59 4.08 2.77 1.92 4.10 3.27 4.39 1.77 3.54 4.52 7.18 3.932.36 1.86 1.85 1.70 1.90 2.32 3.14 2.72 1.58 2.44 3.30 1.67 3.66 3.10 3.34 3.06 2.68 2.80 6.16 3.231.53 1.89 2.52 3.80 1.57 2.11 1.64 1.94 1.55 2.22 2.14 1.96 4.43 1.77 2.69 4.41 1.92 2.44 3.25 1.882.79 2.12 2.50 2.04 2.16 4.34 2.29 1.55 1.52 2.40 4.07 2.13 2.13 2.62 2.10 2.90 2.17 2.68 4.32 3.106.53 4.24 1.75 2.14 2.93 2.00 4.26 2.93 1.63 2.35 2.00 2.44 2.80 2.17 3.31 3.41 1.71 2.61 1.62 4.295.34 4.30 2.17 1.84 6.87 4.09 2.19 5.49 4.94 1.83 4.55 1.99 4.57 3.83 4.10 2.48 1.71 2.45 4.69 2.662.30 5.08 4.96 3.40 2.33 1.61 4.40 2.31 2.11 1.72 2.97 4.39 2.92 1.89 1.85 3.59 3.11 1.95 2.37 4.262.58 5.16 4.18 2.14 2.03 4.10 1.81 2.06 4.11 5.38 1.84 2.68 2.91 1.83 2.34 6.26 5.83 3.60 2.50 2.733.48 4.52 3.12 1.94 1.83 2.22 4.07 2.82 1.53 2.16 4.32 6.31 1.93 5.01 2.38 4.08 2.13 3.36 2.14 2.152.00 5.48 3.32 3.00 4.83 3.32 6.96 1.66 1.55 2.17 2.88 2.99 3.79 3.22 1.52 2.02 2.28 1.93 1.71 2.147.76 2.10 2.24 2.60 1.60 2.31 4.85 5.51 3.31 2.90 4.17 2.97 5.12 2.08 2.05 3.20 2.60 5.08 1.90 2.792.34 2.91 2.00 2.78 2.16 1.67 3.12 4.11 2.59 6.21 3.37 3.23 2.84 3.74 2.24 2.95 2.03 2.37 2.48 3.423.77 2.16 3.23 1.56 1.69 2.82 2.37 4.64 3.35 1.53 3.64 1.90 1.76 3.14 3.22 2.61 3.64 2.51 2.44 3.971.53 1.90 3.60 2.16 3.11 3.60 2.71 4.65 4.28 4.39 1.67 2.78 1.82 5.69 2.41 3.77 2.12 1.58 4.15 2.033.87 2.89 1.86 1.75 4.05 2.90 1.75 1.62 3.36 1.78 2.36 2.26 1.82 2.87 3.59 2.76 1.74 3.29 1.55 1.654.33 1.90 1.71 2.74 1.88 3.16 2.09 2.56 3.94 5.10 1.51 3.66 2.61 1.55 3.99 1.63 2.71 2.26 1.97 1.753.14 5.34 2.07 2.45 2.02 2.19 2.39 3.36 4.53 2.83 2.10 1.89 1.99 4.01 5.40 2.64 3.86 2.35 4.04 2.361.76 4.86 5.00 1.86 3.69 1.98 2.42 1.80 1.78 2.54 1.78 2.22 3.14 4.44 1.77 2.17 3.53 1.56 2.63 3.332.09 2.36 4.09 2.88 2.08 2.39 1.96 3.02 8.14 4.45 1.52 2.80 2.59 3.77 3.23 6.42 1.59 2.28 3.36 4.043.14 1.73 1.97 3.66 1.69 2.48 3.02 1.53 1.62 3.02 5.35 3.98 2.15 4.16 2.89 3.23 2.25 1.84 2.40 3.207.90 2.55 1.64 3.19 2.31 1.63 2.40 1.78 3.27 4.14 2.40 5.72 2.73 3.47 4.22 3.26 2.82 2.13 2.11 1.914.12 3.15 1.68 1.92 2.70 1.60 2.21 1.62 2.56 1.69 3.23 2.85 3.74 3.62 2.65 4.85 1.54 5.59 1.98 1.984.39 3.24 2.45 2.17 3.67 3.08 5.67 3.95 2.23 1.95 2.49 1.95 3.18 4.07 1.56 1.76 3.57 2.05 1.87 3.201.52 1.58 3.06 2.77 2.46 1.62 3.48 2.29 3.64 3.67 6.39 4.75 2.59 3.92 2.01 6.03 2.01 3.74 1.76 1.731.87 1.79 4.77 2.37 3.12 5.96 1.54 5.40 1.62 2.36 1.76 2.39 2.59 2.85 3.44 4.07 1.60 2.35 3.37 1.511.51 2.05 2.33 3.73 3.95 5.87 1.77 2.65 1.66 1.62 2.07 2.04 1.88 2.02 5.11 2.43 1.92 3.45 1.80 6.20

