In France, the viager system is a special type of home sale. Under this type of agree
ment, the buyer agrees to pay the seller a down-payment and/or a whole life annuity in
exchange of the house ownership. Normally, the sellers will be allowed to stay in the sold
property for the rest of their lives. This essentially makes the price of the property a life
This contract is attractive to the house owners, normally retired people because it allows
them to extract values from their asset without making them homeless. For the buyers,
these type of properties are normally priced lower than the market price. Moreover, they
can gamble on the life of owners. If the owner passes away shortly after the sale, the
buyer would have acquire a house with low price. However, the whole life annuity in this
contract has to be payed until the seller pass away even if they out live the buyer. So it
would be unfortunate if the buyer buys the house from Jeanne Calment (Google it).
In this assignment, your group is required to design a simple viager sell and help to set
the payment level.
You are tasked to calculate the payment level for the whole life annuity for a specific
viager house sale and also assess the variance of the whole life annuity’s present value.
The default setting is as follows:
(a) This sale takes place in a city/town of your choice, so not necessarily France. Please
name the place in the first page of your report.
(b) The sale is solely paid for by a whole life annuity, so no down-payment. The specific
of the annuity will be given later but the actuarial present value of the annuity should
be equal to 80% of the market value of a typical apartment/house at the sale location.
(c) To keep the question simple, the effective annual interest rate used for discounting
is assume to be a constant and positive through the whole contract. You may pick
this interest rate yourself but you have to provide support/reasoning.
(d) Use the uniform distribution of deaths assumption for fractional age.
(e) In order to diversify the assignments, the problem settings will be personalized.
Details will be given later in this document.
2 Mortality Models
Use the national life table for the location of your choice in the calculation and provide
proper reference. You are expected to adjust your calculation based on the condition of
the owner for the house. There are three possible owners.
(1) A 50 years old female smoker. It is assumed that the probability of dying in the
following year for all ages (qx; 8x) would be 1.4 times higher than national average.
Hint: Remember to adjust the probability at the limiting age such that the life table
remains a valid probability model.
(2) A 62 years old man who recently has surgery. After assessing the risk involved, the
selected probability of survival after having such a surgery are:
p = 0:7; p+1 = 0:8; p+2 = 0:82; p+3 = 0:85:
and the selection period is 4 years.
(3) The owners are a heterosexual couple, both at 60 years old. The whole life annuity
needs to be paid until both owners have passed away. For simplicity, you may
assume that the mortality of the partners is independent of each others.
3 Payment timing and frequency
All whole life annuity is annuity-due and all payment has the same face value. The pay
ment frequency has the following 4 options:
(2) Every 4 months;
(3) At the first month and the ninth month after closing the deal, repeating the same
pattern every year afterward;
4 Personalized assignment: explanation
By enrolling as a group on Nestor, you will get a group number. A code is associated to
each group number. This code uniquely determines your group assignment. For instance,
if you get 1-1, this means you have to finish the assignment assuming the house owner
is a 50 years old smoker with the payment-due every 6 months. For an overview of the
group numbers and the codes, a coresponding table will be uploaded to Nestor after the
grouping is finished. It is crucial that you do the assignment corresponding to your group
number. We will only grade your assignment if you have made the right one. So double
check your code and make sure that you make the right assignment.
5 Assignment tasks
Here we formulate the assignment tasks in detail:
• Tasks to perform: Explain the exact setting and parameters you used and justify
them. Determine payment according to your assigned questioned and the general
setting. Calculate the variance of the annuity. List out all the formula you have used
in your calculation. Give your recommendation to a couple of both 30 years old if
they should buy the house with your scheme.
• Bonus task: Based on your results or otherwise, should governments encourage
using the viager system as part of the retirement support, justify your answer. Rea
sonable attempt would result in a 0.2 extra to your assignment grade, a well argued
and supported answer would add 0.5 to the final grade.
Points of attention:
− As mentioned above, you have to choose appropriate values of yourself to finish
the assignment. Do this in such a way that all calculation can be performed and
− If you feel that additional assumptions have to be made, then do so and clearly
explain and motivate this in your report.
− We expect you to complete the necessary calculation in R as this is quite compli
cated to calculate by hand.
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