Time Value of Money and the Art of Making Choices PDF

Time value of money and the art of making choices Gherardo Gennaro Caracciolo September 6, 2024 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 1 / 52 Choosing is hard !...

Time value of money and the art of making choices Gherardo Gennaro Caracciolo September 6, 2024 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 1 / 52 Choosing is hard ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 2 / 52 Choosing is hard ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 2 / 52 Choosing is hard ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 2 / 52 What is finance ? Finance: Study of how individuals and firms acquire, spend, and manage scarce resources Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 3 / 52 What is finance ? Finance: Study of how individuals and firms acquire, spend, and manage scarce resources At the most general level, any financial decision is a claim to a stream of cash flow These cash flows differ along three dimensions: size timing risk Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 3 / 52 What is finance ? Finance: Study of how individuals and firms acquire, spend, and manage scarce resources At the most general level, any financial decision is a claim to a stream of cash flow These cash flows differ along three dimensions: size timing risk Let us postpone our treatment of risk until later and focus for now on comparing certain (or risk-free) cash flows differing in size and timing Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 3 / 52 What is finance ? Finance: Study of how individuals and firms acquire, spend, and manage scarce resources At the most general level, any financial decision is a claim to a stream of cash flow These cash flows differ along three dimensions: size timing risk Let us postpone our treatment of risk until later and focus for now on comparing certain (or risk-free) cash flows differing in size and timing Problem: to choose between alternatives, we must find a way to compare cash flows differing in these dimensions Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 3 / 52 “The essence of finance is time travel” (M. Levine) Intuition: Money today ̸= Money tomorrow → If you have the money today you can earn interest on it ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 4 / 52 “The essence of finance is time travel” (M. Levine) Intuition: Money today ̸= Money tomorrow → If you have the money today you can earn interest on it ! It is only possible to compare or combine values at the same point in time ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 4 / 52 “The essence of finance is time travel” (M. Levine) Intuition: Money today ̸= Money tomorrow → If you have the money today you can earn interest on it ! It is only possible to compare or combine values at the same point in time ! Two fundamental techniques Compounding → future value Discounting → present value Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 4 / 52  Compounding and Future Value Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 5 / 52 Compounding and future value: Numerical Example Suppose that you invest $1, 000 in a bank account paying an interest rate of 10%, and that interest is credited to your account once a year... 0 1 2 T... 1,000 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 6 / 52 Compounding and future value: Numerical Example Suppose that you invest $1, 000 in a bank account paying an interest rate of 10%, and that interest is credited to your account once a year Money in the account after one year:... 0 1 2 T... 1,000 1000 × (1 +.1) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 6 / 52 Compounding and future value: Numerical Example Suppose that you invest $1, 000 in a bank account paying an interest rate of 10%, and that interest is credited to your account once a year Money in the account after two years:... 0 1 2 T... 1,000 1000 × (1 +.1) 1, 000 × (1 +.1)(1 +.1) | {z } 1,000×(1+.1)2 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 6 / 52 Compounding and future value: Numerical Example Suppose that you invest $1, 000 in a bank account paying an interest rate of 10%, and that interest is credited to your account once a year Money in the account after T years:... 0 1 2 T... 1,000 1000 × (1 +.1) 1, 000 × (1 +.1)2 1, 000 × (1 +.1)T Continuing this reasoning, you will have $1, 000(1.10)T after T years Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 6 / 52 The Interests on Interests Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 7 / 52 Compounding and Future Value: General Case Suppose now that you invest C dollars in a bank account paying an interest rate r, and that interest is credited to your account once a year... 0 1 2 T... C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 8 / 52 Compounding and Future Value: General Case Suppose now that you invest C dollars in a bank account paying an interest rate r, and that interest is credited to your account once a year Money in the account after one year:... 