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Financial Math Final Exam Review Notes

by: Taylor Patterson

Financial Math Final Exam Review Notes ACCT 2001 001

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Taylor Patterson

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These notes cover ALL material covered in MATH 2610.
Prin. of Financial Accounting
Richard Kochanek (PI)
Study Guide
Math, Financial math, actuarial science
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This 13 page Study Guide was uploaded by Taylor Patterson on Thursday March 3, 2016. The Study Guide belongs to ACCT 2001 001 at University of Connecticut taught by Richard Kochanek (PI) in Fall 2015. Since its upload, it has received 55 views. For similar materials see Prin. of Financial Accounting in Accounting at University of Connecticut.

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Date Created: 03/03/16
Financial Mathematics I Math 2620 Final Exam Review Sheet Chapter 1 – Interest rates  Overview o Interest: the birth of money from money, payment by the borrower for the use of an asset that belongs to the lender over a period of time o Capital: the asset in question o Principal: monetary capital o Interest rate: percent of capital amount o Risk of default: risk that borrower will not be able to repay the loan principal  If some then lender normally requires a higher interest rate  Interest o Interest on saving accounts  Depositor is lender  Bank is borrower  Amount of interest earned from time t to time t+s is AV t+sAV t AV t+1 AV t  Interest rate is AV t o Interest earned on loans  Bank is lender  Individual is borrower  The principal amount of the loan is the amount provided to the borrower when the loan is originated(PV)  Interest begins to accrue when loan is originated and continues until loan is repaid in full  Simple Interest o AV tX(1+ti) o interest earned each year is $Xi  Compound Interest o Interest is earned on interest o AV =X(1+i) t t o simple interest is greater until t=1 then compound is greater  Accumulated Value o Use cash flow timelines o To accumulate further use formulas above o Accumulated Value factor: (1+i)  Present Value o Finds value now, at time 0, of a payment o Discounting a payment −t o Compound: PV tX(1+i) o Simple: PV =X(1+ti) −1 t o Present Value Factor: v = (1+i)−1  Rate of Discount o Discount: interest is being paid in advance o Amount of discount earned from time t to time t+s is AV t+sAV t o Discount rate is AV t+1– AVt AV t+1 o d=iv= i =1-v 1+i t o Compounded Present Value= X(1−d) o Compounded Accumulated Value= X(1−d) −t o Simple Present Value= X(1-td) −1 o Simple Accumulated Value= X(1−td)  Constant Force of Interest o Force of interest: continuously compounded interest rate  Instantaneous change in the account value  Expressed as an annualized percentage of the current value ' AV t o δ= =ln(1+i) AV t o i=e −1 δt o AV = e o PV = e−δt  Varying Force of Interest t 2 AVt 2 o AV = t 1dt= e AVt1 − δ dt o PV = t 1 e  Discrete Changes in Interest Rates o The annual effective interest rate might be x% for the first m years and then y% for the next n years o AV 0,p=X(AVF )(0,n n,m)(AVF m ,p Chapter 2 – Level Annuities  Overview o Level annuity –series of level payments made at uniform periodic intervals o Period certain annuity – annuity where the payments continue for a certain period of time o Contingent annuity – an annuity where the payments continue for an uncertain period  Annuity Immediate o First payment is at time 1 and last payment is at time n years o Payments occur at the end of each time period n o Present Value: X a = 1−v n¬i i  Valued at time 0 1+i ¿ ¿ o Accumulated Value: ¿n−1 ¿ Xs n¬i¿  Valued at time n 1+i o ¿ ¿ sn¬i=¿ n o sn¬iv =a n¬i  Annuity Due o First payment is at time 0 and last payment is at time n-1 years o