Earthquakes

1.50 1.58 1.67 1.77 1.86 1.95 2.06 2.17 2.31 2.44 2.60 2.80 2.97 3.20 3.41 3.69 3.98 4.30 4.95 5.831.50 1.58 1.67 1.77 1.87 1.95 2.06 2.17 2.31 2.44 2.60 2.80 2.98 3.20 3.42 3.69 3.99 4.32 4.96 5.831.51 1.58 1.67 1.77 1.87 1.96 2.07 2.17 2.32 2.44 2.60 2.80 2.98 3.20 3.44 3.69 3.99 4.32 4.98 5.861.51 1.58 1.67 1.77 1.87 1.96 2.07 2.17 2.32 2.44 2.61 2.80 2.98 3.22 3.45 3.69 4.01 4.32 5.00 5.871.51 1.58 1.68 1.77 1.87 1.96 2.08 2.17 2.33 2.45 2.61 2.80 2.99 3.22 3.45 3.70 4.02 4.33 5.00 5.881.51 1.58 1.68 1.78 1.87 1.96 2.08 2.18 2.33 2.45 2.61 2.81 2.99 3.23 3.46 3.70 4.04 4.33 5.01 5.891.51 1.59 1.68 1.78 1.88 1.97 2.08 2.18 2.33 2.45 2.61 2.81 3.00 3.23 3.46 3.70 4.04 4.33 5.02 5.891.51 1.59 1.69 1.78 1.88 1.97 2.09 2.18 2.33 2.46 2.62 2.82 3.00 3.23 3.46 3.70 4.05 4.34 5.02 5.961.52 1.59 1.69 1.78 1.88 1.98 2.09 2.18 2.33 2.46 2.63 2.82 3.02 3.23 3.47 3.71 4.05 4.39 5.04 5.981.52 1.59 1.69 1.78 1.88 1.98 2.09 2.19 2.33 2.46 2.63 2.82 3.02 3.23 3.48 3.73 4.07 4.39 5.06 6.031.52 1.60 1.69 1.78 1.88 1.98 2.09 2.19 2.34 2.46 2.64 2.83 3.02 3.23 3.48 3.74 4.07 4.39 5.08 6.141.52 1.60 1.69 1.79 1.88 1.98 2.09 2.19 2.34 2.47 2.64 2.83 3.04 3.23 3.48 3.74 4.07 4.39 5.08 6.161.52 1.60 1.69 1.79 1.88 1.99 2.10 2.19 2.35 2.48 2.64 2.83 3.04 3.24 3.48 3.74 4.07 4.40 5.10 6.161.52 1.60 1.69 1.79 1.88 1.99 2.10 2.20 2.35 2.48 2.65 2.83 3.05 3.24 3.49 3.74 4.07 4.41 5.11 6.171.52 1.60 1.69 1.79 1.88 1.99 2.10 2.21 2.35 2.48 2.65 2.84 3.05 3.25 3.50 3.74 4.08 4.41 5.12 6.201.52 1.61 1.70 1.79 1.88 1.99 2.10 2.21 2.35 2.48 2.65 2.84 3.05 3.25 3.51 3.75 4.08 4.43 5.13 6.211.53 1.61 1.70 1.80 1.89 1.99 2.11 2.21 2.35 2.48 2.66 2.85 3.06 3.26 3.52 3.75 4.09 4.43 5.16 6.211.53 1.61 1.70 1.80 1.89 1.99 2.11 2.21 2.35 2.48 2.66 2.85 3.06 3.26 3.53 3.76 4.09 4.43 5.16 6.261.53 1.62 1.71 1.80 1.89 2.00 2.11 2.22 2.35 2.48 2.67 2.85 3.06 3.27 3.54 3.77 4.09 4.44 5.22 6.311.53 1.62 1.71 1.81 1.89 2.00 2.11 2.22 2.36 2.48 2.68 2.85 3.07 3.27 3.55 3.77 4.10 4.45 5.25 6.371.53 1.62 1.71 1.81 1.89 2.00 2.12 2.22 2.36 2.49 2.68 2.86 3.08 3.27 3.55 3.77 4.10 4.46 5.26 6.381.53 1.62 1.71 1.81 1.89 2.00 2.12 2.22 2.36 2.49 2.68 2.87 3.10 3.27 3.56 3.77 4.10 4.46 5.27 6.391.53 1.62 1.71 1.81 1.90 2.00 2.13 2.22 2.36 2.49 2.68 2.87 3.10 3.29 3.56 3.79 4.11 4.52 5.31 6.421.53 1.62 1.71 1.81 1.90 2.00 2.13 2.23 2.36 2.50 2.69 2.88 3.10 3.30 3.57 3.79 4.11 4.52 5.31 6.461.53 1.62 1.72 1.81 1.90 2.00 2.13 2.23 2.36 2.50 2.69 2.88 3.10 3.30 3.57 3.79 4.11 4.53 5.34 