much I would be willing to pay or invest today such that I earn a necessary

# Much i would be willing to pay or invest today such

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much I would be willing to pay (or invest) today such that I earn a necessary rate of return (or interest) The “opportunity cost” represents the next best alternative rate of return for an investment of similar risk and time duration The present value represents what I am willing to sacrifice today in order to invest in an asset (stock, bond, property, business, etc.), such that I (at the least) earn my opportunity cost FIN 300 - TVM Lump Sum 18 Lump-Sum Discounting Let’s say that you buy a “piece of paper” (some sort of a financial security or asset) that promises to pay (a single lump-sum of) exactly \$10k in exactly 6 years Also, let’s say that you require a 6% annual rate of return for investment in assets of a similar risk What would you be willing to pay today for this piece of paper – in other words, what is its “value” FIN 300 - TVM Lump Sum 19 Lump-Sum Discounting Do the work for the previous… 𝑃𝑃 = \$10,000 1.06 6 = \$7,049.61 It would be valued at \$7, 049.61 Note that you would be thrilled to get it cheaper than that But, as long as you don’t pay any more than \$7,049.61, you are earning at least as good a return as you could get elsewhere for the same risk FIN 300 - TVM Lump Sum 20 Lump-Sum Discounting If we were to diagram the previous example FIN 300 - TVM Lump Sum 21 t=0 t =6 Invest \$7,049.61 (negative cash-flow) Cash Out \$10,000 (positive cash-flow) Discounting Backward in Time Compounding Periods Assume that you will invest \$1 You are told that you will earn 12% “APR” (annual percentage rate) with annual compounding How much will you have in 1 year? Pretty simple answer…..\$1.12! What if it said that the \$1 investment earns 12% APR with “semi-annual” compounding? How much will you have in 1 year? FIN 300 - TVM Lump Sum 22 APR APR (annual percentage rate) is a “quoted rate” Must be paired with the frequency of compounding The APR is not a true, honest-to-goodness interest rate – the APR is the number of compounding periods per year multiplied by the effective rate per period 𝐴𝑃𝐴 = 𝑚 ∗ 𝑖 (where m = # of compounding periods/yr. and the i = rate of return per period (aka. period rate) FIN 300 - TVM Lump Sum 23 APR Back to investing \$1 for year at a 12% APR, with semi-annual compounding This doesn’t mean that you earn 12%/year This does mean that the actual (or, effective) rate is 6% (1/2 of 12%) per semi-annual period Go back to the compound-interest formula \$1(1.06)(1.06) = \$1.1236 Note that your actual (or, “effective”) rate of return for the year is 12.36% This 12.36% is termed the EAR (or, effective annual rate) FIN 300 - TVM Lump Sum 24 APR vs. EAR Continuing the previous with more frequent compounding and 12% APR, investing \$1 for a year Quarterly: really earning 3% per 3-month period \$1(1.03) 4 =\$1.125509 So, the EAR (effective annual rate) = 12.5509% Monthly : earning 12%/12=1% per month \$1(1.01) 12 =\$1.126825 EAR = 12.6825% Weekly: earning a period rate of 12%/52 per week \$1(1 + 0.12/52) 52 =\$1.127341 EAR = 12.7341% FIN 300 - TVM Lump Sum 25 APR vs. EAR Continuing the previous with more frequent  #### You've reached the end of your free preview.

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• Fall '08
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