Managerial Finance Session 3 Nicole Hruban

TIME VALUE OF MONEY Future Value of a Single Sum: Note: Compounding Interest Assumption FVN=n=PV0 x(1+ i)n Present Value of a Future Sum: Future Value of a Ordinary Annuity:

TIME VALUE OF MONEY Future Value of an Annuity Due: (1+i) Present Value of an Ordinary Annuity:

TIME VALUE OF MONEY Present Value of an Annuity Due: Future Value of a Constant Growth Annuity: Present Value of a Constant Growth Annuity:

TIME VALUE OF MONEY Present Value of a Perpetuity: PV=A/i PV of constant growth perpetuity can be written as: PVGAP=A/(i-g) PV of annuity diminishing at a constant rate: PVDAP=A/(i+g) Effective Interest Rate: Nominal Rate = Annual Percentage Rate(APR) Effective Annual Rate = Annual Percentage Yield (APY)

Questions Suppose you want to be able to withdraw $10,000 five years from and $20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals)

Questions Suppose you want to be able to withdraw 10,000 five years from and 20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals) 1 5 6 $10,000 $20,000 X i=5%

INVESTMENT TODAY = $7,835.26 + $14,924.31 = 22,759.57 Questions Suppose you want to be able to withdraw 10,000 five years from and 20,000 six years from now from an account. How much do you have to deposit today if you can earn 5% annually to satisfy your withdrawal needs? (assume there is a zero balance after the withdrawals) FV = $10,000; n=5, i=5% PV = $7,835.26 FV = $20,000; n=6, i=5% PV = = $14,924.31 INVESTMENT TODAY = $7,835.26 + $14,924.31 = 22,759.57 Time Line: ------1-------2-------3-------4-------5-------6 10000 200000 i = 5%

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much is in the account at the end of the first year? How much at the end of the seventh year?

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much is in the account at the end of the first year? 1 11 12 $16,000-$30 = $15,970 Balance – 1.25% = X i = 5.5% per annum or 5.5%/12 per month which compounds charge: 1.25% of balance

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much is in the account at the end of the first year? 5.5%/12 FVN=n=PV0 x(1+ i)n = FV12=15,9700 x(1+ 0.055/12)12 = $16.870.83 Minus Admin Charge: $16.870.83 *0.9875= $16,659.95 OR: FV12 = PV0 x (1+0.055/12)12 * 0.9875

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much is in the account at the end of the first year? How much at the end of the seventh year?

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much at the end of the seventh year? 12 72 84 $15,970 X i = 5.5% per annum or 5.5%/12 per month charge: 1.25% of balance

Questions You decided to invest a $16,000 bonus in a money market account that guarantees a 5.5% annual interest, compounded monthly for 7yrs. A one time fee of $30 is charged to set-up the account . In addition, there is an administrative charge of 1.25% of the balance in the account at the end of the year. How much at the end of the seventh year? For one year: FV12 = PV0 x (1+0.055/12)12 * 0.9875 For seven years: FV12 = PV0 x (1+0.055/12)12*7 * 0.98757 = 15,970 * (1+0.055/12)84 * 0.98757 =$23,449.11*0.98757 =$21.472.67

Questions On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, 2007 he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,990.76 . If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2,203.76. What was ratio of k/j?

Questions On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, 2007 he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,990.76 . If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2,203.76. What was ratio of k/j? 2002 2007 2010 $1,000 Comp. semi-annually Comp. quartely $1,990.76 i = j i = k We also know: if k would be earned for the entire period, the balance would be: $2,203.76

Questions On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, 2007 he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,990.76 . If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2,203.76. What was ratio of k/j? FVN=n=PV0 x(1+ k/4)4*N = > solve for k 2,203.76=1,000 x(1+ k/4)4*8 = 2.20376 = (1+k/4)32 / ^1/32 2.20376^1/32 = 1+k/4 1.025 =1+k/4 => k=0.1