6.481.53 1.62 1.72 1.82 1.90 2.01 2.13 2.23 2.37 2.50 2.70 2.88 3.10 3.30 3.58 3.80 4.12 4.53 5.34 6.501.53 1.62 1.72 1.82 1.90 2.01 2.14 2.24 2.37 2.50 2.71 2.89 3.11 3.31 3.59 3.80 4.12 4.55 5.35 6.531.53 1.63 1.72 1.82 1.90 2.01 2.14 2.24 2.37 2.50 2.71 2.89 3.11 3.31 3.59 3.83 4.12 4.57 5.38 6.541.54 1.63 1.72 1.83 1.91 2.01 2.14 2.24 2.37 2.51 2.71 2.89 3.11 3.32 3.59 3.83 4.13 4.63 5.38 6.581.54 1.63 1.72 1.83 1.91 2.01 2.14 2.24 2.37 2.51 2.71 2.89 3.12 3.32 3.59 3.83 4.14 4.63 5.39 6.691.54 1.63 1.72 1.83 1.92 2.01 2.14 2.24 2.37 2.52 2.72 2.90 3.12 3.32 3.59 3.84 4.14 4.64 5.40 6.731.54 1.63 1.73 1.83 1.92 2.02 2.14 2.25 2.38 2.52 2.72 2.90 3.12 3.33 3.60 3.84 4.15 4.65 5.40 6.791.54 1.63 1.73 1.83 1.92 2.02 2.15 2.25 2.38 2.54 2.73 2.90 3.12 3.33 3.60 3.85 4.16 4.69 5.43 6.811.55 1.64 1.73 1.83 1.92 2.02 2.15 2.25 2.38 2.54 2.73 2.90 3.14 3.34 3.60 3.85 4.17 4.72 5.46 6.821.55 1.64 1.73 1.83 1.92 2.02 2.15 2.25 2.38 2.55 2.73 2.91 3.14 3.34 3.62 3.86 4.17 4.73 5.48 6.871.55 1.64 1.73 1.84 1.92 2.02 2.15 2.25 2.39 2.55 2.73 2.91 3.14 3.34 3.62 3.87 4.18 4.75 5.49 6.891.55 1.64 1.73 1.84 1.92 2.02 2.15 2.26 2.39 2.55 2.74 2.91 3.14 3.35 3.63 3.88 4.19 4.75 5.49 6.961.55 1.64 1.74 1.84 1.92 2.02 2.16 2.26 2.39 2.56 2.76 2.92 3.14 3.35 3.64 3.90 4.20 4.77 5.51 7.021.55 1.64 1.74 1.84 1.93 2.03 2.16 2.26 2.40 2.56 2.76 2.92 3.15 3.35 3.64 3.92 4.22 4.77 5.52 7.061.55 1.64 1.75 1.84 1.93 2.03 2.16 2.28 2.40 2.57 2.76 2.93 3.15 3.36 3.64 3.92 4.24 4.83 5.59 7.101.56 1.65 1.75 1.84 1.93 2.03 2.16 2.28 2.40 2.57 2.76 2.93 3.15 3.36 3.65 3.93 4.24 4.85 5.67 7.111.56 1.65 1.75 1.85 1.93 2.03 2.16 2.28 2.40 2.57 2.76 2.93 3.16 3.36 3.65 3.94 4.26 4.85 5.67 7.111.56 1.65 1.75 1.85 1.94 2.03 2.16 2.29 2.41 2.58 2.77 2.95 3.16 3.36 3.65 3.94 4.26 4.86 5.69 7.171.56 1.66 1.75 1.85 1.94 2.03 2.16 2.29 2.41 2.59 2.77 2.95 3.17 3.37 3.66 3.94 4.27 4.86 5.70 7.181.56 1.66 1.76 1.85 1.94 2.04 2.16 2.29 2.42 2.59 2.78 2.95 3.17 3.37 3.66 3.95 4.28 4.86 5.71 7.331.56 1.66 1.76 1.86 1.94 2.04 2.16 2.30 2.43 2.59 2.78 2.96 3.18 3.37 3.66 3.95 4.28 4.88 5.71 7.631.56 1.66 1.76 1.86 1.94 2.04 2.17 2.30 2.43 2.59 2.78 2.97 3.18 3.38 3.66 3.95 4.28 4.90 5.72 7.761.57 1.66 1.76 1.86 1.95 2.05 2.17 2.30 2.44 2.59 2.78 2.97 3.19 3.39 3.67 3.95 4.29 4.92 5.73 7.901.57 1.67 1.76 1.86 1.95 2.05 2.17 2.30 2.44 2.59 2.79 2.97 3.20 3.40 3.67 3.97 4.29 4.94 5.75 8.141.58 1.67 1.76 1.86 1.95 2.05 2.17 2.31 2.44 2.60 2.79 2.97 3.20 3.41 3.67 3.97 4.30 4.95 5.77 8.93

