Uncertainty Problem Set Flashcards
Exercise 1 The decision maker is faced with two payment plans, plan A and plan B. Under
plan A, they pay $400 now and at the beginning of each of the following two years. Under
plan B, they pay $976 now.
(a) Suppose the annual interest rate is 10%. Which plan is preferable according to their
PDV?
Each year A pays 400 but 400 in one year doesn’t have the same value as 400 today time value of money since you can invest it and make a return on it.
One of the formula’s commonly used to take int o account time is
FV = PV(1+r)^t
From this equation we get
N
PV = Σ FV/ (1+r)^t
t = 0
Then the PV of option A is given by the FV of 400
PV = 400/ 1.1^0 + 400/ 1.1^1 + 400/(1.1)^2
PV = 1094.22
According to their PDV, payment plan B is preferable because you pay less ten option A.
Suppose the annual interest rate is 30%. Which plan is preferable now? Briefly, explain
how the PDV depends on the interest rate
Under plan A, they pay $400 now and at the beginning of each of the following two years. Under
plan B, they pay $976 now.
The annual interest rate is 30%.
N PV = Σ FV/ (1+r)^t
t = 0
PV = 400/ 1.3^0 + 400/1.3^1 + 400/ 1.3^2 = $944.38
$944.38 < 976
Payment plan A is preferable. PVA < PVB. The interest rate determines the opportunity cost
of time. Paying $976 today comes at a higher opportunity cost if the interest rate is high because if they money was paid later it could be invested and would earn a lot of interest.
For what interest rate r do both payment plans have the same PDV?
Under plan A, they pay $400 now and at the beginning of each of the following two years. Under
plan B, they pay $976 now.
−400 + −400/1 + r + −400 (1 + r)^2
= -976
- Rearrange and solve for r
- Multiply LCM bt (1+ r)
- Expand Out
- Simplify
- Let x = r
- Use the quadratic equation
- Use the plus part of the quadratic
- Solve for x which is r
r = 0.25
Exercise 2 An asset pays $10 for the next T years at an annual interest rate of 10% starting
next year.
(a) Suppose T → +∞. What is the PDV of the asset?
PV is the sum of the discounted future payments
PV = FV/ (1+r)^0 + FV/ (1+r)^1 + FV/(1+r)^2 + ………
We can re-write this as
∞
PV = Σ. V/(1 + r)^t = 1+r / r * V
t = 0
∞ PV = Σ. 10/(1 + 0.1)^t
t = 0
∞ PV = 1+ 0.1 / 0.1 * 10 = 110
However the payment stream only starts next year, at t = 1
1/1.1 * 110 = $100
or 110 - 10 = 100
Exercise 2 An asset pays $10 for the next T years at an annual interest rate of 10% starting
next year.
Suppose T = 8 and the price of the asset is $60. Should you buy it?
The Present Value of future payments is:
8
PVD = Σ. 10/(1 + 0.1)^t
t = 0
= (1 + r) ^T - 1 / r(1+r)^T * V
= (1.1)^ 9 - 1 / 0.1(1.1)^8 *10
However this is used when the payment stream starts at t = 0 while here the stream starts at
t = 1
1.1^9 - 1/ 0.1(1.1)^9 *10 = 57.59
57.59 < 60
You shouldn’t buy it, if you pay $60 today you will have a loss because the present value of the future payments is smaller then the price.
There are three decision makers 1, 2 and 3.
U1(x) =2x
U2(x) = ln(2 + x)
U3(x) = ln(1 + x)
(a) Compare the degree of risk aversion between decision makers.
Calculate the degree of risk aversion from their coefficient
- Decision Maker 1:
Coefficient of Risk Aversion for the first decision maker:
U1(x) =2x
R1A(X) = - U”(x)/U’(X)
First Derivative = 2
Second derivative = 0
R1A(X) = - U”(x)/U’(X) = - 0/2 = 0
Since R1A(X) = 0, Decision Maker 1 is Risk Neutral
- Decision Maker 2:
U2(x) = ln(2 + x)
R2A(X) = - U”(x)/U’(X)
First Derivative = 1/2+ x
Secound Derivative = -1/(2+ x)^2
R2A(X) = - U”(x)/U’(X)
= - -1/(2+x)^2 / 1/2+ x
= 1/ 2+x > 0
R2A(X) > 0 , Decision Maker 2 is Risk Adverse
- Decision Maker 3
U3(x) = ln(1 + x)
R3A(X) = - U”(x)/U’(X)
First Derivative = 1/(1+x)
Secound Derivative = -1/(1+x)^2
R3A(X) = - U”(x)/U’(X)
=-1/(1+x)^2/1/(1+x) = 1/1+x
R3A(X) > 0 Decision Maker 3 is risk adverse
Consider lottery L = (p, 1 − p),
p = 1/2over {0, 100}. Calculate the the expected utility
from the lottery for each player.
U1(x) =2x
U2(x) = ln(2 + x)
U3(x) = ln(1 + x)
Remember Expected Utility use the utility function.
