Let Y denote the present value of a ten-year deferred 30-year temporary life annuity- 100, immediate on (30). If i = 0 and mortality is uniformly distributed with w = calculate the probability that the sum of the payments will exceed E (Y).
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- A corporate bond that pays 4% per annum semi-annually has a yield of 3% p.a. withcontinuous compounding and a remaining life of 1.5 years (immediate after couponpayment). The yield on a similar risk-free bond is 2% p.a. with continuous compounding. The risk-free rates are 1% p.a. with continuous compounding for all maturities.Assume that the unconditional probability of default per every six months is a constantand that defaults can happen at the end of every six months (immediate before couponpayment). The recovery rate is 40%. Estimate the unconditional probability of defaultusing the “more exact calculation”.Let X = the time between two successive arrivals at the drive-up window of a local bank. If X has an exponential distribution with λ = 1 Compute P(X≤4) and P(2≤X≤5)After an automobile is 1 year old, its rate of depreciation at any time is proportional to its value at that time. If an automobile was purchased on March 1, 2022, and its values on March 1, 2023 and March 1, 2024, were $7000 and $5800 respectively, what is its expected value on March 1, 2028?
- suppose x has an exponential distribution with probability density function f(x) =2e^-2x, x>0. Then P(X>1)The projected annual net cash flows associated with an investment opportunity are shown below in Table 5. The expected rate of return on the investment is 12%. Year Future Cash Flows Discount Factor Present Values 1 220000 K L 2 250000 0.7972 199300 3 300000 M N 4 200000 0.6355 127100 5 180000 0.5674 102132 Question: If the Net Present Value of the investment opportunity is an unfavourable R11 490, what is the initial outlay?A R811 490B R850 000C R848 510D R860 000For a fully discrete whole life insurance of 100,000 on each of 10,000 indepen- dent lives age 60, you are given the following values from the life table, A60 = 0.36913,2 A60 = 0.17741, i = 0.06. Let π be the annual premium for each insurance policy. (a) Use the normal approximation, calculate π, such that the probability of a positive total loss is 1%, given Φ0.99 = 2.326. (b) Let L be the aggregate present value of future loss random variable at issue. Use the normal approximation, calculate P (L < −10, 000, 000).
- Alex decides to sell his old guitar, and sequentially receives bids from potential buyers. The minimum price that he will accept to sell his guitar for is £500. Let {Xn, n ≥ 0} denote the sequence of independent and identically distributed bids that Alex receives, and assume that each Xn has the following probability density functionfX(x) = (1/400)e-x/400 for x ≥ 0. Let N denote the number of bids that Alex obtains before selling his guitar i.e., Alex sells his laptop to the Nth bid. Showing your full working, (a) find E[N]. (b) find E[XN].Carl decides to sell his old laptop on eBay, and sequentially receives bids from potential buyers. The minimum price that he will accept to sell his laptop for is $600. Let {Xn, n ≥ 0} denote the sequence of independent and identically distributed bids that Carl receives, and assume that each Xn has the following probability density function fX (x) = (1/400)e −x/400 for x ≥ 0. Let N denote the number of bids that Carl obtains before selling his laptop i.e., Carl sellshis laptop to the Nth bid. (a) Find E[N]. (b)Find E[XN ].If X has the exponential distribution given by f(x) =0.5 e−0.5x, x > 0, find the probability that x > 1.
- A company estimates that 0.1% of their product will fail after th original warranty period but within 2 years of the purchase, with a replacement cost of $350. If they offer a 2 year warranty for $18 , what is the company's expected value of each warranty sold?An insurance policy is written to cover a loss, X, where X has a uniform distribution on [0, 1000]. At what level must a deductible be set in order for the expected payment to be 25% of what it would be with no deductible?For the continuous probability function f(x ) = kx^2e^-x when 0≤x≤1. Find (a)k (b)mean (c)variance