Frequency Table

Richter Scale Number of Earthquakes

1.5 ≤ R < 2.5 474

2.5 ≤ R < 3.5 240

3.5 ≤ R < 4.5 158

4.5 ≤ R < 5.5 65

5.5 ≤ R < 6.5 38

6.5 ≤ R < 7.5 20

7.5 ≤ R < 8.5 4

8.5 ≤ R < 9.5 1

How Big? How Many?

Relative Frequency Table

Richter Scale Number of Earthquakes

1.5 ≤ R < 2.5 0.474

2.5 ≤ R < 3.5 0.240

3.5 ≤ R < 4.5 0.158

4.5 ≤ R < 5.5 0.065

5.5 ≤ R < 6.5 0.038

6.5 ≤ R < 7.5 0.020

7.5 ≤ R < 8.5 0.004

8.5 ≤ R < 9.5 0.001

How Big? How Many?

0.474

0.240

0.158

0.065

0.0380.020 0.004 0.001

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

[1.5 - 2.5) [2.5 - 3.5) [3.5 - 4.5) [4.5 - 5.5) [5.5 - 6.5) [6.5 - 7.5) [7.5 - 8.5) [8.5 - 9.5)

Relative Frequency Histogram

Prob. of an event = proportion of observations that corresponds to the event

= percent of observations that corresponds to the event

= portion of area of histogram that corresponds to the event

• Planning for retirement

• Two options for investment

• Each has a track record, the historical rates-of-return over a specified time period

• Each can be used to compute various statistics; e.g., average rate-of-return, etc.

AN INVESTMENT DECISION

AN INVESTMENT DECISION

Expected Value Std. Dev. Variance

A1 5.00% 1.25% 1.5625

A2 5.70% 2.75% 7.5625

Alternative 1

-2

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

time

Alternative 2

-2

0

2

4

6

8

10

12

14

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

time

r

n

n

r

Alternative 1

0

200

400

600

800

1000

1200

-0.30 -0.23 -0.16 -0.09 -0.02 0.05 0.12 0.19 0.26 0.33 0.40

Alternative 2

0

200

400

600

800

1000

1200

-0.30 -0.23 -0.16 -0.09 -0.02 0.05 0.12 0.19 0.26 0.33 0.40

What’s the likelihood of ?

What’s the likelihood of ?

r < 0

r < 0

I don’t want a rate ofreturn < 0!

I want a rate of return > 0!

THE FREQUENCY HISTOGRAM( The “KEY to it ALL’’ )

• The relative frequency histogram over the outcomes contains all relevant information.

• This information allows us to quantify risk.

• This is provides our most powerful tool for risk management.

A QUANTITATIVE

DEFINITION OF RISK

Risk is a COMBINATION of the answers to three questions:

(1) “What can go wrong?”

(2) “How likely is it to go wrong?”

(3) “If it does go wrong, what are the consequences?”

A QUANTITATIVE DEFINITION OF RISK

Adapted from S. Kaplan and B. John Garrick, “On the Quantitative Definition of Risk”, Risk Analysis, Vol.1, no.1, 1981

EXAMPLE: Hinterland Illegal Immigration

What can go wrong? recession;depression;economic collapse

How likely is it to go wrong? chances are 1 in a 10; a 10% chance; PF = .10

If it does go wrong, what happens to Drmecia? large numbers of illegalimmigrants ; increasing crime;failing social services; social unrest;

What can go wrong?

How likely is it to go wrong?

If it does go wrong, what are the consequences?

FuturescenarioF

Probabilityof FPF

Result dueto FY

A QUANTITATIVE DEFINITION OF RISK

THE ANSWER TO THE FIRST QUESTION

1. It all starts with the future scenario, F.

2. The F is uncertain so we need probability, PF.

3. F causes a result, an outcome of concern, Y.

4. Y is a function of F. We need to know this relation! The relation between Y and F is uncertain!!!

BEGINNING – MIDDLE – END

F → X → Y

F1, F2,… → X1 then X2 then… → Y1, Y2,…

X1 then X2 then….. XM

Y1

Y2

Y3

…YN

F1

F2

F3

…FK

THE “SYSTEM”

EXAMPLE: Hinterland Illegal Immigration

Illegal immigration is proportional to the ratio of per capita GDP.

