FINC Financial mathematics Lecture 3

A deposit of $100 earning compound interest at 10% pa, will after 10 years accrue to an amount of

$100(1 + 0.10)10 = $100(2.594) = $259.40

A deposit of $100 earning simple interest at 10% pa, will after 10 years accrue to an amount of?

$100(1 + 10(0.10)) = $100(2) = $200.

A credit union pays interest at 5% p.a. compounded annually. If $1000 is invested for five years, what is the equivalent simple interest rate that the amount will earn?

0.0553, or 5.53%

Elle is 35 years old, and she has decided it is time to plan seriously for her retirement. If the account earns 10 % per year, how much will Ellen have saved at age 65?

1.645 million

Elle is 35 years old, and she has decided it is time to plan seriously for her retirement. If the account earns 10 % per year, how much will Ellen have saved at age 65? 10000 each year and answer in FV

1.645 million

If we have $1,000 today that we put in an account that will generate at rate of return of 12% per year for the next two years, how much will we have in the account in two years?

1000*(1+0.12)^2 = 1254.40

What is the future value of $20,000 paid at the end of each of the following 5 years, assuming your investment returns 8% per year?

117 332

Example: Suppose that benefactor wonders how much it would cost to finance the professorial chair at the Uni for only 20 years rather than for perpetuity (ie $100,000 per year for 20 years)? [r is still assumed to be 10% pa]. What is PV of annuity?

851 4000

Mr Simon de Money wishes to purchase a new Mercedes Benz C250 CDI. The car costs $73,200. Mr de Money has arranged a loan that only covers part of the purchase price. He intends to finance the rest of the purchase with money from his savings. The loan requires payments of $1400 per month for 5 years. The interest rate on the loan is 12% p.a. compounded monthly. How much of his savings must Mr de Money use?

Additional

You are considering buying a new Volkswagen Passat. The car costs $45,000. BankEast Ltd is offering car loans at 7.5% p.a. compounded monthly. a. You plan to make monthly payments on the loan over a 3 year period. How much would you need to repay each month to repay the interest and principal over three years? b. If instead you decided to make weekly repayments on the loan over a 4 year period, how much would each weekly repayment need to be to repay the interest and the principal (calculated on a 52 week year)?

Additional

Suppose person wishes to endow a university so as to provide $100,000 per year in perpetuity for funding of a professorial chair. If the rate of interest is 10% pa, how much must the person donate today?

C = $100,000 r = 0.10 PV of perpetuity = C/r = $100,000/0.10 = $1,000,000

You agree to lease a car for 4 years at $300 per month. You are not required to pay any money up front or at the end of your agreement. If your opportunity cost of capital is 0.5% per month, what is the cost of the lease?

Cost = $12,774.10

What is the FV of $10 000 compounded at 12% annually for 20 years?

FV = 10000*(1.12)^20=96462.93

Suppose you have a choice between receiving $1,000 today or $1,210 in two years. You believe you can earn 10% on the $1,000 today, but want to know what the $1,000 will be worth in two years.

FV=1000*(1+0.10)^2 = 1210

Tiburon Autos offers you "easy payments" of $5,000 per year, at the end of each year for 5 years. If interest rates are 7%, per year, what is the cost of the car?

NPV = 20 501

What is the present value of $100 000 to be received at the end of 25 years given a 5% discount rate?

PV = $29,530

Example: You plan to obtain a $6000 loan from a furniture dealer at 15% annual interest rate that you will pay off in annual payments over four years. Determine the annual payments on this loan and complete the amortisation table.

PV = $6000; r = 0.15; n = 4 C=2101.59

What is the present value of $100 000 to be received at the end of 25 years given a 5% discount rate?

PV = 29530

A 5-year annuity makes annual payments of $100 with the first payment due in one year. What is the present value of the annuity at a discount rate of 12% p.a., compounded monthly?