Payments occur at the start of each time period 1−v n o Present Value: X ´ n¬i d  Valued at time 0 1+i ¿ ¿ o Accumulated Value: ¿n−1 ¿ X´sn¬i¿  Valued at time n year, 1 year after the last payment 1+i  ¿ ¿ sn¬i=¿  ´ v =a´ n¬i n¬i  a´n¬i = (1+i)an¬i  n = a a´n¬i n¬i  a´n¬i = an−1¬i1  Deferred Annuities o Deferred Annuity - An annuity that starts at some point after the first time period o Can be immediate or due o Present value – use a angle formula for n-years and multiply by v^(# years from 0) o Accumulated value – use s angle formula for n-years and multiple by (1+i)^(# of years to final date)  Continuously Payable Annuities o Payments are made continuously over the year n o Present Value: X ´ = 1−v n¬i δ  Valued at time 0 1+i ¿ o Accumulated Value: ¿ ¿n−1 ¿ X´sn¬i¿  Valued at time n o sn¬i =a´n¬i  Perpetuities o Perpetuity – an annuity with payments that continue forever o Perpetuity – Immediate  First payment occurs in 1 year and the annual payments continue indefinitely  Valued at time 0 1  PV = Xa ∞¬i= i o Perpetuity – Due  First payment occurs now and the annual payments continue indefinitely  Valued at time 0 1  PV = Xa∞¬i= d o a´∞¬i1+a ∞¬i o Continuously Payable Perpetuity  Payments are made as a continuous payment stream  Valued at time 0 1  PV = Xa´∞¬i δ o Accumulated values of perpetuities are infinite  Equations of Value o Equation of value – Present Value of Inflows = Present Value of Outflows Chapter 3 – Varying Annuities  Overview o Varying annuity – series of payments that increase or decrease predictably each period  Increasing Annuity – Immediate o First payment occurs in one year and each payment is higher than the previous one at a constant amount a a −nv n o PV = I ¿ or Pa n¬i+Q n¬i ¿ i X¿ s I ¿ sn¬in o AV = or Ps n¬iQ ¿ i X ¿  Increasing Perpetuity – Immediate o Infinite number of payments o Valued at time 0 a o I ¿ or P + Q ¿ i i2 X ¿  Increasing Annuity – Due o First payment is made now and each payment is higher than the previous one at a constant amount a a −nv n o PV = I ¿ or Pa´n¬i+Q n¬i ¿ d X ¿ ´ sn¬in o AV = I ¿ or P´sn¬iQ ¿ d X¿  Increasing Perpetuity – Due ´ o I ¿ ¿ X ¿  Decreasing Annuity – Immediate o First payment is made in 1 year and each payment is less than the previous one by a constant amount a o PV = D¿ ¿ X¿ s o AV = D¿ ¿ X ¿  Decreasing Annuity – Due ´ o PV = D¿ ¿ X ¿ s D¿ o AV = ¿ X¿  Continuously Payable Varying Annuities o Continuously Payable Increasing Annuity - Payment stream where the payments are made at a continuous rate but increase at discrete times a  PV = I ¿ ¿ X¿ ´ I ¿  AV = ¿ X ¿ o Continuously Payable Increasing Perpetuity – pays an amount continuously each year that increases each year forever a  PV = I ¿ ¿ X¿ o Continuously Payable Decreasing Annuity – payments are made at a continuous rate but decrease at discrete times ´  PV = D¿ ¿ X ¿ s D¿  AV = ¿ X¿  Compound Increasing Annuities o Use geometric series o (First term – next term) / (1 – common ratio)  Continuously Varying Payment Streams o Annuities when the amount of payment changes continuously b t −aδsds o PV = ∫ pte dt a b b ∫δsds o AV = p et dt ∫a t  Continuously Increasing Annuities o Continuously increasing and continuously payable payment stream ´ ´  PV = I ¿ ¿ X ¿ s  AV = I ¿ ¿ X¿ o Continuously increasing and continuously payable perpetuity a I ¿  PV = ¿ X ¿  Continuously Decreasing Annuity o Continuously payable and continuously decreasing annuity ´ D¿ o PV = ¿ X ¿ ´ D¿ o AV = ¿ X¿ Chapter 4 – Non-Annual Interest Rates and Annuities  Non-annual interest rates and discount rates (1+i) 1 o Effective interest rate to nominal rate: p(¿ ¿ −1 ) p ¿ 1 o Effective discount rate to nominal rate: p(1− 1(d ) )p  Nominal Rates m d ( ) 1− m o ¿ ¿ i )p (1+ ) =1+i=¿ p  Annuities o Bottom rate is nominal o Top rate is regular Chapter 5 – Project Appraisals and Loans  