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5744Multiplying by ElevenMultiply 73 by 11 in your head using the digit-insertion shortcut.脑筋急转弯简单数值题未尝试免费5745Multiplying by Twenty-FiveMultiply 864 by 25 in your head using the divide-by-four shortcut.脑筋急转弯简单数值题未尝试免费5746What Percent Is X of Y45 is what percent of 360? Work it out mentally.脑筋急转弯简单数值题未尝试免费5747Up Then Down by the Same PercentA price rises 20% and then falls 20%. What is the net percentage change from the start?脑筋急转弯中等数值题未尝试免费5748Odds to Implied ProbabilityAn event is quoted at 3-to-1 odds against. What is the implied probability that it happens?脑筋急转弯简单数值题未尝试免费5749Dividing by NineDivide 5000 by 9 mentally and give the result to two decimal places.脑筋急转弯中等数值题未尝试免费5750Quick Cube-Root EstimateEstimate the cube root of 30,000 to one decimal place mentally.脑筋急转弯中等数值题未尝试免费5751Kelly Fraction From Edge and OddsA binary bet pays net odds of 3-to-1 (you net 3 units per unit staked on a win, lose your stake on a loss). Your win probability estimate is 0.40. What full-Kelly fraction of capital should you stake?金融与交易简单数值题未尝试免费5752Even-Money Kelly FractionYou can repeatedly take an even-money bet (win 1 unit per unit staked, or lose your stake) that you believe wins with probability 0.55. What full-Kelly fraction of capital maximizes long-run growth, and what simple expression gives it for any even-money win probability p?金融与交易简单数值题未尝试免费5753Log-Growth Rate at Full KellyFor a repeatable even-money bet that wins with probability 0.60, you bet the full-Kelly fraction. What is the resulting expected log-growth rate per round (use natural log)?金融与交易中等数值题未尝试免费5754Growth Retained at Half-KellyFor a repeatable even-money bet winning with probability 0.60, full Kelly stakes 0.20 of capital. If instead you bet half-Kelly (0.10 of capital), what fraction of the full-Kelly expected log-growth rate do you retain? Give the ratio as a decimal.金融与交易中等数值题未尝试免费5755Kelly With Asymmetric Win and Loss SizesA single bet, if it wins, gains 2 units per unit staked; if it loses, it loses only 0.5 units per unit staked. Win and loss are each 50% likely. What full-Kelly fraction of capital maximizes expected log growth?金融与交易中等数值题未尝试免费5756Growth Rate at a Chosen Overbet FractionA repeatable even-money bet wins with probability 0.60 (full-Kelly fraction is 0.20). A trader instead stakes a fixed 0.30 of capital every round. What is his expected log-growth rate per round, and is it above or below the full-Kelly rate? Give the growth rate as a decimal.金融与交易中等数值题未尝试免费5757Kelly Capped by a Max-Loss LimitA binary bet pays net odds 2-to-1 and you estimate a win probability of 0.60, giving a full-Kelly fraction of 0.40. House policy forbids risking more than 25% of capital on a single position. Since a loss costs the full stake, what fraction should you actually bet, and how is it determined?金融与交易中等数值题未尝试免费5758Sensitivity of Kelly Fraction to the EdgeFor a binary bet at net odds b=2, the full-Kelly fraction is f*=(bp-q)/b. By how much does f* change per 0.01 increase in the estimated win probability p? Give the derivative df*/dp and the change for a 0.01 move.金融与交易中等数值题未尝试免费5759Kelly Sizing When the Win Probability Is UncertainFor an even-money bet you are unsure of the true win probability: it is equally likely to be 0.52 or 0.62. A colleague plugs the average estimate 0.57 into the Kelly formula and bets 0.14 of capital. To maximize expected log growth you should instead average the realized growth over the two possible true probabilities. Set up the correct objective and state, with a brief reason, whether the growth-optimal stake is at, above, or below 0.14.金融与交易困难数值题未尝试面试订阅5760Semiannual Coupon Bond PriceA 2-year bond with face 100 pays a 6% annual coupon rate in semiannual installments. Its yield to maturity is 5% per year (compounded semiannually). What is the bond's price?金融与交易中等数值题未尝试免费5761Clean Price From Dirty PriceA bond pays an annual coupon of 5. At settlement, 0.4 of the coupon period has elapsed and the invoice (dirty) price is 103.5. What is the quoted clean price?金融与交易简单数值题未尝试免费5762Macaulay Duration Of A 3-Year BondA 3-year annual-coupon bond has face 100, coupon rate 7%, and yield to maturity 6%. What is its Macaulay duration in years?金融与交易中等数值题未尝试免费5763Price Change From Modified DurationA bond trades at 100 with modified duration 6.2. Using the first-order (duration-only) approximation, estimate the new price if the yield rises by 25 basis points.金融与交易简单数值题未尝试免费