Chapter 2 - Value - How to Calculate Present Values

How are asset values determined?

Is the present value positive? - The value today exceeds the investment that is required

A positive net present value implies that the rate of return on investment is higher than your opportunity cost of capital.

Perpetuity investment - steady stream of cash flows forever

Annuity investment - steady stream for a limited period

Money has a

*time value*- $ x (1 + r)^{t}
Value of investment

1 year $ x (1 +

*r*)
2 year $ x (1 + r)

How much do you need to invest today to get a certain amount at the end of a year?

Kahn Academy

Alanis Academy

Asset -Business entity, property, plant, equipment, knowledge, reputation, research and development.

An asset is a

Numeraire - a standard by which we measure everything - a base - use what money is equal to today

Cash flows at different points in time are like different currencies

Time₀ - cash flows can then be converted into present value

NPV - net present value operator - date 0 value of the cash flows - you have to figure out how to convert all the numbers to the same value, today's dollars

We can value cash flows

-we know the cash flows in advance

-we know the exchange rates (market)

-there is no friction in the exchange rates

1: Present Value Learning Objectives Review of Concepts Concepts oCompounding/discounting oPV/FV oR l ea vs. nomi l na rate oAnnuities and perpetuities Examples oCD oAuto loan oScholarship Scholarship fund oProject planning 2010 / Yichuan Liu 2 Review: Compounding / Discounting We can… o move money forward in time by compounding. o move money backward in time by discounting. x (1+r)m‐n x (1+r)s‐n (1+r)m‐n∙Y Y (1+r)s‐n∙Y t = m n s Note: o Only relative time matters o Multiplying by (1+r)m‐n = dividing by (1+r)n‐m. 2010 / Yichuan Liu 3 Review: APR vs. EAR Annual percentage percentage rate (APR) vs. equivalent equivalent annual return (EAR): APR N EAR 1 APR 1 (N = comp. freq.) N N t o e: oalways use the EAR when compounding and discounting oDue to interest compounding, the EAR is higher than the APR whenever the compounding frequency is higher than once a year. 2010 / Yichuan Liu 4 Continuous Compounding (optional) Given a fixed APR, higher compounding compounding frequency frequency leads to higher EAR. Suppose we take compounding frequency to infinity, then EAR lim1 APR N 1 eAPR 1. n N e 2.71828183... The continuously continuously compounded compounded EAR is the highest highest possible EAR for a given APR. 2010 / Yichuan Liu 5 Review: PV / FV Cash flow: 0 1 2 … … T‐1 T T+1 … … periods C0 C1 C 2 … … CT‐1 CT CT … … PV0 FVT Present value (PV): C PV0 C0 C1 1 C2 2 1 r 1 r Future value (FV) : FVT C0 1 rT C1 1 rT 1 0 1 CT 1 r0 CT 1 1 r1 2010 / Yichuan Liu 6 Review: Nominal vs. Real Interest Rate Nominal Nominal‐real interest interest rate conversion: conversion: 1 rreal 1 rnominal 1 i Nominal‐real cash flow conversion: Cnominal Creal 1 i When you discount or compound, oEither use the nominal nominal cash flow and the nominal nominal interest interest rate o Or use the real cash flow and the real interest rate oDo not mix and match 2010 / Yichuan Liu 7 Review: Annuity/Perpetuity Annuity: C C … … C C 0 1 2 … … T‐1 T periods C 1 PV0 r 1 1 rT Perpetuity: C C C … … 0 1 2 3 … … periods C PV0 r 2010 / Yichuan Liu 8 Review: Growing Annuity/Perpetuity Growing Growing Annuity: Annuity: C C(1+g) … … C(1+g)T‐2 C(1+g)T‐1 0 1 2 … … T‐1 T periods C 1 gT PV0 r g 1 1 rT . Growing Perpetuity (r > g): C C(1+g) C(1+g)2 … … 0 1 2 3 … … periods C PV0 . r g 2010 / Yichuan Liu 9 Example 1: CD You can invest $10,000 $10,000 in a CD offered offered by your bank. The CD matures in 5 years and the bank quotes you a rate of 4.5%. How much will you have in 5 years, if the 4.5% is a) EAR b) Quarterly APR c) Monthly APR 2010 / Yichuan Liu 10 Example 1: CD Answer: a) 10,0001.0455 $12,461.82 b) rEAR 1 0.045 4 1.04576 4 10 0001.045765 10,000 $12,507.51 c) rEAR 1 0.045 12 1.04594 12 10,0001.045945 10,000 $12,517.96 2010 / Yichuan Liu 11 Example 2: Auto Loan You would like to buy a new car for $22,000. $22,000. The dealer requires a down payment of $10,000 and offers you 6% APR financing (compounded monthly) for 5 years for the remaining balance. What is your monthly payment? 2010 / Yichuan Liu 12 Example 2: Auto Loan Answer: Answer: let C be the monthly monthly payment, payment, then C 1 22000 1 125 10000. 0.06 /12 1 0.06 /12 0 06 /12 1 0 06 /12 C $231.99. 2010 / Yichuan Liu 13 Example 3: Scholarship Fund You would like to establish establish a scholarship scholarship fund that will help outstanding students with financial difficulties pay their college tuition. oStarting today, you hope to give 50 students $20,000 each in today’s money (i.e., adjusted for inflation) every year. oThe effective effective nominal nominal interest interest rate is 5%/yr. oInflation is 2%/yr. How much money do you need now if you want the fund to last forever? 2010 / Yichuan Liu 14 Example 3: Scholarship Fund Answer: o Method 1: nominal amount + nominal interest rate 1m 1.02 1m 35 m. 1.05 1.02 o Method 2: real amount + real interest rate 1.05 1m rreal 1 2.9412% 1m 35m. 1.02 0.029412 oNote: same answer! You need $35 million today. 2010 / Yichuan Liu 15 Example 4: Project Planning GeneriCorp is considering considering whether whether or not to expand into a new market. The company faces the following cash flow (in $million) if it decides to expand: ‐200 ‐400 ‐300 +100 +500 +600 t = 0 1 2 3 4 5 years A committee appointed by the CEO determined that the appropriate appropriate discount discount rate is 9%. Should the company take on the expansion project? 2010 / Yichuan Liu 16 Example 4: Project Planning Answer: 400 300 100 500 600 NPV 200 1 09 1.092 1 09 3 4 5 1.09 1 09 1.09 1 0.9 1 0.9 $1.91m. P i ostive NPV = t ka e the proj t ec ; though NPV is dangerously close to zero.