GDPD/popD

GDPH/popH

Yillegal immigration =

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

8.0

9.0

10.0

1975 1980 1985 1990 1995 2000 2005 2010 2015 2020 2025

Year

Relative GDP per capita

0.0E+00

2.0E+04

4.0E+04

6.0E+04

8.0E+04

1.0E+05

1.2E+05

1.4E+05

1.6E+05

1980 1985 1990 1995 2000 2005 2010 2015 2020 2025

Annual Illegal Immigration

F =

G. H. Hanson (2009), “The Economics and Policy of Illegal Immigration in the U.S.”, Washington, D.C.: Migration Policy Institute

THE ANSWER TO THE SECOND QUESTION

X1 then X2 then….. XM PF PY

→ Math Model →Probability Distribution

for Future Scenarios

Probability Distributionfor

Outcomes of Interest

THE “SYSTEM”

SIMULATIONMODELING

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Val

ues

x 10

-̂6

I llegal Immigration [Millions]

Total Illegal Immigration (Baseline)

THE ANSWER TO THE SECOND QUESTION

PY

What number of illegal immigrants do you most want to avoid? 10000; 100000; 1000000; 10000000; 20000000.

What outcome do you most prefer to avoid: minor economic strain; substantial strain; or collapse of government social/educational services?

HOW YOU FEEL (about the possible Y)

= PREFERENCE

THE ANSWER TO THE THIRD QUESTION

1. It all starts with the future scenario, F.

2. The F is uncertain so we need probability, PF.

3. F causes a result, an outcome of concern, Y.

4. Y is a function of F. Given PF we can derive PY

5. How do you feel about the probable outcomes? Do you prefer to avoid some Y more than other Y?

THE ANSWER TO THE THIRD QUESTION

• Preferences < = > value function < = > v(Y) (1) v(Y) > 0 if Y is “good”

(2) v(Y) < 0 if Y is “bad”

• Value Function Charcteristics(1) reference point [defining GAINS from LOSSES]

(2) loss aversion [losses MORE IMPORTANT than GAINS]

(3) decreasing marginal values

THE ANSWER TO THE THIRD QUESTION

v(Y)

Losses ( - )convex

Gains ( + )concave

Illegal Immigration

Reference Point

A QUANTITATIVE DEFINITION OF RISK

1. It all starts with the future scenario, F.

2. The F is uncertain so we need probability, PF.

3. F causes a result, an outcome of concern, Y.

4. Y is a function of F. Given PF we can derive PY

5. Your preference info, v(Y), is the LAST PIECE!defines the consequences!

ProbabilityDistribution

(Outcome)

Decision MakerPreferencesAND

0.0

1.0

2.0

3.0

4.0

5.0

6.0

7.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Val

ues

x 10

^-6

I llegal Immigration [Millions]

Total Illegal Immigration (Baseline) v(Y)

Illegal Immigration, Y

AND

PY v(Y)AND

Risk

Fg(F)

Y

PF

g(F)

PY

v(Y)

Hinterland Economy Collapse Prob. of Economic Collapse

How does Drmecia “feel” about the Y?

Number of IllegalImmigrants

Prob. Dist. IllegalImmigrants

Y

v(Y)SPECIAL CASE OF PREFERENCE

Y

v(Y)

“I can’t bear the thought of experiencing loss! “

“Experiencing loss would be a catastrophe!”

In the limit the weight we assign to all outcomes <=> a losstends to - ∞.

In this case risk is very simple to quantify risk.

SPECIAL CASE OF PREFERENCE

ASSESSING THE RISK

PY

v(Y)

-8

-7

-6

-5

-4

-3

-2

-1

0

1

2

-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.5

+ -

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Valu

es x 1

0̂-6

Values in Millions

Total Illegal Immigration (Collapse)

0.20.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

1.0

2.0

3.0

4.0

5.0

6.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Valu

es x

10-̂

6

Values in Millions

Total Illegal Immigration (Collapse)

ASSESSING THE RISK

PY

v(Y)

-105

-85

-65

-45

-25

-5-3.5 -2.5 -1.5 -0.5 0.5 1.5 2.5 3.50.20.1 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0.0

1.0

2.0

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5.0

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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Valu

es x 1

0̂-6

Values in Millions

Total Illegal Immigration Collapse)

RISK = P{ Y correspond to loss }

RISK = P{ Y ≥ reference point }

RISK = P{ Y you prefer to avoid }

RISK = P{ unacceptable Y }

ASSESSING THE RISK (SPECIAL CASE)

OVERALL C-RATING System for Readiness:

C-1 = MAE > 89% P{not capable} ≤ 0.11C-2 = MAE 80-89% 0.11 ≤ P{not capable} ≤ 0.20C-3 = MAE 70-79% 0.21 ≤ P{not capable} ≤ 0.30C-4 = MAE 50-69% 0.31 ≤ P{not capable} ≤ 0.50C-5 = MAE < 50% 0.50 ≤ P{not capable}

Senate Armed Services Committee, terminology used in arguments before the committee, Feb. 1997

Who uses this stuff?.......