0 1 2 T... C C × (1 + r ) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 8 / 52 Compounding and Future Value: General Case Suppose now that you invest C dollars in a bank account paying an interest rate r, and that interest is credited to your account once a year Money in the account after two years:... 0 1 2 T... C C × (1 + r ) C × (1 + r )(1 + r ) | {z } C ×(1+r )2 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 8 / 52 Compounding and Future Value: General Case Suppose now that you invest C dollars in a bank account paying an interest rate r, and that interest is credited to your account once a year Money in the account after T years:... 0 1 2 T... C C × (1 + r ) C × (1 + r )2 C × (1 + r )T Continuing this reasoning, you will have $C (1 + r )T after T years This quantity is the future value in T years of C dollars invested at a rate of r compounded annually → FVT = C (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 8 / 52 Compounding: Checkpoint Question 1 Kira is 30 years old and plans to retire at 65 (in 35 years) So far, she has saved $80, 000 toward retirement How much will Kira have when she retires if the money is invested at 12% per year? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 9 / 52 Solution We start by drawing our timeline: C=$80,000 T = 65 − 30 = 35 r = 12% 0 1 2... 35... 80,000 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 10 / 52 Solution After 1 year: 0 1 2... 35... 80,000 80, 000 × (1 +.12) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 10 / 52 Solution After 2 years: 0 1 2... 35 80,000 80, 000 × (1 +.12) 80, 000 × (1 +.12)2... Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 10 / 52 Solution When Kira is 65: 0 1 2... 35 80,000 80, 000 × (1 +.12) 80, 000 × (1 +.12)2... 80, 000 × (1 +.12)35 Using the formula from slide 8 to find the future value of Kira’s investment: FV35 = $80, 000(1.12)35 = $4.224million Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 10 / 52  Discounting and Present Value Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 11 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Let PV denote this amount 0 1 2 3... T PV C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Let PV denote this amount 0 1 2 3... T PV C So PV must satisfy: PV (1 + r ) = C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Let PV denote this amount 0 1 2 3... T PV C So PV must satisfy: PV (1 + r ) = C , or equivalently: C PV = 1+r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Let PV denote this amount 0 1 2 3... T PV C So PV must satisfy: PV (1 + r ) = C , or equivalently: C PV = 1+r PV is the present value of C dollars delivered in one year from now, and r is the discount rate Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over One Period Suppose you want to have C dollars in your account in one year, and that the current annual interest rate is r. How much do you have to invest today? Let PV denote this amount 0 1 2 3... T PV C So PV must satisfy: PV (1 + r ) = C , or equivalently: C PV = 1+r PV is the present value of C dollars delivered in one year from now, and r is the discount rate As long as you can borrow and lend at the rate r, you would be indifferent between receiving PV dollars now, or C dollars in one year Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 12 / 52 Discounting Over Multiple Periods Now, suppose you want to have in your account C dollars in T years, and that the current annual interest rate is r. How much do you have to invest today? Again, let PV denote this amount: 0 1 2 3... T PV C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 13 / 52 Discounting Over Multiple Periods Now, suppose you want to have in your account C dollars in T years, and that the current annual interest rate is r. How much do you have to invest today? Again, let PV denote this amount: 0 1 2 3... T PV C So PV must satisfy: PV (1 + r )T =C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 13 / 52 Discounting Over Multiple Periods Now, suppose you want to have in your account C dollars in T years, and that the current annual interest rate is r. How much do you have to invest today? Again, let PV denote this amount: 0 1 2 3... T PV C So PV must satisfy: PV (1 + r )T = C or, equivalently: C PV = (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 13 / 52 Discounting Over Multiple Periods Now, suppose you want to have in your account C dollars in T years, and that the current annual interest rate is r. How much do you have to invest today? Again, let PV denote this amount: 0 1 2 3... T PV C So PV must satisfy: PV (1 + r )T = C or, equivalently: C PV = (1 + r )T PV is the present value of C dollars delivered in T years from now, and r is the discount rate Again, as long as you can borrow and lend at the rate r, you would be indifferent between receiving PV dollars now, or C dollars in T years Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 13 / 52 Discounting Multiple Cash Flows Suppose you want to receive C1 dollars from your account in one year, C2 in two years,... , CT in T years. Suppose also that the current annual interest rate is r. How much do you have to invest today? 