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## Investment Tools – Time Value of Money

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**Time Value of Money**Concepts Covered in This Section • Future value • Present value • Perpetuities • Annuities • Uneven Cash Flows • Rates of return**0**1 2 3 i% CF0 CF1 CF2 CF3 Interest Rate Time lines show timing of cash flows. Cash Flows • Tick marksat ends of periods. • Time 0 is today; Time 1 is the end of Period 1; or the beginning of Period 2. • 90% of getting a Time Value problem correct is setting up the timeline correctly!!!**0**1 2 3 10% 100 What’s the FV of an initial $100 after 3 years if i = 10%? Future Values FV = ? • Finding FVs (moving to the right on a time line) is called compounding. • Compounding involves earning interest on interest for investments of more than one period.**FV3 = (1+i)3PV**FV2 = (1+i)2PV FV1 = (1+i)PV PV 0 1 2 3 FV1 FV2 FV3 PV = FV1/(1+i) PV = FV2/(1+i)2 0 1 2 3 PV = FV3/(1+i)3 Single Sum - Future & Present Value • Assume can invest PV at interest rate i to receive future sum, FV • Similar reasoning leads to Present Value of a Future sum today.**FVn = PV(1 + i )n for given PV**3 1 PV = $100 1.10 = $75.13. = $100 0.7513 Single Sum – FV & PV Formulas PV Calculation for $100 received in 3 years if interest rate is 10%**Ex 1.An investor wants to have $1 million when she retires**in 20 years. If she can earn a 10 percent annual return, compounded annually, on her investments, the lump-sum amount she would need to invest today to reach her goal is closest to: A. $100,000. B. $117,459. C. $148,644. D. $161,506. This is a single payment to be turned into a set future value FV=$1,000,000 in N=20 years time invested at r=10% interest rate. PV =[ 1/(1+r) ]N FV PV = [ 1/(1.10) ]20 $1,000,000 PV10 = [0.14864]($1,000,000) PV10 = $148,644 Question on PV of a given FV**A**A A A A A A 0 1 2 3 4 5 6 7 PV1 = A/(1+r) PV2 = A/(1+r)2 PV3 = A/(1+r)3 PV4 = A/(1+r)4 etc. etc. Perpetuities Perpetuity is a series of constant payments, A, each period forever. PVperpetuity = [A/(1+i)t] = A [1/(1+i)t] = A/i Intuition: Present Value of a perpetuity is the amount that must invested today at the interest rate i to yield a payment of A each year without affecting the value of the initial investment.**Annuities**• Regular or ordinary annuity is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that occurs one period from now. • An annuity due is a finite set of sequential cash flows, all with the same value A, which has a first cash flow that is paid immediately.**0**1 2 3 Time line for an ordinary annuity of $100 for 3 years. Ordinary Annuity Timeline i% $100 $100 $100**Ordinary Annuity**0 1 2 3 i% PMT PMT PMT Annuity Due 0 1 2 3 i% PMT PMT PMT PV FV Ordinary Annuity vs. Annuity Due Difference between an ordinaryannuity and an annuitydue?**1. Perpetuity of A per period in Period 0 -- PV1 = A/i**A A A A A A A A A A A A A A 0 2 4 6 8 10 12 14 2. Perpetuity of A per period in Period 8 -- PV8 = [1/(1+i)]8x (A/i) A A A A A A 0 2 4 6 8 10 12 14 3. Annuity of A for 8 periods -- PV =PV1 – PV8 = (A/i) x { 1 – [1/(1+i)]8 } A A A A A A A A 0 2 4 6 8 10 12 14 Annuity Formula and Perpetuities Intuition: Formula for a N-period annuity of A is: PV of a Perpetuity of A today minus PV of a Perpetuity of A in period N**Rather than memorize the annuity formula I find it easier to**calculate it as the difference between two perpetuities with the same payment. PV of an N-period annuity of $A per period is: PVN = (A/i) x { 1 – [1/(1+i)]N} Calculating the PV of an annuity has 3 steps: Calculate (A/i) PV of a Perpetuity with payments of $A per period. Calculate [1/(1+i)]N Discount factor associated with end of the annuity. Calculate PVN = (A/i) x { 1 - [1/(1 + i)]N } I think this is easier under pressure than memorizing the formula. Annuities & Perpetuities Again**Ex 2.An individual deposits $10,000 