AR 220 – 1 (2010), AFI 10-201 (2006), SORTS (US Department of Defense)

A QUANTITATIVEAPPROACH TO

RISK MANAGEMENT

The History of Risk Management

• 1950 B.C. – Code of Hamurabi – formalization of bottomry contracts containing a risk premium for chance of loss of ships and cargo.

• 750 B.C. – Greece – the use of bottomry contracts.

• 1285 A.D. – King Edward - forbids use of soft coal in kilns to manage air pollution in London.

• 1583 A.D. – 1st life insurance policy issued in England.

• 19th and 20th century – water and garbage sanitation, building codes, fire codes, boiler inspections, railroads, steamboats, autos.

• 1959 A.D. – H. Markowitz, stock portfolio diversification.

Identify Risks

Assess Risks

ImplementMonitor

Management Action

RISK MANAGEMENT PROCESS

What can go wrong F?

What is F and PF?

What are the outcomes [Y, and PY]?

What are the consequences, v(Y)?

What is the risk [quantified]? Prevent

Mitigate

Negotiate

Tools of Risk Management

• Prevention.

• Mitigation.

• Hedging.

• Diversification.

Why not plot P{ Y ≥ y* } versus y*, for any y* ?THE RISK CURVE

ASSESSING THE RISK

National policy often specifies a reference point.

Definition depends on a reference point.

Not everyone has the same reference point.

Determining the Outcome Distribution

• theoretical derivation

• direct assessment

• simulationGenerating your own data

EARTHQUAKES

Total cost

Number ofearthquakes

Cost perearthquake

Earthquakepolicy

Program Cost

FIXED

Earthquake cost

VARIABLESize of

earthquake

1.00 0.652 0.11 0.012 0.002 0.00P{ Total Cost ≥ y* }

THE RISK CURVE

0.00

0.20

0.40

0.60

0.80

1.00

0 10 20 30 40 50 60 70

y*

P{Y>

y*}

A1 : Do Nothing

A2 : New Building Codes

A3 : Retro-Fit & New Codes

The RISK CURVES

compared

P{ Total Cost ≥ x }

A1 (red)

A2 (blue)

A3 (green)

0.065

0.328

0.652

ACCEPTABLE RISK

“The perennial question free people ask with regard to defense is:

‘How much is enough?’ To this there can be no precise answer.

A country’s security is a function of the DEGREE OF RISK A

COUNTRY IS WILLING TO ACCEPT.”

Hitch & McKean, The Economics of Defense in the Nuclear Age, Atheneum, 1986

ACCEPTABLE RISK

Risk

Cost

AcceptableRisk

Required Budget

Too Risky

Proposed Budget

Ultimately, policy makers must decide how much the United States is willing to pay to lower the risks associated with de-

ploying forces abroad. But some might argue that defense planners occasionally focus on absolute requirements – the

minimum number of forces that they believe will meetDoD’s military needs – without fully weighing the

relative risks and costs of alternative levels.

Moving U.S. Forces: Options for Strategic MobilityCongressional Budget Office, Feb. 1997

Who uses this stuff?.......

“Our armed forces remain capable, within an acceptable level of risk, of meeting the demands of our strategy.”

Maj. Gen. John J. Maher,Vice Director for Operations, Joint Staff:

testimony before House National Security readiness subcommittee,

Feb. 1997

Who uses this stuff?.......

“Computer security is basically risk management.”

Stephen H. Wildstrom,review of the book “Secrets and Lies by Bruce Schneier,

Businessweek Sept. 2000

Who uses this stuff?.......

“…. Managers have to decide what they are trying to protect and how much they are willing to spend, both in cost and

convenience, to defend it.”

“…we continue to believe the federal government can benefit from risk management.”

Raymond J. Decker,Director, Defense Capabilities and Management, GAO,

Testimony before the Senate Committee on Governmental Affairs Oct. 2001

Who uses this stuff?.......

“…. An effective risk management approach includes a threat assessment, a vulnerability assessment and a

criticality assessment ...”