^{2}^{in year 2 you earn interest on both your initial investment and the previous year's interest - compound interest}^{ }^{Calculating Present Values }How much do you need to invest today to get a certain amount at the end of a year?

Kahn Academy

Alanis Academy

__Lecture__

*Lecture starts at 19:17**Professor continues to mention that we won't understand everything right away and that we have to build knowledge. It will all come together and we will have an epiphany of understanding.**Cash Flow - money that is coming or going away from you*

Asset -Business entity, property, plant, equipment, knowledge, reputation, research and development.

An asset is a

__sequence__of cash flows - a combination of present and future cash flows
Assetₜ = (CFₜ, CFₜ₊₁, CFₜ₊₂, …)

Value of an Assetₜ = Vₜ(CFₜ, CFₜ₊₁, CFₜ₊₂, …)

Draw a line to visualize the timing of cash flows

Cash flows at different points in time are like different currencies

Time₀ - cash flows can then be converted into present value

NPV - net present value operator - date 0 value of the cash flows - you have to figure out how to convert all the numbers to the same value, today's dollars

We can value cash flows

-we know the cash flows in advance

-we know the exchange rates (market)

-there is no friction in the exchange rates

**We want to move money back and forth through time - any two dates you have to have an exchange rate**

__Recitation__1: Present Value Learning Objectives Review of Concepts Concepts oCompounding/discounting oPV/FV oR l ea vs. nomi l na rate oAnnuities and perpetuities Examples oCD oAuto loan oScholarship Scholarship fund oProject planning 2010 / Yichuan Liu 2 Review: Compounding / Discounting We can… o move money forward in time by compounding. o move money backward in time by discounting. x (1+r)m‐n x (1+r)s‐n (1+r)m‐n∙Y Y (1+r)s‐n∙Y t = m n s Note: o Only relative time matters o Multiplying by (1+r)m‐n = dividing by (1+r)n‐m. 2010 / Yichuan Liu 3 Review: APR vs. EAR Annual percentage percentage rate (APR) vs. equivalent equivalent annual return (EAR): APR N EAR 1 APR 1 (N = comp. freq.) N N t o e: oalways use the EAR when compounding and discounting oDue to interest compounding, the EAR is higher than the APR whenever the compounding frequency is higher than once a year. 2010 / Yichuan Liu 4 Continuous Compounding (optional) Given a fixed APR, higher compounding compounding frequency frequency leads to higher EAR. Suppose we take compounding frequency to infinity, then EAR lim1 APR N 1 eAPR 1. n N e 2.71828183... The continuously continuously compounded compounded EAR is the highest highest possible EAR for a given APR. 2010 / Yichuan Liu 5 Review: PV / FV Cash flow: 0 1 2 … … T‐1 T T+1 … … periods C0 C1 C 2 … … CT‐1 CT CT … … PV0 FVT Present value (PV): C PV0 C0 C1 1 C2 2 1 r 1 r Future value (FV) : FVT C0 1 rT C1 1 rT 1 0 1 CT 1 r0 CT 1 1 r1 2010 / Yichuan Liu 6 Review: Nominal vs. Real Interest Rate Nominal Nominal‐real interest interest rate conversion: conversion: 1 rreal 1 rnominal 1 i