Discounted Cash Flow Analysis Cashinflow(t) Cashoutflow(t) o NPV = ∑ t −∑ t (1+r) (1+r) o r = required return, opportunity cost of capital o time period for r and t have to be consistent o want NPV at the highest o IRR = the rate at which the inflows = the outflows or NPV is 0  Nominal vs. Real Interest Rates o Inflation rate(r) – rate at which prices have increased o Real rate of interest – calculated by removing inflation (1+i)  I(real) = -1 (1+r) o Nominal interest rate – interest rate that takes inflation into account  (1+i) = (1+I)(1+r) nominalCashFlow o Real Cash Flow = t (1+r) nominalCF nominalCF realCF t =¿ ∑ t t= ∑ t o PV = (1+i) (1+r) (1+I) (1+I) ∑ ¿  Investment Funds B – A–CFs o DW = i = A− ∑ CF(1−t) F 1 F2 F 3 FT o TW = (1+i) = F 0× F1+c1 × F 2+c2 ×…× Fn+cn  Allocating Investment Income o Portfolio Method  Uses average portfolio rate for each year o Investment Year Method  Uses investment year rate for the increasing year  Loans: the amortization method o The loan amount is gradually reduced during term o Calculating loan balance  Prospective-> loan balance = PV(future loan payments)  Retrospective -> loan balance = AV(loan) – AV(loan payments made to date) L o Calculating level loan payment -> a =X n¬i o Interest portion of loan payment -> It=i t−1 L o Principal portion of a loan payment -> Pt= −I t an¬i  Loans: the sinking fund method o Pay service payments of interest and put a little bit of fund into a sinking fund which helps protect the lender and accumulates to the total balance owed on loan o SFP = L sn¬ j o Net amount of sinking fund loan = L - SFP st¬ j o L(1+i) =SP n¬i+¿ SFP sn¬ j Chapter 6 – Financial Instruments  Overview o Real assets – productive i.e. factories o Financial assets – claims on the income from real assets o One party’s financial asset is a financial liability to the other party o Financial instruments – provide financial capital and trade on financial markets  Types of Financial Instruments o Money Market Instruments – cash like investments that consist of short term, liquid, low-risk debt securities o Fixed income market – income generated is fixed at issue  Treasury Bills  Short term debt instruments issued by the U.S. government  Investors purchase T-bills at a discount to face value and earn an investment return from the difference between the purchase price and the face value  Price = (Face amount) [1-(discount yield) (n/360)] Faceamount Price  Annual Effective Yield on a T-Bill -> ¿ ¿ y=¿  Certificates of Deposit  Time deposit with a bank that cannot be withdrawn on demand  Investor receives interest and principal at the end of the fixed term  Cannot access deposit before maturity without penalty  FDIC insures up to $100,000  Commercial Paper  Short-term unsecured debt note, usually issued by a large corporation  Face amount is usually a multiple of $100,000  Relatively safe because term is short o Bonds  Debt instrument that requires issuer to repay the initial principal amount plus interest  Issuer (gov’t or corporation) borrows face amount from bond holder, pays coupon payments (interest payments) until maturity date, then pays redemption amount at maturity  Note – bond with term of 1 to 10 years  Issuer must pay off bondholders before stockholders  Default risk – risk that bond issuer is not able to make the coupon or principal payments  Types of bonds  Fixed-Rate Bonds: interest rate is fixed over life of bond  Floating-Rate Bonds: interest rate is allowed to fluctuate  Zero-coupon Bonds: does not pay interest until the maturity of the bond  Callable Bonds: gives the issuer the option to repay the principal before the maturity date  Convertible Bonds: includes the option to exchange the bond for a fixed number of shares of common stock of the company that issued the bond  Investment-grade Bonds: low expected default rate  High-yield