0 1 2 3... T PV C1 C2 C3 CT Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 14 / 52 Discounting Multiple Cash Flows Suppose you want to receive C1 dollars from your account in one year, C2 in two years,... , CT in T years. Suppose also that the current annual interest rate is r. How much do you have to invest today? 0 1 2 3... T PV C1 Using arguments similar to those on pages 1.13 and 1.14, we should find C1 PV = 1+r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 14 / 52 Discounting Multiple Cash Flows Suppose you want to receive C1 dollars from your account in one year, C2 in two years,... , CT in T years. Suppose also that the current annual interest rate is r. How much do you have to invest today? 0 1 2 3... T PV C1 C2 Using arguments similar to those on slides 12 and 13, we should find C1 C2 PV = + 1+r (1 + r )2 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 14 / 52 Discounting Multiple Cash Flows Suppose you want to receive C1 dollars from your account in one year, C2 in two years,..., CT in T years. Suppose also that the current annual interest rate is r.How much do you have to invest today? 0 1 2 3... T PV C1 C2 C3 CT Using arguments similar to those on pages 12 and 13, we should find T C1 C2 CT Ct PV = + + + = X... 1+r (1 + r )2 (1 + r )T t=1 (1 + r )t PV of a sequence of cash flows is the sum of the present values of each individual cash flow (value additivity) → Cannot add cash flows that occur at different points in time ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 14 / 52 Discounting: Example Two years ago, you put $30, 000 in a savings account earning an annual interest rate of 8% At the time, you thought that these savings would grow enough for you to buy a new car five years later (i.e., in three years from now) However, you just re-estimated the price that you will have to pay for the new car in three years at $54, 000 Question: How much more money do you need to put in your savings account now for it to grow to this new estimate in three years? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 15 / 52 Discounting: Example Let us first figure out how much money FV is now in the account Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 16 / 52 Discounting: Example Let us first figure out how much money FV is now in the account Now, the account should have an amount PV in it for it to grow to $54, 000 in three years Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 16 / 52 Discounting: Example Let us first figure out how much money FV is now in the account Now, the account should have an amount PV in it for it to grow to $54, 000 in three years So, you need to put $42, 866.94-$34, 992.00 = $7, 874.94 in the account Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 16 / 52  Useful Present Value Formulas Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 17 / 52 1. Perpetuities A perpetuity is an investment paying a fixed sum C at the end of every year forever 0 1 2 3...... PV C C C From our general formula (on slide 14), we know that the present value of the perpetuity is given by: C C C PV = + + +... (1 + r ) (1 + r ) 2 (1 + r )3 This sum nicely simplifies to C PV = r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 18 / 52 [OPTIONAL] The Perpetuity Formula We are looking to simplify C C C PV = + + +... (1) (1 + r ) (1 + r )2 (1 + r )3 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 19 / 52 [OPTIONAL] The Perpetuity Formula We are looking to simplify C C C PV = + + +... (1) (1 + r ) (1 + r )2 (1 + r )3 Step 1: Dividing both sides by 1 + r PV C C C = + + +... (2) (1 + r ) (1 + r ) 2 (1 + r ) 3 (1 + r )4 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 19 / 52 [OPTIONAL] The Perpetuity Formula We are looking to simplify C C C PV = + + +... (1) (1 + r ) (1 + r )2 (1 + r )3 Step 1: Dividing both sides by 1 + r PV C C C = + + +... (2) (1 + r ) (1 + r ) 2 (1 + r ) 3 (1 + r )4 Step 2: Subtract (2) from (1) PV C C C PV − = + + +... (1 + r ) (1 + r ) (1 + r ) 2 (1 + r )3 C C −[ + +...] (1 + r )2 (1 + r )3 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 19 / 52 [OPTIONAL] The Perpetuity Formula We are looking to simplify C C C PV = + + +... (1) (1 + r ) (1 + r )2 (1 + r )3 Step 1: Dividing both sides by 1 + r PV C C C = + + +... (2) (1 + r ) (1 + r ) 2 (1 + r ) 3 (1 + r )4 Step 2: Subtract (2) from (1) PV C C  C  PV − = +  +  +... (1 + r ) (1 + r ) (1 + r )2 (1 + r )3   C  C  −[  +  +...] (1 + r )2 (1 + r )3   Notice that all but one term on the right-hand side of the above equation can be canceled out Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 19 / 52 [OPTIONAL] The Perpetuity Formula (cont’d) We can now solve for PV as follows PV C PV − = ⇔ PV (1 + r ) − PV = C ⇔ PV × r = C (1 + r ) (1 + r ) The present value of the perpetuity of C paid at the end of every year forever is therefore C PV = r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 20 / 52 Practical Example: Deferred Perpetuity A rich entrepreneur would like to set up a foundation that, every year, will pay $25, 000 in the form of a scholarship to one deserving student The first such scholarship is to be awarded in 3 years, and a scholarship will be awarded in perpetuity every year after that (even after the entrepreneur’s death) How much money should the entrepreneur put in the foundation’s account today, if the account earns 8% compounded annually? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 21 / 52 Practical Example: Deferred Perpetuity (cont’d) First, let us calculate how much money will need to be in the account at the end of the second year; let us denote that amount by PV2... 0 1 2 3 4 5... 25,000 25,000 25,000 25,000 PV2 = 0.08 = 312, 500 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 22 / 52 Practical Example: Deferred Perpetuity (cont’d) First, let us calculate how much money will need to be in the account at the end of the second year; let us denote that amount by PV2... 0 1 2 3 4 5... 25,000 25,000 25,000 25,000 PV2 = 0.08 = 312, 500 For the account to be worth this much in two years, the amount that the entrepreneur needs to contribute initially is PV2 312, 500 PV = = = 267, 918 (1.08)2 (1.08)2 The perpetuity starts in 3 years, but the exponent on the discount factor is a 2 This is because the perpetuity formula used in the first step calculates the value of a stream of cash flows starting a year later Only two more years of discounting are needed after that Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 22 / 52 Shortcuts to Calculating PVs: Growing Perpetuities A growing perpetuity is an investment paying a growing sum (at a rate g) every t1 year forever, i.e., the amount paid in year t is C (1 + g)t−1 0 1 2 3...... PV C C (1 + g) C (1 + g)2 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 23 / 52 Shortcuts to Calculating PVs: Growing Perpetuities A growing perpetuity is an investment paying a growing sum (at a rate g) every t1 year forever, i.e., the amount paid in year t is C (1 + g)t−1 0 1 2 3...... PV C C (1 + g) C (1 + g)2 From our general formula (on slide 14), we know that the present value of the growing perpetuity is given by C C (1 + g) C (1 + g)2 PV = + + +... (1 + r ) (1 + r )2 (1 + r )3 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 23 / 52 Shortcuts to Calculating PVs: Growing Perpetuities A growing perpetuity is an investment paying a growing sum (at a rate g) every t1 year forever, i.e., the amount paid in year t is C (1 + g)t−1 0 1 2 3...... PV C C (1 + g) C (1 + g)2 From our general formula (on slide 14), we know that the present value of the growing perpetuity is given by C C (1 + g) C (1 + g)2 PV = + + +... (1 + r ) (1 + r )2 (1 + r )3 As long as r > g, this infinite sum simplifies to C PV = r −g Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 23 / 52 [Optional] The Growing Perpetuity Formula We seek to simplify the following expression: C C (1 + g) C (1 + g)2 PV = + + +... (1) (1 + r ) (1 + r )2 (1 + r )3 1+g Let us multiply both sides of equation 1+r 1+g C (1 + g) C (1 + g)2 C (1 + g)3 ! PV = + + +... (2) 1+r (1 + r )2 (1 + r )3 (1 + r )4 Let’s now subtract (2) from (1) 1+g C (1 + g) C (1 + g)2 ! C PV − PV = + + +... 1+r (1 + r ) (1 + r )2 (1 + r )3 C (1 + g) C (1 + g)2 " # + +... (1 + r )2 (1 + r )3 Notice that all but one term on the right-hand side of this last equation can be cancelled out Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 24 / 52 [Optional] The Growing Perpetuity Formula (cont’d) We can now solve for PV as follows (provided that g < r ) 1+g ! C PV − PV = ⇔ PV (1 + r ) − PV (1 + g) = C ⇔ PV (r − g) = C 1+r (1 + r ) The present value of the growing perpetuity is therefore C PV = r −g Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 25 / 52 Exercise: An Endowed Chair A benefactor wishes to endow a chair in finance at the Beedie School of Business The aim is to provide an amount equaling $150, 000 in the first year and growing at a rate of 5% each subsequent year in order to adjust for the expected growth in salaries Suppose that the interest rate is 10%. How much should the benefactor donate? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 26 / 52 Exercise: An Endowed Chair A benefactor wishes to endow a chair in finance at the Beedie School of Business The aim is to provide an amount equaling $150, 000 in the first year and growing at a rate of 5% each subsequent year in order to adjust for the expected growth in salaries Suppose that the interest rate is 10%. How much should the benefactor donate? Solution We have r = 10%, g = 5%, C = 150, 000 C 150, 000 PV = = = 3, 000, 000 r −g 0.10 − 0.05 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 26 / 52 2. Annuities An annuity is an investment that pays a fixed sum C at the end of each year for T years... T+1 T+2... 0 1 2 T-1 T...... PV C C C C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 27 / 52 2. Annuities An annuity is an investment that pays a fixed sum C at the end of each year for T years... T+1 T+2... 