Nominal‐real cash flow conversion: Cnominal Creal 1 i When you discount or compound, oEither use the nominal nominal cash flow and the nominal nominal interest interest rate o Or use the real cash flow and the real interest rate oDo not mix and match 2010 / Yichuan Liu 7 Review: Annuity/Perpetuity Annuity: C C … … C C 0 1 2 … … T‐1 T periods C 1 PV0 r 1 1 rT Perpetuity: C C C … … 0 1 2 3 … … periods C PV0 r 2010 / Yichuan Liu 8 Review: Growing Annuity/Perpetuity Growing Growing Annuity: Annuity: C C(1+g) … … C(1+g)T‐2 C(1+g)T‐1 0 1 2 … … T‐1 T periods C 1 gT PV0 r g 1 1 rT . Growing Perpetuity (r > g): C C(1+g) C(1+g)2 … … 0 1 2 3 … … periods C PV0 . r g 2010 / Yichuan Liu 9 Example 1: CD You can invest $10,000 $10,000 in a CD offered offered by your bank. The CD matures in 5 years and the bank quotes you a rate of 4.5%. How much will you have in 5 years, if the 4.5% is a) EAR b) Quarterly APR c) Monthly APR 2010 / Yichuan Liu 10 Example 1: CD Answer: a) 10,0001.0455 $12,461.82 b) rEAR 1 0.045 4 1.04576 4 10 0001.045765 10,000 $12,507.51 c) rEAR 1 0.045 12 1.04594 12 10,0001.045945 10,000 $12,517.96 2010 / Yichuan Liu 11 Example 2: Auto Loan You would like to buy a new car for $22,000. $22,000. The dealer requires a down payment of $10,000 and offers you 6% APR financing (compounded monthly) for 5 years for the remaining balance. What is your monthly payment? 2010 / Yichuan Liu 12 Example 2: Auto Loan Answer: Answer: let C be the monthly monthly payment, payment, then C 1 22000 1 125 10000. 0.06 /12 1 0.06 /12 0 06 /12 1 0 06 /12 C $231.99. 2010 / Yichuan Liu 13 Example 3: Scholarship Fund You would like to establish establish a scholarship scholarship fund that will help outstanding students with financial difficulties pay their college tuition. oStarting today, you hope to give 50 students $20,000 each in today’s money (i.e., adjusted for inflation) every year. oThe effective effective nominal nominal interest interest rate is 5%/yr. oInflation is 2%/yr. How much money do you need now if you want the fund to last forever? 2010 / Yichuan Liu 14 Example 3: Scholarship Fund Answer: o Method 1: nominal amount + nominal interest rate 1m 1.02 1m 35 m. 1.05 1.02 o Method 2: real amount + real interest rate 1.05 1m rreal 1 2.9412% 1m 35m. 1.02 0.029412 oNote: same answer! You need $35 million today. 2010 / Yichuan Liu 15 Example 4: Project Planning GeneriCorp is considering considering whether whether or not to expand into a new market. The company faces the following cash flow (in $million) if it decides to expand: ‐200 ‐400 ‐300 +100 +500 +600 t = 0 1 2 3 4 5 years A committee appointed by the CEO determined that the appropriate appropriate discount discount rate is 9%. Should the company take on the expansion project? 2010 / Yichuan Liu 16 Example 4: Project Planning Answer: 400 300 100 500 600 NPV 200 1 09 1.092 1 09 3 4 5 1.09 1 09 1.09 1 0.9 1 0.9 $1.91m. P i ostive NPV = t ka e the proj t ec ; though NPV is dangerously close to zero.

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