Financial Math, candidates are expected to have a high degree of understanding of time value of money principles, security valuation and basic statistics. Formulas are provided on the exam, therefore do not memorize them only know how to use them! Candidates report that a lot of relatively easy marks are often missed because of a lack of familiarity with basic Math skills and most surprising simply not knowing how to use a financial calculator correctly. If you are weak in Math and have never used a financial calculator, we suggest you use the Texas Instruments BA II PLUS.do not buy the HP 10B or 12C.you will spend weeks just learning how to turn it on! The BA II PLUS is a much simpler machine to operate, it can do all of the required calculations and it costs about one half the price of the HPs. www.examsuccess.ca 1 Email: info@examsuccess.ca
Present Value of an Annuity The present value of an annuity tells us how much money we need to have today in order to fund a specific dollar amount of income over a specific period of time paid at a specific time interval. If that sounds like a mouthful, let`s look at an example to clarify exactly what we are talking about. Consider the following example: Assume that Jarrod would like to be able to withdraw $1,000 per year for the next 3 years from his TFSA. If the local bank pays interest on savings held in a TFSA of 3.25% and Jarred makes his withdrawals at the end of the year, how much will Jarred need to have in his account today? The answer is $2,815.07 Time Lines Time lines can help visualize the amounts and the timing of the cash flows. In Jarred s case the time line would look like this, cash flows occur at the end of the time period: -$2,815.07 $1,000 $1,000 $1,000 Today End of End of End of the first the second the third year year year www.examsuccess.ca 2 Email: info@examsuccess.ca
We can use the following formula to solve for the present value of an annuity : PVA = PMT x 1 - (1 + i) -n i Where: PVA = Future Value of the Annuity PMT = Annuity amount i = stated annual interest rate n = number of time periods In Jarred s case we can calculate exactly how much he needs to have in order to withdraw $1,000 per year for the next 3 years, in just one step by using the formula: PVA = PMT x 1 - (1 + i) -n i = $1,000 x 1 - (1 + 3.25%) -3 3.25% = $2,815.07 We could also use the time value of money buttons on a financial calculator to calculate this as well: END MODE FV = $0 Remember the future value of an annuity is ALWAYS 0! i = 3.25 n = 3 PMT = $1,000 Solve for PV = $2,815.07 Enter the interest rate as a whole number, do not hit % or make it a decimal Enter the number of time periods Payments refers to the amount of income paid out each time period. You get a negative number indicating an outflow or deposit www.examsuccess.ca 3 Email: info@examsuccess.ca
Timing of the Cash Flows You should be aware of the END MODE and BEG MODE functions on your calculator and how they affect your calculation. Recall the time line above, the cash flows occur at the end of the time period. What do you think would happen if the cash flows occurred at the beginning of the time period? BGN MODE FV = $0 Remember the future value of an annuity is ALWAYS 0! i = 3.25 n = 3 PMT = $1,000 Solve for PV = $2,906.56 Enter the interest rate as a whole number, do not hit % or make it a decimal Enter the number of time periods Payments refers to the amount of income paid out each time period. You get a negative number indicating an outflow or deposit With the payments at the beginning of the time period you get a different answer! www.examsuccess.ca 4 Email: info@examsuccess.ca
Quick Summary Time Value of Money Concepts A. Types and measures of investment returns Time Value of Money Lump-Sum Formulas on the left and the corresponding calculator key strokes on the right, followed by an example. Present value of a lump sum payment Enter the following information given in the question: PV = FV FV n I PMT solve for: PV ( 1 + I ) n Example: You require $10,000 in 5 years to payoff your car loan. If your savings can grow annually at 8% (ignoring taxes), how much must you deposit in your savings account today? PV = $10,000 Enter -$10,000 FV, 5 n, 8 I, 0 PMT (1.08) 5 Solve for PV = $6,805.83 = $6,805.83 Future value of a lump sum payment Enter the following information given in the question: FV = PV( 1 + I ) n PV n I PMT solve for: FV Example: If your savings can grow annually at 11% (ignoring taxes), how much will $3,500 grow to in 15 years? FV = $3,500 (1.11) 15 Enter -$3,500 PV, 15 n,11 I, 0 PMT = $16,746.06 Solve for FV = $16,746.06 www.examsuccess.ca 5 Email: info@examsuccess.ca