0 1 2 T-1 T...... PV C C C C Using our general formula from slide 14, we can write the present value of the annuity as C C C PV = + + ··· + (1 + r ) (1 + r )2 (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 27 / 52 2. Annuities An annuity is an investment that pays a fixed sum C at the end of each year for T years... T+1 T+2... 0 1 2 T-1 T...... PV C C C C Using our general formula from slide 14, we can write the present value of the annuity as C C C PV = + + ··· + (1 + r ) (1 + r )2 (1 + r )T Although we can use a spreadsheet to calculate this finite sum, pages 28-30 show that there is a simple formula for the present value of an annuity: 1 " # C PV = 1− r (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 27 / 52 [Optional] The annuity Formula To reach the formula on slide 27, first observe that the cash flows from the annuity equal the difference between the cash flows of two perpetuities: one starting at time 1;... T+1 T+2... 0 1 2 T-1 T...... PVA C C C C C C Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 28 / 52 [Optional] The annuity Formula To reach the formula on slide 27, first observe that the cash flows from the annuity equal the difference between the cash flows of two perpetuities: one starting at time 1; the other starting at time T + 1... T+1 T+2... 0 1 2 T-1 T...... PVA C C C C C C... PVB C C The present value of the first perpetuity is PVA = r , C as shown on slide 18. What about the second perpetuity, which is deferred for T years? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 28 / 52 [Optional] The annuity Formula (cont’d) Let us first calculate the value of that perpetuity at the end of T years. We call this value PVB (at T)... T+1 T+2... 0 1 2 T-1 T...... C C PVB (at T)= C r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 29 / 52 [Optional] The annuity Formula (cont’d) Let us first calculate the value of that perpetuity at the end of T years. We call this value PVB (at T)... T+1 T+2... 0 1 2 T-1 T...... C C PVB (at T)= C r Since PVB (at T) is the value in T years from now, we need to discount this value to time 0 to get the value of the perpetuity... T+1 T+2... 0 1 2 T-1 T ÷(1 + r )T... PVB PVB (at T) C C PVB (at T) C /r ,→ PVB = = (1 + r )T (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 29 / 52 [Optional] The annuity Formula (cont’d) The calculation of the present value of the annuity then simply involves a difference of two perpetuities C C /r PV = PVA − PVB = − r (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 30 / 52 [Optional] The annuity Formula (cont’d) The calculation of the present value of the annuity then simply involves a difference of two perpetuities C C /r PV = PVA − PVB = − r (1 + r )T After simplification, the present value of the annuity is therefore given by 1 " # C PV = 1− r (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 30 / 52 [Optional] The annuity Formula (cont’d) The calculation of the present value of the annuity then simply involves a difference of two perpetuities C C /r PV = PVA − PVB = − r (1 + r )T After simplification, the present value of the annuity is therefore given by 1 " # C PV = 1− r (1 + r )T Intuition C r would be the present value if the payments went on forever The term in square brackets, which is always smaller than 1, accounts for the fact that the payments stop (i.e., the PV is not as large as that of a perpetuity) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 30 / 52 Exercise: Mortgage Payments You have decided to buy a house for $500, 000, with an initial down-payment of $50, 000 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 31 / 52 Exercise: Mortgage Payments You have decided to buy a house for $500, 000, with an initial down-payment of $50, 000 To finance the balance, you have negotiated a 30-year mortgage at an annual rate of 6% Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 31 / 52 Exercise: Mortgage Payments You have decided to buy a house for $500, 000, with an initial down-payment of $50, 000 To finance the balance, you have negotiated a 30-year mortgage at an annual rate of 6% Assume that your mortgage calls for equal payments at the end of every year (and that the 6% is compounded annually) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 31 / 52 Exercise: Mortgage Payments You have decided to buy a house for $500, 000, with an initial down-payment of $50, 000 To finance the balance, you have negotiated a 30-year mortgage at an annual rate of 6% Assume that your mortgage calls for equal payments at the end of every year (and that the 6% is compounded annually) What is your annual payment? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 31 / 52 Exercise: Mortgage Payments (cont’d) Let us denote the annual payment by C. Because a down-payment of $50, 000 has been made on the house, the present value of this annuity must be $450, 000 0 1 2 3... 