Risk

Cost

RISK MANAGEMENT

old

new

Risk

Cost

RISK MANAGEMENT

Proposed Budget

new

Risk

Cost

RISK MANAGEMENT

AcceptableRisk

new

APPLICATION I

ENTERPRISE BUSINESSRISK

Correct

Correct

Error

P(correct) = 0.8

P(corrected) = 0.9

P(error) = 0.2

P(uncorrected) = 0.1

Data Entry

Reconciliation Check

P = .02

P = .18

P = .80SECNAV M-5200.35 March 2007

En

d U

ser

Ad

min

.S

up

po

rtS

po

nso

red

Pro

gra

mP

urc

has

eA

gen

tDirect or

Indirect/Reimb.?

Define Needs, prepare PurchaseRequisition Form

Forward to SPFA

Forwardto ASA

SPFA Reviews PR

FundsAvailable, etc.?

ASA reviews PR, confirms funds,

obtains approval

FundsAvailable, etc.?

SPFA assigns PR number and

form to PA

NO

NO

DIR.

YES

YES

INDIR.

En

d U

ser

Ad

min

.S

up

po

rtS

po

nso

red

Pro

gra

mP

urc

has

eA

gen

t

Purchaser reviews for

completeness of

documentation

All required info.present and adequate to

make procurement?

ASA /OA will Assign req., number and

task Purchaser

Clarify requirementswith end user

Screen request for mandatory

sources of supply,

prohibited or special items,

and authority to buy

NO

YESBuy from

mandatory source orgo open market?

En

d U

ser

Ad

min

.S

up

po

rtS

po

nso

red

Pro

gra

mP

urc

has

eA

gen

t

Place order with source, direct delivery point, and provide estimated

delivery date

Order completeand accurate?

Receive order (if delivery point)

Receive ordered items

and sign acknowledging

Reconcile with vendor

NO

YES

STOP

ReimburseOr

Direct Funds

0.9

0.9

0.9

0.9

0.9

0.1

0.1

0.1

0.1

0.1

ERROR

ERROR

ERROR

ERROR

ERROR

P = .1

P = .1

P = .09

P = .081

P = .0729

NOERROR

P = .3439

P = .6561

ASA

SPFA

Purchaser

ScreenRequest

ReceiptReview

What can go wrong?

How likely is it to go wrong?

ReimburseOr

Direct Funds

0.95

0.95

0.9

0.9

0.9

0.05

0.05

0.1

0.1

0.1

ERROR

ERROR

ERROR

ERROR

ERROR

P = .05

P = .05

P = .095

P = .0855

P = .07695

NOERROR

P = .30745

P = .69255

SPFA

ASA

Purchaser

ScreenRequest

ReceiptReview

ReimburseOr

Direct Funds

0.95

0.95

0.95

0.95

0.95

0.05

0.05

0.05

0.05

0.05

ERROR

ERROR

ERROR

ERROR

ERROR

P = .05

P = .05

P = .0475

P = .0451

P = .04287

NOERROR

P = .1855

P = .8145

SPFA

ASA

Purchaser

ScreenRequest

ReceiptReview

ReimburseOr

Direct Funds

0.99

0.99

0.99

0.95

0.99

0.01

0.01

0.01

0.01

0.01

ERROR

ERROR

ERROR

ERROR

ERROR

P = .01

P = .01

P = .0099

P = .009801

P = .009703

NOERROR

P = .0394

P = .9606

SPFA

ASA

Purchaser

ScreenRequest

ReceiptReview

APPLICATION II

COST RISKASSESSMENT

ESTIMATING SHIPBOARD HELICOPTER O&M COSTS

Life-cycle cost estimates for the helicopter are needed. The cost analysis staff is organized into four groups, one each for the four main components of the life-cycle cost: (1) R&D; (2) Procurement; (3) Operations and Maintenance; and (4) Salvage/Residual. As leader of the O&M cost estimating group you have decided to use a factor cost estimates since:

1. Relevant O&M cost data produce reliable CERs for the three components of

the O&M cost [POL, Parts, and “Other”] as functions of the procurement cost.

2. The helicopter is a recently developed model and procurement cost is expected to be $3.7 million (+/- 3%).

3. Why not just use an O&M cost factor approach: annual O&M cost = 10% of acquisition cost?

POL ($/hr) vs Acquisition ($M): Summary Output

Regression StatisticsMultiple R 0.736R Square 0.542Adjusted R Square 0.522Standard Error 28.224Observations 25.000

Coefficients Standard Error t Stat P-valueIntercept 112.845 22.447 5.027 0.000

Acquisition cost 30.155 5.778 5.219 0.000

Parts ($/hr) vs Acquisition ($M): Summary Output

Regression StatisticsMultiple R 0.922R Square 0.851Adjusted R Square 0.844Standard Error 24.425Observations 25.000

Coefficients Standard Error t Stat P-value

Intercept -84.956 19.426 -4.373 0.000Acquisition cost 57.215 5.000 11.442 0.000

Other ($/hr) vs Acquisition ($M): Summary Output

Regression StatisticsMultiple R 0.685R Square 0.470Adjusted R Square 0.446Standard Error 21.751Observations 25

