### General information

First some general information regarding the exam.

Partial credits were assigned according to

Problem 1: a - 2, b - 4, c - 4,

Problem 2: a - 7, b - 3,

Problem 3: a - 6, b - 4,

Probem 5: a - 4, b - 6.

In problem 2, strictly speaking there are no arbitrage opportunities, even though we have a negative interest rate for the first cash flow time. However, those who have argued that one could keep the money in oneâ€™s pocket - that is, use a contract with discount factor one for any time - have been awarded full credit. Note however that since no such bond/contract is available in the given market, there is a set of discount factors consistent with the prices and thus no arbitrage.

In problem 3, you can be awarded full credit if you use the (erroneous) face value 250 for Bond C. The requirement is that you have commented on the then strange behavior of the discount factors - they do not follow the pattern we know discount factors / interest rates to (typically) have.

Lastly, for problem 4, what is needed for full credit is (i) compute the joint distribution of the bonds (or associated indicators if you prefer), (ii) formulate the maximization problem using the given utility function, and (iii) write down some form of the optimality conditions / equation system that will give the solution.

Grades have been assigned according to the following scores (incl. bonus points):

A: > 42

B: 38 - 42

C: 32 - 37

D: 28 - 31

E: 24 - 27

Fx: 22 - 23

F: < 22

For those of you who have received Fx, please contact Pierre. Your task for obtaining a passing grade will involve the topics on which you performed poorly on the exam (different for different individuals). Note that scores close to a passing grade have been re-checked after the initial grading. Thus, for any complaints on such scores you should make sure that you can make a strong case for why you should be awarded more points.

From the statistics (see next section) it is clear that problems 4 and 5 were more difficult than expected. In particular, the most common mistakes on both problems concerned basic probability, such as computing expectations and covariances, rules for conditioning and similar.

The other somewhat unexpected outcome was the poor performance on problem 3b. This was mainly due to many of you not formulating your intuitive thoughts on the problem in proper mathematical terms. Most of you, especially students who passed the exam, seemed to have an idea of how the solution should go, but did not / were not able to formulate it in a precise manner.

### Statistics

Total number of students that took the exam:

## [1] 54

Number of students that passed:

## [1] 31

Histogram for individual final scores (including bonus points):

A box plot for the score on the five problems:

Summary statistics for all students:

##      Prob 1           Prob 2           Prob 3           Prob 4
##  Min.   : 0.000   Min.   : 0.000   Min.   : 0.000   Min.   : 0.000
##  1st Qu.: 5.250   1st Qu.: 3.000   1st Qu.: 0.000   1st Qu.: 0.000
##  Median : 9.000   Median : 5.500   Median : 4.000   Median : 2.000
##  Mean   : 7.444   Mean   : 5.593   Mean   : 3.519   Mean   : 2.333
##  3rd Qu.:10.000   3rd Qu.: 8.750   3rd Qu.: 6.000   3rd Qu.: 3.750
##  Max.   :10.000   Max.   :10.000   Max.   :10.000   Max.   :10.000
##      Prob 5           Bonus            Exam            Sum
##  Min.   : 0.000   Min.   :0.000   Min.   : 0.00   Min.   : 2.00
##  1st Qu.: 0.000   1st Qu.:1.000   1st Qu.:17.00   1st Qu.:20.25
##  Median : 4.000   Median :2.500   Median :22.50   Median :26.00
##  Mean   : 4.481   Mean   :2.778   Mean   :23.37   Mean   :26.15
##  3rd Qu.:10.000   3rd Qu.:5.000   3rd Qu.:30.00   3rd Qu.:31.75
##  Max.   :10.000   Max.   :5.000   Max.   :50.00   Max.   :55.00

Summary statistics for students who passed the exam:

##      Prob 1           Prob 2           Prob 3           Prob 4
##  Min.   : 2.000   Min.   : 1.000   Min.   : 0.000   Min.   : 0.000
##  1st Qu.: 8.500   1st Qu.: 5.000   1st Qu.: 2.500   1st Qu.: 0.000
##  Median :10.000   Median : 7.000   Median : 5.000   Median : 3.000
##  Mean   : 8.903   Mean   : 6.935   Mean   : 4.806   Mean   : 3.419
##  3rd Qu.:10.000   3rd Qu.:10.000   3rd Qu.: 7.000   3rd Qu.: 5.500
##  Max.   :10.000   Max.   :10.000   Max.   :10.000   Max.   :10.000
##      Prob 5           Bonus            Exam            Sum
##  Min.   : 0.000   Min.   :0.000   Min.   :20.00   Min.   :24.00
##  1st Qu.: 3.000   1st Qu.:2.000   1st Qu.:25.00   1st Qu.:29.00
##  Median : 8.000   Median :4.000   Median :30.00   Median :30.00
##  Mean   : 6.355   Mean   :3.355   Mean   :30.42   Mean   :33.77
##  3rd Qu.:10.000   3rd Qu.:5.000   3rd Qu.:33.00   3rd Qu.:37.50
##  Max.   :10.000   Max.   :5.000   Max.   :50.00   Max.   :55.00

Summary statistics for students who failed the exam:

##      Prob 1           Prob 2           Prob 3          Prob 4
##  Min.   : 0.000   Min.   : 0.000   Min.   :0.000   Min.   :0.0000
##  1st Qu.: 2.000   1st Qu.: 1.000   1st Qu.:0.000   1st Qu.:0.0000
##  Median : 6.000   Median : 3.000   Median :1.000   Median :0.0000
##  Mean   : 5.478   Mean   : 3.783   Mean   :1.783   Mean   :0.8696
##  3rd Qu.: 8.000   3rd Qu.: 5.000   3rd Qu.:4.000   3rd Qu.:2.0000
##  Max.   :10.000   Max.   :10.000   Max.   :5.000   Max.   :4.0000
##      Prob 5           Bonus        Exam            Sum
##  Min.   : 0.000   Min.   :0   Min.   : 0.00   Min.   : 2.00
##  1st Qu.: 0.000   1st Qu.:0   1st Qu.: 9.00   1st Qu.: 9.50
##  Median : 0.000   Median :2   Median :16.00   Median :18.00
##  Mean   : 1.957   Mean   :2   Mean   :13.87   Mean   :15.87
##  3rd Qu.: 3.000   3rd Qu.:4   3rd Qu.:19.00   3rd Qu.:21.50
##  Max.   :10.000   Max.   :5   Max.   :23.00   Max.   :23.50

The grade distribution was as follows:

## A:  9.259259 %
## B:  5.555556 %
## C:  25.92593 %
## D:  20.37037 %
## E:  11.11111 %
## Fx:  11.11111 %
## F:  31.48148 %