Examples: Your client stated that she will need $10,000 in 4 years to pay for her daughter s wedding. Your client would like to know the minimum amount of money she would need to invest in a time deposit in order to achieve her goal? Assume the current rate on 4 year time deposits is 4%. The answer is $8,548.04. Your client has $5,250 in cash in an RRSP account. What will the account balance be at the end of 10 years if you were to invest the entire amount is a Government of Canada Strip Bond that is yielding 6% per annum. The answer is $9,401.95. Exam Tip: Make sure you are comfortable with your financial calculator. Many exam candidates make mistakes because they do not know how to use their calculator correctly! Simple time value of money questions do show up on the Exam and are easy points! www.examsuccess.ca 6 Email: info@examsuccess.ca
15. Yield to maturity (YTM) The YTM for a bond represents an expected return over the life of the bond assuming that all coupons are reinvested at that same YTM! This assumption is quite unrealistic, as interest rates are likely to change over time. (Link to reinvestment risk) Bond dealers use YTM as a convenient way to determine the market value of a bond. Yield to Maturity = Coupon + (1,000 - Market Price B ) n (1,000 + Market Price B ) / 2 Example: Calculate the YTM for a 6 year annual pay 10% GOC bond purchased at 975. 100 + (1,000-975) YTM = 6 Enter $100 PMT, 6 n, -975 PV, 1,000 FV (1,000 + 975) / 2 = 10.55% Solve for I = 10.58% 16. Market value of a bond 17. Market value of a strip bond The market value of a bond, or its price, is equal to the present value of the future coupon payments and the present value of the par or face value of the bond. A Strip Bond has only one future cash flow, this could be the par value with no coupons (Pstrips) or any one of the individual coupons (C-strips), thus to find the market value of a Strip Bond we simple calculate the present value of a lump-sum formula! www.examsuccess.ca 7 Email: info@examsuccess.ca
Bond Pricing General Relationship: As Yields rise Bond Prices fall Bond Price = PV (Cash Flows) + PV (Maturity Value) this is the clean price of the bond. Looks like our old friends PV of an annuity and PV of a lump sum! Our suggestion is to calculate this in three steps: Step 1: PV (Coupon Payments) Present value of an annuity payment END MODE: Enter the following information given in the question: END PVA = PMT x 1 - ( 1 + I ) -n FV n I PMT solve for: PV I Step 2: PV (Maturity Value) Present value of a lump sum payment Enter the following information given in the question: PV = FV FV n I PMT solve for: PV ( 1 + I ) n Step 3: Add your results in steps 1 and 2 together to get your Bond Price. P B = PV (Coupon Payments) + PV (Maturity Value) www.examsuccess.ca 8 Email: info@examsuccess.ca
Example: Calculate the current market price of a 5 year annual pay 6.25% Government of Canada bond, if similar 5 year bonds are yielding 8%. Step 1: PVA = 62.5 x 1 - (1.08) -5 Enter -$62.5 PMT, 5 n, 8 I, 0 FV 0.08 = $249.54 Solve for PV = $249.54 Step 2: PV = $1,000 Enter -$1,000 FV, 5 n, 8 I, 0 PMT (1.08) 5 Solve for PV = $680.58 Step 3: = $680.58 P B = PV (Coupon Payments) + PV (Maturity Value) = $249.54 + $680.58 = $930.12 Additional Examples: 1. Assume the Government of Canada issues a 20 year semi-annual pay bond with a coupon of 6%. A quick scan of the internet reveals, current market interest rates (or yields) for similar bonds are 7.5%. Calculate the bond s current market price 2. Assume that market interest rates have changed and the bond is now valued at $1,020, calculate the Yield to Maturity for this bond. 3. Assume that an investment dealer strips off the coupons and principal to sell to investors as C-strips and a P-strip. Using the initial information as above, calculate the current market value of the P-strip. www.examsuccess.ca 9 Email: info@examsuccess.ca
Instructor/Author Profile: Brian Y. Gordon, CFA, CFP, CIM, MBA, FCSI, is a former tenured Professor in the School of Business at Centennial College in Toronto where he has taught Economics, Financial Accounting, Corporate Finance, the Canadian Securities Course, Personal Financial Planning and Investment Management. Prof. Gordon is also a part-time faculty member at Concordia University in Montreal, where he teaches Economics and Investment Management courses at the MBA level. Prof. Gordon has also lectured at Ryerson University in Toronto teaching Corporate Finance. Since 1999, Prof. Gordon has been a featured lecturer and workshop facilitator for CFP and CFA review programs offered across Canada. Prior to entering academia, Prof. Gordon developed his expertise in the discount brokerage, full service brokerage and banking industries, specializing in investment management, business development, strategic sales and marketing, and wealth management training. Prof. Gordon holds a BA in Economics from the University of Toronto, an MBA from Heriot- Watt University in the UK, and was awarded his CFA charter in 1999. In 1995, Prof. Gordon was granted a fellowship from the Canadian Securities Institute, earning the prestigious FCSI designation. Prof. Gordon successfully challenged the CFP Professional Proficiency Examination and was awarded the right to use the CFP designation in 2005. www.examsuccess.ca 10 Email: info@examsuccess.ca