Payments made annually => require effective rate for one year ( ie effective annual interest rate EAR ). r=(1+(0.12/12))^12 -1 = 0.1268 N=5 C=100 PV of annuity = 354.47

A 5-year annuity makes monthly payments of $100 with the first payment due in one month. What is the present value of the annuity at a discount rate of 12% p.a., compounded monthly?

Payments made monthly => require effective rate for 1 month (this is also the periodic rate). r= 0.12/12=0.01 N=12*5=60 C=100 PV of annuity=4495.50

A 5-year annuity makes monthly payments of $100 with the first payment due in one month. What is the present value of the annuity at a discount rate of 12% p.a., compounded quarterly?

Payments made monthly => require effective rate for one month r=(1+(0.12/4))^4*(1/12) -1 = (1+(0,12/4))^1/3 -1 = 0.0099 N=12*5=60 C=100 PV of annuity = 4507.58

A 5-year annuity makes quarterly payments of $100 with the first payment due in three months. What is the present value of the annuity at a discount rate of 12% p.a., compounded monthly?

Payments made quarterly => require effective rate for one quarter (3 months). r= (1+(0.12/12))^3 -1= 0.0303 N= 4*5= 20 C= 100 PV of annuity= 1483.62

Suppose your credit card has nominal interest rate of 14% pa and interest is charged daily

Periodic (daily) interest rate: 0.14/365= 0.0003836 or 0.03836% Effective annual rate of interest: EAR = 0.1502 or 15.02%

The state lottery advertises a jackpot prize of $295.7 million, paid in 25 installments over 25 years of $11.828 million per year, at the end of each year. If interest rates are 5.9% what is the true value of the lottery prize?

Value = 152.6 m

A bank has quoted you a savings rate of 5% p.a. compounded monthly. i. What is the annual percentage rate (APR)? ii. What is the periodic rate (monthly)? iii. What is the EAR (i.e. effective annual interest rate for 12 months)? iv. What is the effective interest rate for 3 months?

i. The APR is 5%. ii. The periodic rate (monthly) is 0.417% (= 0.05/12 = 0.00416667). iii. The EAR is 5.12% iv. The effective interest rate for 3 months is 1.26%

Your firm plans to buy a warehouse for $100,000. The bank offers you a 30-year loan with equal annual payments and an interest rate of 8% per year. The bank requires that your firm pay 20% of the purchase price as a down payment, so you can borrow only $80,000. What is the annual loan payment?

we can solve for the loan payment, C, given N = 30, r = 8% (0.08) and PV=$80,000. Your firm will need to pay $7,106.19 each year to repay the loan. • The bank is willing to accept these payments because the PV of 30 annual payments of $7,106.19 at 8% interest rate per year is exactly equal to the $80,000 it is giving you today

Suppose you expect to receive $400,000 in 1 year: C1 = 400,000 The discount rate is 7% pa. r = 0.07 What is the PV of this future cash flow?

PV of C1 = 400 000 at 7 % 400000/1.07 = 373832

Previous Example: If benefactor to university has to take account of an annual growth in salaries of 4% pa, how much must be donated to university today?

PV of growing perpetuity = C/(r-g) = $100,000/(0.10 - 0.04) = $1,666,667

What is the present value of an asset that commences paying cash flows of $2 million in two years' time for four years, when the interest rate is 5%?

Step 1: Calculate the present value of $2 million payments at start of yr 2 (= end of yr 1). =7.09m Step 2: Calculate the present value of this lump sum cash flow occurring at the end of period 1 back to time 0. PV0 = 7.09/1.05=6.75

Ellen considered saving $10,000 per year for her retirement. Although $10,000 is the most she can save in the first year, she expects her salary to increase each year so that she will be able to increase her savings by 5% per year. With this plan, if she earns 10% per year on her savings, how much will Ellen have saved at age 65?

This example involves a 30-year growing annuity with a growth rate of 5% and an initial cash flow of $10,000. = $10, 000 15.0463 = $150, 463 today Ellen's proposed savings plan is equivalent to having $150,463 in the bank today. To determine the amount she will have at age 65, we need to move this amount forward 30 years: FV = 150.463*1.10^30 = 2.625 million in years