Coefficients Standard Error t Stat P-value

Intercept 45.2104 12.857 3.516 0.002Acquisition cost 10.1249 3.310 3.059 0.006

POL[$/hr] = 112.84 + 30.16 × ACQ + error

Other[$/hr] = 45.21 + 10.12 × ACQ + error

Parts[$/hr] = -84.96 + 57.21 × ACQ + error

PROBABILISTIC COST ESTIMATING

y = a + b x + eCost estimateis a RANDOMVARIABLE

There always is the model residual error

Slope coefficient issubject to estimation error

Intercept issubject toestimationerror

Future explanatory variable is not always known withcertainty

PROBABILISTIC COST ESTIMATING

y = a + b x + eWhat is the resulting distribution function?

What is the most appropriate distribution function?

What is the most appropriate distribution function?

What is the most appropriate distribution function?

What is the most appropriate distribution function?

PROBABILISTIC COST ESTIMATING

Triangular Density

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.000 1.000 2.000 3.000 4.000 5.000 6.000

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Empirical Distribution Function

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10.0 12.0

X

Empirical Distribution Function

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0.0 2.0 4.0 6.0 8.0 10.0 12.0

X

y = a + b x + e

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Lognormal Distributions

0.00

0.20

0.40

0.60

0.80

1.00

1.20

1.40

1.60

0.00 0.50 1.00 1.50 2.00 2.50

x

?

?

cPOL = a1 + b1×ACQ + e1

cParts = a2 + b2×ACQ + e2

cOther = a3 + b3×ACQ + e3

CO,M&S = [cPOL + cParts + cother ] × H

$164K ≤ CO,M&S ≤ $676

$164K ≤ CO,M&S ≤ $672

$158K ≤ CO,M&S ≤ $511

$190K ≤ CO,M&S ≤ $463K

APPLICATION III

PROJECTMANAGEMENT

FACILITIES FOR THE FLIGHT SIMULATOR DEPARTMENT(ACTIVITIES, TIME ESTIMATES, AND DEPENDENCIES)

#

ACTIVITY DESCRIPTION

REQUIRED

PRECEDING ACTIVITIES

ACTIVITY

DURATION DAYS

A

Demolish areas 1 & 2

----

10

B

Demolish area 3

----

20

C

Dismantle Basic Simulator

----

10

D

Construct Bomber Simulator area

A

70

E

Construct Fighter Simulator area

A

40

F

Upgrade utilities

B

60

G

Construct Basic Simulator area

B

6

H

Reassemble Basic Simulator

C

48

I

Install Bomber Simulator

D

10

J

Install Fighter Simulator

E,F

27

K

Install Basic Simulator

G,H

40

New Facilities for the Flight Simulator Department

ACTIVITY DESCRIPTION a m b te s 2 te s 2

A Demolish Areas 1 & 2 8 17 28 17.3 11.1 0.0 0.0B Demolish Areas 3 30 60 180 75.0 625.0 75.0 625.0C Dismantle Basic Simulator 7 15 22 14.8 6.3 0.0 0.0D Construct Bomber Sim Area 85 120 206 128.5 406.7 0.0 0.0E Construct Bomber Fighter Sim Area 37 50 62 49.8 17.4 0.0 0.0F Upgrade Utilities 75 90 135 95.0 100.0 95.0 100.0G Construct Basic Sim Area 4 8 11 7.8 1.4 0.0 0.0H Reassemble Basic Sim 58 70 102 73.3 53.8 0.0 0.0I Install Bomber Sim 9 15 22 15.2 4.7 0.0 0.0J Install Fighter Sim 38 52 100 57.7 106.8 57.7 106.8K Install Basic Sim 37 50 62 49.8 17.4 0.0 0.0

Project estimated completion time = 227.7Variance = 831.8

Standard Deviation = 28.84

Critical Path

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

140 160 180 200 220 240 260 280 300 320 340 360

Shortest Time to Completion

P = 0.482

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

140 160 180 200 220 240 260 280 300 320 340 360

Shortest Time to Completion

P = 0.048

SUMMARY

• RISK is a factor in every decision with significant uncertainty

• RISK is a combination of the answers to 3 questions– what can go wrong?– how likely is it to go wrong?– if it does go wrong, what are the conse-

quences?

• RISK is quantified using PROBABILITY.

– use it to express the riskiness an

alternative.

– use it to find the least risky alternative.

• THINK about the RISK vs COST

tradeoff curve.

SUMMARY

• MANAGING RISK requires the information provided by the tradeoff curve!

– THINK about where you want to be on the curve.

– THINK about changing the tradeoff curve!

• USE THE MODEL to help find how to change things!

SUMMARY