Decision Maker 1:
U1(x) =2x
E(U) = 1/2( 2(0) + 1/2(2(100)
E(U) = 100
Decision Maker 2:
U2(x) = ln(2 + x)
E(U)
= 1/2 [ ln(2+0)] + 1/2[ln(100+0]
= 2.65905
Decision Maker 3:
U2(x) = ln(1 + x)
E(U)
= 1/2 [ ln(1+0)] + 1/2[ln(100+1]
= 2.3075
Consider the willingness to pay of each decision maker for the lottery.
Info Needed:
Decision Maker 1:
U1(x) =2x
E(U) = 100
Decision Maker 2:
U2(x) = ln(2 + x)
E(U) = 2.65905
Decision Maker 3:
U2(x) = ln(1 + x)
E(U) = 2.3075
This means calculate the certainty equivalent.
Certainty equivalent is the guaranteed level of wealth that provides the same level of utility as the expected utility of risk.
Decision Maker 1:
100 = 2x
x = 50
CE1 = 50
Decision Maker 2:
2.65905 = ln(2+x)
e^2.65905 = 2+ x
x = 12.282
CE2 = 12.282
Decision Maker 3:
2.3075 = ln(1+x)
e^2.3075 = 1+x
x = 9.0492
CE3 = 9.0492
Calculate the risk premium for each decision maker. How do the risk premium of decision maker 2 and 3 compare? Explain
Decision Maker 1:
CE1 = 50
Decision Maker 2:
CE2 = 12.282
Decision Maker 3:
CE3 = 9.0492
Risk premium is the amount the risk adverse person is willing to pay to avoid risk.
- Calculate the expected value (without using the utility function)
E(V) = 0.5(0) + 0.5(100) = 50
Decision Maker 1:
Risk premium = E(V) = CE1
= 50 - 50 = 0
Decision maker 1 is not risk adverse so they have no risk premium they are risk neutral.
Decision Maker 2:
Risk Premium = E(V) - CE2
= 50 - 12.28
= 37.72
Decision Maker 3:
Risk Premium = E(V) - CE3
= 50 - 9.04
= 40.96
Uncertainty II
Exercise 1 The decision maker is faced the lottery L = (1/2, 1/2) over
X = {$0,$100}.
The decision maker has no income other than from the lottery. The utility function of the decision maker is ln(x). They can buy a unit of insurance at price $q. Denote the amount
of insurance the decision maker buys by I.
(a) Set up the expected utility function of the decision maker
1/2(ln(1-q)I) + 1/2(ln(100-qI)
Uncertainty II
Exercise 1 The decision maker is faced the lottery L = (1/2, 1/2) over
X = {$0,$100}.
The decision maker has no income other than from the lottery. The utility function of the decision maker is ln(x). They can buy a unit of insurance at price $q. Denote the amount
of insurance the decision maker buys by I.
Argue that the decision maker will not not choose to be uninsured regardless of what q is.
If I = 0 and x = $0 (which happens with 50% probability), the utility
is ln(0) → −∞. Buying some small amount of insurance even if its price is very
high is always better than that
Suppose I = 2 and q = 1/2. Calculate the expected utility of the decision maker. Furthermore, calculate the certainty equivalent CE. What is the interpretation of
CE?
E(U) =
1/2(ln(1-q)I) + 1/2(ln(100-qI)
Substitute the I = 2 and q = 1/2
E(U) =
= 1/2(ln(1-1/2)(2) + 1/2(ln(100-(1/2)(2))
E(U) = 2.2975
Calculate the Certainty Equivalent:
E(U) = ln(x)
2.2975 = ln(x)
x = 9.949
CE = 9.95
Certainty equivalent is the guaranteed level of wealth provides the same amount of utility as the expected utility of risk.
The decision maker is indifferent between receiving 9.95 with 100% probability and receiving $1 and $99 with 50% probability
Derive the amount of insurance I* the decision maker purchases as a function of q
The expected utility function:
1/2(ln(1-q)I) + 1/2(ln(100-qI)
max (1/2ln(1-q)I + 1/2ln(100-
I. qI)
- Find the first order condition
d/ dI = 1/2(1-q/I-Iq) + 1/2(-q/100-qI)
0 = 1/2I - 1/(q/100 - qI)
Solve for I * as a function of q
I* = 100/2q
From I* = 100/2q derivation,
Suppose q = 3/4. What is the expected profit of the insurance company?
Expected Profit
= 1/2 (3/4 -1)I* + 1/2(3/4)I*
1. Find I* value
I* = 100/2q = 100/2(3/4) = 66.66
- Plug I* into expected profit
= 1/2 (3/4 -1)I* + 1/2(3/4)I*
= = 1/2 (3/4 -1)(66.66) + 1/2(3/4)(66.66)
Expected Profit = 16.66
Suppose q = 1/2. Is the insurance actuarially fair? Show that the decision maker is fully insured.
Fully Insured I = 100
Calculating Expected Profit:
1/2(1/2 - 1)I + 1/2(1/2) I =0
The insurance is actuarily fair if the expected profit is 0.
1/2(1/2 - 1)100 + 1/2(1/2) 100 =0
To show this:
If x = $0 the decision maker receives net insurance fees
0 + (1-q)I
= 0 + (1-1/2)100 = 50
if x = $100 the decision maker receives
100 - (1-q) * I
100 - (1-1/2) *100 = 50
Either way the decision maker receives $50 they are fully insured.
