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Advance Finance
Question 1: Periodic Interest Rates
Calculating Periodic Rate and Effective Annual Interest Rate
Applied Formula by Fouque and Papanicolaou (2011):
Effective interest rate per period, (i) = ( 1 + ( r / m ) )m – 1
Effective interest rate for t periods, it = ( 1 + i )t - 1 or a single equation it = ( 1 + ( r / m ) )mt - 1.
The rate per compounding period P = R / m, in percent.
Where: r = R/100 and i = I/100 (p. 124)
Filled Table (Rounded to two decimal places)

Period | Annual percentage rate (APR) | Compounding per Period (m) | Periodic rate/Periodic Interest Rate (P) | Effective Annual Rate
(i) | Semiannual | 9% | 2 | 4.50% | 9.20% | Quarterly | 10% | 4 | 2.50% | 10.38% | Monthly | 8.5% | 12 | 0.71% | 8.84% | Daily | 3.25% | 365 | 0.01% | 3.30% |

Question 2: Periodic Interest Rate
Given:
This is a Saving Account, no adding or withdrawing from the account
Which is favorable?
Daily compound rate of 0.045%
Weekly compounded rate of 0.285%
Monthly compounded rate of 1.15%
Quarterly compounded rate of 4.325%
Semiannual compounded rate of 8.5%
(i) An interest rate compounded annually is favorable, since it yields the highest Effective Annual Rate of Interest, and will accumulate more interest after the given period.

Period | Annual percentage rate (APR) | Compounding per Year | Periodic ratePeriodic Interest Rate (P) | Effective Annual Rate
(i) | Annually | 16% | 1 | 16% | 16% | Semiannually | 8.5% | 2 | 4.25% | 8.68% | Quarterly | 4.25% | 4 | 1.06% | 4.32% | Monthly | 1.15% | 12 | 0.10% | 1.16% | Weekly | 0.285% | 48 | 0.01% | 0.29% | Daily | 0.045% | 365 | 0.00% | 0.05% |

(ii) Therefore Effective Annual Rate (in 2 decimal places) = 0.05%

Period | Annual percentage rate (APR) | Compounding per Year | Periodic ratePeriodic Interest Rate (P) | Effective Annual Rate
(i) | Daily | 0.045% | 365 | 0.00012% | 0.045% |

Question 3: EAR

Period | Annual percentage rate (APR) | Compounding per Year | Periodic ratePeriodic Interest Rate (P) | Effective Annual Rate
(i) | Semiannually | 6.75% | 2 | 3.38% | 6.86% |

Effective Annual Rate (i) with semiannual payments = 6.86%

Question 4: EAR

Period | Annual percentage rate (APR) | Compounding per Year | Periodic ratePeriodic Interest Rate (P) | Effective Annual Rate
(i) | Semiannually | 9.5% | 2 | 4.75% | 9.73% |

Effective Annual Rate (i) with semiannual payments = 9.73%

Question 5: Future Value with Periodic Rates
Using, Future Value Calculator, from: http://www.calculatorsoup.com/calculators/financial/future-value-calculator.php
Future Value (FV) of the Ordinary Annuity
Future Value for R1 = 5.5% is $ 2,368.79
Future Value for R2 = 9% is $ 2,813.21
Future Value for R3 = 12% is $ 3,279.17
Question 6: Savings with Periodic Rates * 1 Period = 1 Year * Present Value Investment PV = 0 * Number of Periods t = 3 (years) * Rate per period R = 6% * Compounding 12 times per period (monthly) m = 4 * Growth Rate per Period G = 0 * Payment Amount PMT =? * Payments per Period q = 4 (monthly) * Future Value (FV) of the Ordinary Annuity = $8000 8000 = 0 + PMT/0.015[(1+0.015)12-1]
Payment Amount PMT = $613.44

Question 7: Inflation, Nominal Interest Rates, and Real Rates
Nominal rate = real interest rate + inflation rate = 4.96+2.71 = 7.67
Real interest rate =

r = Real Interest Rate i = Nominal Interest Rate π = Inflation Rate

Fishers Equation Real Interest Rate r= 1+Norminal Interest Rate i 1+Inflation Rate (π) -1 i = 4.96% or 0.0496 π = 2.71% or 0.0271
Real Interest Rate r= 1+0.0496 1+0.0271-1

r = Real Interest Rate = 0.02191 r = Real Interest Rate = 0.02 (to two decimal places)

Question 8: Historical Interest Rates
Effective interest rate per period, (i) = ( 1 + ( r / m ) )m – 1
Effective interest rate for t periods, it = ( 1 + i )t - 1 or a single equation it = ( 1 + ( r / m ) )mt - 1.
The rate per compounding period P = R / m, in percent.

Effective interest rate for Treasury Bond periods = ( 1 + ( r / m ) )mt - 1. m = 1 t = 10 r = {(1+0.0412)10 – 1} r = 0.084097
Treasury bond periods in 10 years = 0.084097

Effective interest rate for AAA Corporate Bonds periods = ( 1 + ( r / m ) )mt - 1. m = 1 t = 10 r = {(1+0.0566)10 – 1} r = 0.11640

Treasury AAA Corporate Bonds in 10 years = 0.11640

Risk Premium/Default Premium The US Treasury Bond (T-bond) is normally used as the risk free rate for calculations, hence:

ra = AAA Corporate Bond = 0.11640 rf = Treasury Bond = 0.084097
Default Premium = 0.03 (Two Decimal Places)