30 C C C C PV=450, 000 With an annual interest rate of 6%, we seek to solve 1 " # C 450, 000 = 1− 0.06 (1.06)30 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 32 / 52 Exercise: Mortgage Payments (cont’d) Let us denote the annual payment by C. Because a down-payment of $50, 000 has been made on the house, the present value of this annuity must be $450, 000 0 1 2 3... 30 C C C C PV=450, 000 With an annual interest rate of 6%, we seek to solve 1 " # C 450, 000 = 1− = C × 13.765 0.06 (1.06)30 450,000 The annual payment is C = 13.765 = 32, 692 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 32 / 52 Shortcut to Calculating PVs: Growing Annuities A growing annuity is an investment that pays a growing sum (at a rate g) at the end of every year, and stops at the end of year T... 0 1 2 T-1 T... PV C C (1 + g) C (1 + g)T −2 C (1 + g)T −1 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 33 / 52 Shortcut to Calculating PVs: Growing Annuities A growing annuity is an investment that pays a growing sum (at a rate g) at the end of every year, and stops at the end of year T... 0 1 2 T-1 T... PV C C (1 + g) C (1 + g)T −2 C (1 + g)T −1 There is also a shortcut to compute the present value of a growing annuity C C (1 + g) C (1 + g)T −1 PV = + + · · · + 1+r (1 + r )2 (1 + r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 33 / 52 Shortcut to Calculating PVs: Growing Annuities A growing annuity is an investment that pays a growing sum (at a rate g) at the end of every year, and stops at the end of year T... 0 1 2 T-1 T... PV C C (1 + g) C (1 + g)T −2 C (1 + g)T −1 There is also a shortcut to compute the present value of a growing annuity C C (1 + g) C (1 + g)T −1 PV = + + · · · + 1+r (1 + r )2 (1 + r )T leads to !T # 1+g " C PV = 1− r −g 1+r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 33 / 52 Excel’s Annuity Functions Excel has several functions for annuities Unfortunately no functions for perpetuities → need to use the formula Excel’s annuity functions = PV (r , T , C ): present value (at time 0) = FV (r , T , C ): future value (at time T ) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 34 / 52 Excel’s Annuity Functions Excel has several functions for annuities Unfortunately no functions for perpetuities → need to use the formula Excel’s annuity functions = PV (r , T , C ): present value (at time 0) = FV (r , T , C ): future value (at time T ) = PMT (r , T , PV0 ): constant periodic payment amount (C) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 34 / 52 Excel’s Annuity Functions Excel has several functions for annuities Unfortunately no functions for perpetuities → need to use the formula Excel’s annuity functions = PV (r , T , C ): present value (at time 0) = FV (r , T , C ): future value (at time T ) = PMT (r , T , PV0 ): constant periodic payment amount (C) = NPER(r , C , PV 0): number of periods (T) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 34 / 52 Excel’s Annuity Functions Excel has several functions for annuities Unfortunately no functions for perpetuities → need to use the formula Excel’s annuity functions = PV (r , T , C ): present value (at time 0) = FV (r , T , C ): future value (at time T ) = PMT (r , T , PV0 ): constant periodic payment amount (C) = NPER(r , C , PV 0): number of periods (T) = RATE (T , C , PV 0): rate (r) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 34 / 52 Excel’s Annuity Functions: Example Let us solve the mortgage payment example from slide 31 using Excel’s PMT (r , T , PV0 ) function The interest rate is 6% → r = 0.06 The mortgage is for 30 years → T = 30 We need to borrow $450, 000 → PV0 = 450000 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 35 / 52 Excel’s Annuity Functions: Example Let us solve the mortgage payment example from slide 31 using Excel’s PMT (r , T , PV0 ) function The interest rate is 6% → r = 0.06 The mortgage is for 30 years → T = 30 We need to borrow $450, 000 → PV0 = 450000 The answer is the same as that on page 1.37 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 35 / 52  Net Present Value (an Introduction...) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 36 / 52 Making Choices:The Concept of Net Present Value Now that we have seen the rules of “time travel” (compounding and discounting), we can use them to make financial decisions The idea is compare the costs and benefits in present-value terms For this purpose, let us define the net present value (NPV) of an investment as NPV = PV (benefits) − PV (costs) The investment should be made when NPV> 0, as the benefits then exceed the costs (again, in PV terms) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 37 / 52  More frequent compounding Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 38 / 52 More frequent compounding Sometimes, an annual rate is compounded more than once a year → interest are credited more rapidly Semiannual compounding: Suppose that you invest $1,000 in a bank account paying an interest rate of 10%, and interest is credited to your account twice a year Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 39 / 52 More frequent