between exact ages 5, 10. In both parts (a), (b) of the problem, assume
that the interest rate is fixed at 5%, and assume wherever necessary
that the individuals distribution of death-time is uniform
elapsed, and if he is alive at the end of n years he receives $15,000. This
contract is evidently a superposition of a n-year pure endowment with
face value $15,000 and a n-year temporary life annuity
result in the removal of a factor m from the right-hand sides of the last
two equations). Two other applications of the balancing-equation
principle can be made in calculating level premiums for insur
payment at time n in the case of a finite term n over which the
annuitant survives) as (life) annuities-immediate. The present value of
the insurance companys payment under the life annuity contract i
expression for the level annual pure-risk premium for the policy, in
terms of standard actuarial and interest functions. (17). Prove that for
every m, n, x, k, the net single premium for an n-year ter
except in the very special (completely artificial) case where (x+k) has
the same constant value for all x, k. In the latter case, where T is an
exponential random variable, it is easy to check from (5
at time Tm x + 1 m following policy initiation, if death occurs at T
between x and x + n. The present value of the insurance companys
payment under the contract is evidently F(T x) v Tmx+1/m if x T <
policy are therefore calculated using the conditional probability
distribution of T given that T x, which has density f(t)/S(x) at all times t
x. Define from the random variable T the related discret
CALCULATION Thus the premium amount to be paid at each payment
time is $ 47, 563 / 1.7589 = $27, 041 Alternatively, as a second example,
suppose that the purchaser is in effect taking out his
annuity/
can be obtained by combining (superposing or subtracting) these in
various ways. A further possibility, which we address in Chapter 10, is
4.1. EXPECTED PAYMENT VALUES 97 to restrict payments to some
solution P = F(0) A(m)1 x:ne m a (m) x:ne 1 2 A(m)1 x:ne It remains
only to remark what is the effect of loading for administrative expenses
and profit on insurance premium calculation. If all amounts
expand these functions in Taylor series about 0 up to quadratic terms.
Use the resulting expressions to approximate the coefficients (m),
(m) which were derived in the Chapter. Hence justify the so-ca
interest on the annuity value for one-half year is 38201(0.940.5 1) =
1200 . 5.7. USEFUL FORMULAS FROM CHAPTER 5 145 5.7 Useful
Formulas from Chapter 5 P([T] = x + k, T [T] < t| T x) = Z x+k+t x+k
f(y
assumption (i), Tm [T] is a discrete random variable taking on the
possible values 0, 1, . . . ,(m1)/m each with probability 1/m. Disregard
the interest and present-value discounting on the excess amo
fraction 1/m of a year during which death occurs, and life-annuities pay
regularly m times per year until the annuitant dies. The term or
duration n of the contract will always be assumed to be an int
nEx and is evidently equal to A 1 x:ne = nEx = v n npx (4.6) The other
contract frequently referred to in beginning actuarial texts is the
Endowment Insurance, which for a life aged x and term n is si
Labor Markets
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI8
The Labor Supply Decision
how consumers decide which goods they buy
and how they allocate their scarce time to
leisure and labor ar
The Firms Costs
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI6
lecture by Veronika Mikov
1
Profits Costs and Factors of Production
Profits,
profits are equal to the money the business
recieves
Usingg Demand and Supply
pp y
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI4
lecture by Veronika Mikov
1
The Concept of the Elasticity
elasticity = sensitivity to change
the effect of change
Demand, Supply
pp y and Price
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI3
The Role of Prices
Prices are how agents in the economy
communicate.
Prices provide information and incentive.
Pr
Modern Economics
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI1
What is Economics?
Economics studies how individuals, firms, the
government, and other organizations make
choices and how those c
The Consumption
p
Decision
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI5
Economic Decision Making
consumer defines an opportunity set - what is
possible given the constraints he faces
consum
The Effi
Th
Efficiency
i
off
Competitive Markets
Stiglitz, Walsh (2006)
Stiglitz
Economics
Chapter MI10
The General Equilibrium
it occurs when prices, wages and interest
rates are such that demand is
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5. North Korea must have its nuclear weapons program dismantled because it is an
irrational and unpredictable regime.
Two prospective:
For International / World
Its nuclear weapons are being improved
contract is an agreement to pay a scheduled payment to the
policyholder at every interval 1/m of a year while the annuitant is alive,
up to a maximum number of nm payments. Again the payment
amounts a
cumulative sum of the Dx column: Nx = X y=x Dy The expected
present value for the finite-duration life-annuity due is obtained as a
simple difference ax:ne = Xn1 k=0 v k+x lx+k Dx = Nx Nx+n Dx There
i
integer p. 64 tpk = S(k) t(S(k + 1) S(k) S(k) = 1 t qk under (i) p. 66
tpk = S(k + t) S(x) = e (k) t = (1 qk) t under (ii) p. 66 tpk = S(k + t)
S(k + 1) S(k + 1) S(k) = 1 qk 1 (1 t)qk under (iii) 94 C