Question 9: Profits
Profit or Loss = (Selling Price of Investment - Original Cost of Investment) + Distributions Received Investment | Original Cost of Investment | Selling Price of Investment | Distributions Received | Dollar Profit/(Loss) | CD | $200 | $210 | $0 | $10 | Stock | $37 | $30 | $4 | -$3 | Bond | $930 | $960 | $50 | $80 | Car | $33,000 | $20,000 | $0 | -$13,000 |

Question 10: Returns
Percentage Return = [(Selling Price of Investment - Original Cost of Investment) + Distributions Received - Costs]/ Original Cost of Investment

Investment | Original Cost of Investment | Selling Price of Investment | Distributions Received | Percentage Return | CD | $600 | $680 | $0 | 13.33% | Stock | $32 | $22 | $1 | -28.13% | Bond | $1,100 | $1,200 | $120 | 20.00% | Car | $45,000 | $17,000 | $0 | -62.22% |

Question 11: Comparison of Returns
Annualized Return = (End / Beginning)(1 / Num Years) – 1

Investment | Period (y) | Cost of Investment | Promised Payout | Effective annual rate of return | 1 | 5 | $320 | $640 | 14.87% | 2 | 7 | $320 | $896 | 15.85% | 3 | 10 | $320 | $1,536 | 16.98% |

Investment 3 is the most viable it has the highest annual rate of return

Question 12: Internet Exercise Year | Open Price(Jan) | End year prices(Close Price) -Dec | Dividend | Years return
(Profit or Loss) | Annual rate of return | 2000 | 139.44 | 131.19 | 2.291 | -5.959 | -4.27% | 2001 | 132.00 | 114.30 | 1.424 | -19.124 | -14.49% | 2002 | 115.11 | 88.23 | 1.498 | -25.382 | -22.05% | 2003 | 88.85 | 111.28 | 1.63 | 24.06 | 27.08% | 2004 | 111.74 | 120.87 | 2.197 | 11.327 | 10.14% | 2005 | 121.56 | 124.51 | 2.149 | 5.099 | 4.19% | 2006 | 125.19 | 141.62 | 2.446 | 18.876 | 15.08% | 2007 | 142.25 | 146.21 | 2.701 | 6.661 | 4.68% | 2008 | 146.53 | 90.24 | 2.721 | -53.569 | -36.56% | 2009 | 90.44 | 111.44 | 2.177 | 23.177 | 25.63% | 2010 | 112.37 | 125.75 | 2.266 | 15.646 | 13.92% | 2011 | 126.71 | 125.50 | 2.576 | 1.366 | 1.08% | 2012 | 127.76 | 142.41 | 3.103 | 17.753 | 13.90% | 2013 | 145.11 | 184.69 | 3.51 | 43.09 | 29.63% | | | | | | |
Total profit or loss = End year prices (Close Price) - Open Price (Jan) + Dividend
Annual rate of return = Total Profit or Loss/ Open Price (Jan)
Arithmetic annual average rate of return = (Sum/Number) = 4.85%

Year | Annual rate of return | Mean = x | Rate -mean | (x-x)2 | 2000 | -4.27% | 4.85% | -9.12 | 83.1744 | 2001 | -14.49% | 4.85% | -19.34 | 374.0356 | 2002 | -22.05% | 4.85% | -26.9 | 723.61 | 2003 | 27.08% | 4.85% | 22.23 | 494.1729 | 2004 | 10.14% | 4.85% | 5.29 | 27.9841 | 2005 | 4.19% | 4.85% | -0.66 | 0.4356 | 2006 | 15.08% | 4.85% | 10.23 | 104.6529 | 2007 | 4.68% | 4.85% | -0.17 | 0.0289 | 2008 | -36.56% | 4.85% | -41.41 | 1714.7881 | 2009 | 25.63% | 4.85% | 20.78 | 431.8084 | 2010 | 13.92% | 4.85% | 9.07 | 82.2649 | 2011 | 1.08% | 4.85% | -3.77 | 14.2129 | 2012 | 13.90% | 4.85% | 9.05 | 81.9025 | 2013 | 29.63% | 4.85% | 24.78 | 614.0484 | | | | | 4747.1196 |

= √(4747.1196/14)
Standard deviation = 18.41%

Question 13: Beta of a Portfolio

Beta of Portfolio 1 | | Beta | Weights | Beta (Weights | Stock G | 0.42 | 0.25 | 0.105 | Stock H | 0.71 | 0.25 | 0.1775 | Stock I | 1.23 | 0.25 | 0.3075 | Stock J | 1.59 | 0.25 | 0.3975 | | Portfolio 1 Beta = | 0.9875 | | |

Beta of Portfolio 2 | | Beta | Weights | Beta (Weights | Stock G | 0.42 | 0.30 | 0.126 | Stock H | 0.71 | 0.40 | 0.284 | Stock I | 1.23 | 0.20 | 0.246 | Stock J | 1.59 | 0.10 | 0.159 | | Portfolio 2 Beta = | 0.815 | | |

Beta of Portfolio 3 | | Beta | Weights | Beta (Weights | Stock G | 0.42 | 0.10 | 0.042 | Stock H | 0.71 | 0.20 | 0.142 | Stock I | 1.23 | 0.40 | 0.492 | Stock J | 1.59 | 0.30 | 0.477 | | Portfolio 3 Beta = | 1.153 | | |

| Weight in Stock G | Weight in Stock H | Weight in Stock I | Weight in Stock J | Beta Portfolios | Portfolio 1 | 25% | 25% | 25% | 25% | 0.99 |

References
Fouque, J., & Papanicolaou, G. (2011). Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives. Cambridge: Cambridge University Press.

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