compounding Sometimes, an annual rate is compounded more than once a year → interest are credited more rapidly Semiannual compounding: Suppose that you invest $1,000 in a bank account paying an interest rate of 10%, and interest is credited to your account twice a year Each six months, you receive in your account 10% = 5% interest Money in the account after 6 months Money in the account after one year When r is compounded x times a year, we refer to it as an annual percentage rate (APR) r compounded x times a year Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 39 / 52 General Case Generic investment Principal P annual rate: r Frequency of compounding: T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 40 / 52 General Case Generic investment Principal P annual rate: r Frequency of compounding: T 0 P General Case Generic investment Principal P annual rate: r Frequency of compounding: T 0 P P(1 + Tr ) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 40 / 52 General Case Generic investment Principal P annual rate: r Frequency of compounding: T 0 T ); P P(1 + Tr ) r r P(1 + )(1 + ) T T | {z } P(1+ r )2 T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 40 / 52 General Case Generic investment Principal P annual rate: r Frequency of compounding: T 0 year 1 T ); P P(1 + Tr ) r r P(1 + Tr )T P(1 + )(1 + ) T T | {z } P(1+ r )2 T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 40 / 52 General Case Generic investment Principal P annual rate: r Frequency of compounding: T 0 year 1 T ); P P(1 + Tr ) r r P(1 + Tr )T P(1 + )(1 + ) T T | {z } P(1+ r )2 T General formula: r T V = P(1 + ) T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 40 / 52 Example: Mortgage Payment Revisited In everyday finance (mortgages, credit cards, student loans), monthly compounding is typical Let’s go back to our mortgage example You buy a house for $500,000, with an initial downpayment of $50,000 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 41 / 52 Example: Mortgage Payment Revisited In everyday finance (mortgages, credit cards, student loans), monthly compounding is typical Let’s go back to our mortgage example You buy a house for $500,000, with an initial downpayment of $50,000 Let’s now change the set up a bit Your mortgage calls for equal payments at the end of every month The annual rate (APR) of 6% is compounded monthly What’s the mortgage rate ? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 41 / 52 Example: Mortgage Payment Revisited (continued) Let’s denote the monthly payment by C The downpayment is $50,000, hence the present value of this annuity must be $450,000 Also, there are 30 × 12 = 360 months in 30 years Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 42 / 52 Example: Mortgage Payment Revisited (continued) Let’s denote the monthly payment by C The downpayment is $50,000, hence the present value of this annuity must be $450,000 Also, there are 30 × 12 = 360 months in 30 years Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 42 / 52 Example: Mortgage Payment Revisited (continued) Let’s denote the monthly payment by C The downpayment is $50,000, hence the present value of this annuity must be $450,000 Also, there are 30 × 12 = 360 months in 30 years Because an annual rate of 6% compounded monthly is really an effective monthly rate of 12 = 0.5%, we seek to solve 6% Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 42 / 52 Example: Mortgage Payment Revisited (continued) Solving the equation gives us 450,000 Hence, our monthly payment C= 166.79 = 2, 698 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 43 / 52 Effective Annual Rate In the mortgage example, the annual percentage rate (APR) r = 6% is compounded monthly The compounding interval is one month and there are 12 month in a year, so m = 12 The interest rate you pay per month is r m = 6% 12 = 0.5% Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 44 / 52 Effective Annual Rate In the mortgage example, the annual percentage rate (APR) r = 6% is compounded monthly The compounding interval is one month and there are 12 month in a year, so m = 12 The interest rate you pay per month is r m = 6% 12 = 0.5% Since compounding is monthly, this is equivalent to paying a higher interest rate per year (1 + 0.5%)12 = 1.0617 = 1 + 6.17% This rate (of 6.17%) is called the effective annual rate (denoted rEAR ) or the equivalent annual rate In general, for an APR of r compounded m times a year, it can be computed by solving r m 1 + rEAR = (1 + ) m Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 44 / 52 Effective Annual Rate: Example Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 45 / 52 Effective Annual Rate: Example (continued) Why do you think these certificates of deposits (CD’s) seem to offer two different interest rates? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 46 / 52 Effective Annual Rate: Example (continued) Why do you think these certificates of deposits (CD’s) seem to offer two different interest rates? The small print says that the interest rate is a daily compounded rate. This means that a dollar invested in these CD’s will grow to 0.065 365 FV1 = (1 + ) = 1.0672 365 Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 46 / 52 Effective Annual Rate: Example (continued) Why do you think these certificates of deposits (CD’s) seem to offer two different interest rates? The small print says that the interest rate is a daily compounded rate. This means that a dollar invested in these CD’s will grow to 0.065 365 FV1 = (1 + ) = 1.0672 365 The “annual percentage yield” is how some financial institutions refer to the effective annual rate rEAR , which can be found as follows 1 + rEAR = 1.0672 ⇒ rEAR = 6.72% Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 46 / 52 annual rates vs annual effective rates Three possible scenarios: Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 47 / 52 annual rates vs annual effective rates Three possible scenarios: 1 interest compounded more than once per ’year’ → effective rate > nominal rate Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 47 / 52 annual rates vs annual effective rates Three possible scenarios: 1 interest compounded more than once per ’year’ → effective rate > nominal rate 2 interest compounded less than once per ’year’ → effective rate < nominal rate Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 47 / 52 annual rates vs annual effective rates Three possible scenarios: 1 interest compounded more than once per ’year’ → effective rate > nominal rate 2 interest compounded less than once per ’year’ → effective rate < nominal rate 3 interest compounded once per ’year’ → effective rate = nominal rate Example Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 47 / 52 Nominal vs Effective: i nominal = 6% Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 48 / 52  What interest rate ? Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 49 / 52 The Risk-Free interest rate In our simplified risk-free world → all cash flows are certain Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 50 / 52 The Risk-Free interest rate In our simplified risk-free world → all cash flows are certain At the Risk-free interest rate the supply of savings equals the demand for borrowing Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 50 / 52 The Risk-Free interest rate In our simplified risk-free world → all cash flows are certain At the Risk-free interest rate the supply of savings equals the demand for borrowing Interest rate is the ’exchange rate’ that allows us to convert money from one point in time to another Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 50 / 52 The Risk-Free interest rate In our simplified risk-free world → all cash flows are certain At the Risk-free interest rate the supply of savings equals the demand for borrowing Interest rate is the ’exchange rate’ that allows us to convert money from one point in time to another There’s a unique interest rate Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 50 / 52 The Risk-Free interest rate In our simplified risk-free world → all cash flows are certain At the Risk-free interest rate the supply of savings equals the demand for borrowing Interest rate is the ’exchange rate’ that allows us to convert money from one point in time to another There’s a unique interest rate → This will change once we introduce risk ! Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 50 / 52 Conclusions Decision making requires comparing the cash flows of different courses of action Only values at the same point in time are comparable Valuation and NPV decision making thus require discounting or compounding cash flows that occur at different points in time Simple formulas for valuing perpetuities and annuities are useful in practice Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 51 / 52 Simple Formulas Compounding and future value: FV = C (1 + r )T Discounting and present value: PV = C1 (1+r ) + C2 (1+r )2 + ··· + CT (1+r )T Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 52 / 52 Simple Formulas Compounding and future value: FV = C (1 + r )T Discounting and present value: PV = C1 (1+r ) + C2 (1+r )2 + ··· + CT (1+r )T Perpetuity Constant: PV = C r Growing: PV = C r −g Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 52 / 52 Simple Formulas Compounding and future value: FV = C (1 + r )T Discounting and present value: PV = C1 (1+r ) + C2 (1+r )2 + ··· + CT (1+r )T Perpetuity Constant: PV = C r Growing: PV = C r −g Annuity " !# Constant: PV = C r 1− 1 (1+r )T " !T # Growing: PV = C r −g 1− 1+g 1+r Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 52 / 52 Simple Formulas Compounding and future value: FV = C (1 + r )T Discounting and present value: PV = C1 (1+r ) + C2 (1+r )2 + ··· + CT (1+r )T Perpetuity Constant: PV = C r Growing: PV = C r −g Annuity " !# Constant: PV = C r 1− 1 (1+r )T " !T # Growing: PV = C r −g 1− 1+g 1+r Net present value: NPV=PV(Benefits)-PV(Costs) Gherardo Gennaro Caracciolo Time value of money and the art of making choices September 6, 2024 52 / 52