Questions We also know: On January 1st, 2002 Jack deposited $1,000 into Bank X to earn an interest rate of j per year compounded semi-annually. On January 1st, 2007 he transferred his account to Bank Y to earn an interest rate of k per annum compounded quarterly. On January 1st, 2010 the balance at Bank Y was $1,990.76 . If Jack could have earned the interest rate of k per annum compounded quarterly from January 1st, 2002 to January 1st, 2010 his balance would have been $2,203.76. What was ratio of k/j? We also know: We have an initial investment, that earned j compounded semi-annually for five years FVN=n=PV0 x(1+ j/2)2*N = BALANCE07=1,000 x(1+ j/2)2*5 The BALANCE07 will earn a rate of 10% (k) per annum compounding quarterly

Questions We also know: We have an initial investment, that earned j compounded semi-annually for five years FVN=n=PV0 x(1+ j/2)2*N = BALANCE07=1,000 x(1+ j/2)2*5 The BALANCE07 will earn a rate of 10% (k) per annum compounding quarterly for 3 years: BALANCE10 = BALANCE07 x (1+0.1/4)3*4 1,990.76 = 1,000 x(1+ j/2)2*5 x (1+0.1/4)3*4 1,990.76 = 1,000x(1+j/2)10 x 1.02512 1.48025 = 1,000x(1+j/2)10 / ^1/10 then solve for j => j = 0.08 THEREFORE: k/j = 0.1/0.08 = 1.25

Questions The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments). During the payout period, interest is credited at an annual rate of 3%. What is the value of X?

Questions The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments), with the first payment starting immediately. During the payout period, interest is credited at an annual rate of 3%. What is the value of X? 2002 2007 $10,000 i=3% i=5% 120 X X

Questions The proceeds of a $10,000 death benefit are left on deposit with the insurer at an effective annual interest rate of 5%. The balance at the end of 7 years is paid to the beneficiaries in equal monthly payments of X for 10 years (120 payments), with the first payment starting immediately. During the payout period, interest is credited at an annual rate of 3%. What is the value of X? We can calculate the balance at the end of year 7: FV7=PV0 x(1+ i)7 = FV7=10,000 x(1+ 0.05)7 = $14,071 We can then calculate the annuity payment by: Solving for A With PVA = $14,071 i = 0.03/12 = 0.0025 N=120 (i.e. 12*10) ANNUITY DUE ANNUITY: $119.01

Questions Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be?

Questions Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be? 1 19 20 $1,000,000 Constant $ X Constant $ X Constant $ X i = 9% g = 3% $0

Questions Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be? rNOM 9.00% Initial sum 1,000,000 Inflation 3.00% Years 20 A=$88,530.31

Questions Your father now has $1,000,000 invested in an account that pays 9.00%. He expects inflation to average 3%, and he wants to make annual constant dollar (real) end-of-year withdrawals over each of the next 20 years and end up with a zero balance after the 20th year. How large will his initial withdrawal (and thus constant dollar (real) withdrawals) be?

Question You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today. What is the current value of the offer if the prevailing rate of interest is 7% compounded annually? What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly?

Question You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today. What is the current value of the offer if the prevailing rate of interest is 7% compounded annually? 𝑃𝑉𝐴=𝐴 ∗( (1+𝑖) 𝑁 −1 𝑖 ( 1+𝑖) 𝑁 ) ) = 50 * ( (1+0.07) 3 −1 0.07 ( 1+0.07) 3 )=131.2158 The difference is that section 1 we were increasing both the payment frequency and the compounding frequency. Here we are increasing only the compounding frequency. We aren't getting our payments any sooner but we are compounding more frequently so less front money (i.e., a smaller PV) is required in order to grow to the specified future value.

Question You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today. What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly? Step1: Translate 7% rate into effective interest rate EAR= (1+0.07/12)^12 – 1 = 0.07229

Question You have a coin you wish to sell. A potential buyer offers to purchase the coin from you in exchange for a series of three annual payments of $50 starting one year from today. What is the current value of the offer if the prevailing rate of interest is 7% compounded monthly? 𝑃𝑉𝐴=𝐴 ∗( (1+𝑖) 𝑁 −1 𝑖 ( 1+𝑖) 𝑁 ) ) = 50 * ( (1+0.07229) 3 −1 0.07229 ( 1+0.07229) 3 )=130.67 The difference is that section 1 we were increasing both the payment frequency and the compounding frequency. Here we are increasing only the compounding frequency. We aren't getting our payments any sooner but we are compounding more frequently so less front money (i.e., a smaller PV) is required in order to grow to the specified future value.