at the beginning of each**of the next 10 years, starting today, into an account paying 9 percent interest compounded annually. The amount of money in the account at the end of 10 years will be closest to: A. $109,000. B. $143,200. C. $151,900. D. $165,600. This is an annuity due of A=$10,000 for N=10 years at i=9% interest rate. Annuity due must be adjusted by (1+i) to reflect payment is made at beginning rather than end of period. Also must adjust PV formula by (1+i)N for FV of annuity. PVN = (1+i)N(1+i)[(A/i) { 1 – [1/(1+i)]N}] PV10 = (1.09)11 ($10K/.09) {1 – [1/1.09]10} PV10 = (2.58)($111,111){1 – [0.42]} PV10 = $165,601 Question on FV of Annuity Due**0**1 2 3 i% Time line for uneven CFs: $100 at end of Year 1 (t = 1), $200 at t=2, and$300 at the end of Year 3. Uneven Cash Flows $300 $100 $200**Ex 3.An investment promises to pay $100 one year from today,**$200 two years from today, and $300 three years from today. If the required rate of return is 14 percent, compounded annually, the value of this investment today is closest to: A. $404. B. $444. C. $462. D. $516. This is a set of unequal cash flows. You could do it as a sum of annuities but it is easier to calculate it directly in this case. Interest rate is i =14%. PV = [ 1/(1+i) ]t FVt PV = $100/(1.14) + $200/(1.14)2 + $300/(1.14)3 PV = $87.72 + $153.89 + $202.49 PV = $444.10 Question on Uneven Cash Flows**1. Uneven cash Flows over 10 periods – PV = PV10 + PV45**$100 $100 $100 $100 $100 $500 $500 $500 $500 $100 0 2 4 6 8 10 12 14 2. Annuity of $100 per period for 10 periods -- PV10 = { 1 - [1/(1+i)]10 } x (A/i) $100 $100 $100 $100 $100 $100 $100 $100 $100 $100 0 2 4 6 8 10 12 14 3. Annuity of $400 per period for 4 periods from period 5 -- PV45 = [1/(1+i)]5x [ (A/i) x { 1 – [1/(1+i)]4 } ] $400 $400 $400 $400 0 2 4 6 8 10 12 14 Uneven Cash Flows Intuition: PV of uneven cash flows is equal to the sum of the PV’s of regular cash flows that sum to the uneven cash flows.**Interest Rate Definitions**• Stated Annual interest rate or quoted interest rate is = m x ip where ip is the periodic interest rate times the number of periods in a year, m. • Stated in contracts. • Does not account for effects of compounding within the year. • Periodic interest rateip = is /m x ip where is is the stated annual interest rate divided by the number of periods in a year, m. • Used in calculations, shown on time lines. • Effective Annual interest rate or EFF–the amount to which a $1 grows to in year with compounding taken into account. • Use EAR or EFF% only for comparisons when payment periods differ between investments. • Given a stated annual interest rate,iS, the periodic rate is iP = iS/m, where m is the number of periods a year. • Effective annual interest rate is computed as = (1 + ip)m – 1**0**1 2 3 10% 100 133.10 0 1 2 3 4 5 6 0 1 2 3 5% 100 134.01 Comparison of Compounding Periods Annually: FV3 = $100(1.10)3 = $133.10. Semiannually: FV6 = $100(1.05)6 = $134.01.**Develop an approach to problems on Time Value.**Draw the Time line for the cash flows. Put in the cash flows from the problem. Identify if single payment, annuity, annuity due, or perpetuity. If uneven cash flows can you break it into sums of annuities? Identify what is to be calculated –PV, FV, N or i ? Write out the appropriate formula, put in values for the variables, and calculate. Best Study Tip: Do the problems, and then do some more and then do some more!! Practice using your calculator!! Questions on Time Value**Possible Time Value Questions**• Present Value Formula • Given FVN, i, N – solve for PVN • Given PVN , i, N – solve for FVN • Given PVN, FVN, N – solve for i • Given PVN, FVN, i – solve for N • Perpetuity Formula • Given A, i – solve for PVper • Given PVper, i – solve for A • Given PVper, A – solve for i • Annuity Formula • Given A, i, N – solve for PV • Given A, i, N – solve for FV • Given PV, i, N – solve for A