Bonds/Junk Bonds: greater risk of default  Debentures: bonds that are not secured o Common Stocks  A share represents ownership in the issuing company  Equity  Dividends - owner of share is entitled to receive a share of the company’s profits  Residual – shareholders get paid after obligations to debt holders have been paid  Vote at company’s meetings on issues of corporate governance o Preferred Stocks  Hybrid between equity and debt  Fixed dividends  Paid before common dividends and after debt o Mutual Funds  Pools together deposits of many investors and places deposits under the control of a professional money manager  Each investor buys shares in the mutual fund  Active – money manager tries to outperform the market  Passive/index fund – money manager tries to match the market  Bond Valuation o Final payment – redemption amount + final interest payment o Bond Notation:  P – Price paid  F – face amount (nominal amount, par value)  r – coupon rate per payment period in terms of the face amount  g – coupon rate per payment period in terms of redemption amount  i – yield-to-maturity rate per payment period  C – redemption value (equal to F unless otherwise stated)  K – present value of redemption value (K = C(1+i) −n )  G – base amount (amount that if invested now would generate interest equal to the coupon payments indefinitely) (G = Fr/i)  n – number of coupon payments  Fr = Cg o Bond Price: n  P=Fr a n¬iCv or P=Fra n¬i+K  Be consistent and work in either semiannual or annual payment periods o Premium and Discount  Premium bond – P > C  Premium: P – C  g > i  Discount bond – C > P  Discount: C – P  g < i o Amortization of Premium and Accrual of Discount: constant yield method  Value at time of purchase (t=0), is equal to purchase price (P)  Value at time of maturity (t=n), is equal to redemption value (C)  Book value( BV t – value of bond at time t  Based on “book yield” (i) – yield at which bond is purchased  Does not change in response to changes in the market yield that occur after the bond is purchase  Prospective -> BV tFr a (n−t)¬i v(n−t)= Cga (n−t)¬i v(n−t) o BV 0P o BV nC t t  Retrospective -> BV tP(1+i) −Fr s t¬i = P(1+i)−Cg s t ¬i  Interest Earned: I =i BV t t−1  Premium Amortization – amount by which the BV decreases in value over the th time period t  Premium Bonds – positive amount (amortization of premium)  Discount Bonds – negative amount (accumulation of discount)  PA tBV t−1BV t  PA tCg−I t  PA =C(g−i)v n−t+1 t o All Book Value Formulas  BV =Cga +C v (n−t) t (n−t)¬i  BV =t(1+i) −Cgs t¬i  BV =tV t−11+i )−Cg  BV =BV +I −Cg t t−1 t  For premium: BV tBV t−1−PA t  For discount: BV =tV t−1(−PA ) t o Determining i from two premium amortization amounts:  Gives periodic effective yield PA t+k k  =(1+i) PA t o Amortization of Premium and Accrual of Discount: straight-line method  The BV is increased for discount bonds and reduced for premium bonds linearly between the price paid and the redemption amount over the term of the bond  BV =P 0  BV =n  BV −0V n P−C PA t n = n  I tCg−PA t  BV =BV −PA t t−1 t  Stock Valuation o Stockholders receives dividends which can vary over time with the profitability of the issuing company o No maturity date o If the stock is sold at some point in the future, the investor receives the market price at that time ∞ ¿ t o General Discounted Cash Flow (DCF): PV stock= ∑ t t=1(1+r)  r = risk adjusted required return o Level dividends:  If dividends will remain constant over time ¿1  PV stock= r o Constant dividend growth: ¿  PV stock= 1 r−g  r = annual effective interest rate, g = annual growth rate o Non-constant dividend growth:  A high growth rate can’t be assumed for very long o Price-to-earnings (P/E) ratio:  Earnings Per Share (EPS) = Net Income Numberof OutstandingShares  Sometimes referred to as net income per share Stock PricePerShare P0  P/E ratio = EarningsPer Share = EPS  High P/E ratio indicates that the price is high relative to the company’s earnings, expected growth  As P/E approaches zero or becomes negative, it loses meaning Chapter 8 – The Term Structure of Interest Rates  Overview o Term structure of interest rates – relationship between interest rates and their associated terms until maturity o Spot rates and forward rates  are derived from market prices of assets traded in the financial markets  allows us to identify whether an asset is priced consistently with other assets o Arbitrage opportunity- if an asset is undervalued or overvalued according to the term structure of the interest rates, then you can exploit the situation to make risk-free profit  Yield-to-Maturity o “Yield” equal to its Internal Rate of Return (IRR)  IRR – interest rate that equates the present value of an asset’s cash flows with its price CF t  P= ∑ t t>0(1+y)  P = asset price  y = annual effective yield o Yield Curve  Time on x- axis, yield on y-axis  On-the-run yield curve: yield curve constructed using the yields on the most recently issued US government securities  Par yield curve: theoretical construction, each bond’s coupon rate is assumed to be equal to the bond’s yield  Spot Rates ( st¿ o Interest rate that can be earned on a deposit made now and left to accumulate for a specified period of time o Used to discount a single cash flow occurring at a particular point in time CF t o Can be used to calculate the price of a security: P= ∑ t t>0(1+st) o When calculating a bond’s price use multiple spot rates o Derive Spot Rates:  Zero-coupon n-year bond: the n-year spot rate ( sn ) is the IRR  Bootstrapping:  Calculate the 1-year spot rate from the price of a 1-year bond then use this info to calculate the 2-year spot rate from the price of a 2- year bond, and so on  Based on the fact that the yields and the spot rates must produce the same prices  Otherwise arbitrage is possible o Ex. The PV of a bond based on yield is higher than the PV of the bond based on spot rates, buy bond from someone that is using spot rates and sell it to someone that is suing yield rates Fr Fr Fr+C  P= + 2+…+ n 1+s 1 (1+s 2 (1+s n o A bond’s yield-to-maturity can be viewed as a weighted average of the spot rates associated with each of the bond’s cash flows o The subscript (t) denotes the end point of the t-year time interval to which the spot rate applies, the beginning of the time interval to which the spot rate applies is 0  Forward Rates ( f ¿ t o Rate of interest that can be locked in now and that applies from one specified time to another specified time in the future o A set of spot rates implies a set of forward rates o Forward rate beginning at time t years (t-year forward rate): an interest rate specified now for an investment beginning at time t and lasting until time (t+1) P= CF t o Can be used to calculate the price of a security: ∑t>0(+ f )(+ f )(1+ f ) 0 1 t−1 o The subscript t denotes the beginning of the time interval to which the forward rate applies, can add a second subscript to denote the end point of the time interval f o Ways to refer to t :  The t-year forward rate  The forward rate applicable from time t to time (t+1)  The forward rate applicable to the th year (t+1)  The t-year deferred one-year forward rate  The one-year forward rate for year t  The t-year forward one-year rate o Derive Forward Rates: Fr Fr Fr+C  P= 1+ f +(1+ f )(1+ f ) +…+ 1+ f 1+ f …(1+ f ) 0 0 1 ( 0)( 1) n−1 o Flat Yield Curve: special case when the yields are the same for each maturity, if any curve is flat then all three must be flat  Relationships between Forward Rates and Spot Rates t o (1+s t = (+ f 0)(+ f 1)1+ f t−1 t o st= √( f 0)(1+ f1)(1+ f t−1)−1 (1+s )t o ft−1 t −1 (1+s t−1)−1


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