Ch. 4: Time Value of Money: Valuing Cash Flow Streams

1. r is specified as a monthly rate 2. number of periods is expressed in months; FV = C x (1 +r) n = 1000 x (1+.02) ^6 = 1126.16

Non-Annual Cash Flows: When do annual cash flows behave the same as monthly cash flows? Ex: You have a credit card that charges 2% interest per month. You have a $1000 balance on card today, and make no payments for six months, your future balance in one year's time will be?

1.645 million = 10,000/.1 (1.1^30 - 1); 2.625 million in 30 years (or $150,463 today) - PV = 10000/.1-.05 x (1 - 1.05/1.10 ^30)

At the end of each year until she is 65, Ellen will save $10,000 in a retirement account. If the account earns 10% per year, how much will Ellen have in her account at age 65? What if although Ellen can only save $10,000 her first year, 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 in her account at age 65?

2735.54 PV or 3641 FV

Ex: We save $1000 today and at the end of the next two years. At a fixed 10% interest rate, how much will we have in the bank three years from today?

r = rate of interest per compounding period; n = number of periods r = APR / compounding interval and n = number of years x compounding interval

What are r and n? how do you find r from APR and compounding?

Rate of return

the rate at which the present value of the benefits exactly offsets the cost

loan, inverting the annuity formula = C = P / [ (1/r) x (1 - 1/(1+r)^n] ; $7106.19 each year; 60,000 = FV (annuity) = C x 1/.07 x (1.07^10 -1) = $4343

In general, when solving for a _______ payment, think of the amount borrowed (the principal) as the present value of the payments. If the loan payments are an annuity, how can we solve for the payment of the loan? Example: Your firm plans to buy a warehouse for $100,000. The bank offers you a 30-yr loan with equal annual payments and an interest rate of 8% per year. The bank requires that your firm pays 20% of the purchase price as a down payment, so you can borrow only $80,000. What is the annual loan payment? Suppose you would like to have $60,000 saved in 10 years. If you can earn 7% per year on your savings, how much do you need to save each year to meet your goal?

An annuity can be a perpetuity, depending on how it is set up. An annuity is an investment that makes regular payments throughout the year. A perpetuity is a type of annuity that is set up so that the payments never end. As long as an investor owns a perpetuity, he will keep receiving payments. When the investor dies, the perpetuity will pass on to his heirs and keep making payments as normal. If the investor sells the perpetuity, the new owner will receive the payments. Most annuities eventually stop making payments. They might stop making payments after a set number of years or after the contract owner dies. However, if an annuity is set up so that it never stops making payments, then it is a perpetuity. In other words, all perpetuities are annuities, but not all annuities are perpetuities. Because of their extremely long, potentially infinite time frame, perpetuities are relatively rare investments. Annuity companies don't sell perpetuities. The closest example of a true perpetuity is a type of bond from the British government known as a consol. These bonds have no maturity date and keep making interest payments forever - or at least as long as the British government is in existence.

What is the difference between an annuity and a perpetuity?

the annuity is not due right away;

What's a deferred annuity?

PV (48-period annuity of $500) = $500/.005 (1 - 1/ 1.005^48) = 21,290

You are about to purchase a new car and have two payment options. you can pay $20,000 in cash immediately, or you can get a loan that requires you to pay $500 each month for the next 48 months (four years). If the monthly interest rate you earn on your cash is .5%, which option should you choose?

Option (a) gives $11.16 million PV = 1 million + 1 million/0.08 (1 - 1/1.08^29); Option (b) gives $15 million PV

You are the lucky winner of a $30 million state lottery. You can take your prize money either as (a) 30 payments of $1 million per year (starting today) or (b) $15 million paid today. If the interest rate is 8%, which option should you choose?

PV = C/r = 30,000/0.08 = $375,000 today; we must find the PV of 375,000 a year from now -> PV = 375,000/1.08 = 347222 today; 30,000 / (0.08 - 0.04) = 750,000 today

You wan to endow an annual party that has a $30,000 budget per year forever for the party. If the university earns 8% per year on investments, and if the first party is in one year, how much will you need to donate to endow the party? What if the first party will be held two years from today? What if the student association has asked that you increase the donation to account for the effect of inflation on the cost of the party in future years. Although 30,000 is adequate for next year's party, the students estimate that the party's cost will rise by 4% per year thereafter. How much do you need to donate now?

You can guess by plugging in plausible r's to the pv (annuity) equation or use excel/calculator to calculate

Your firm needs to purchase a new forklift. The dealer gives you two options: (1) a price for the forklift if you pay cash and (2) the annual payments if you take out a loan from the dealer. Given the loan payment that the dealer quotes, how do you compute the interest rate charged by the dealer?

consol

a bond that promises its owner a fixed cash flow every year, forever (British government bond)

stream of cash flows

a series of cash flows lasting several periods

growing perpetuity; PV (growing perpetuity) = C / (r-g); r>g

a stream of cash flows that occurs at regular intervals and grows at a constant rate forever; what is the formula (and what can you assume about it)?

growing annuity; n-1 periods of growth; PV of a growing annuity = C/ (r-g) x [1 - [(1+g)/(1+r)]^n]

a stream of cash flows, growing at a constant rate and paid at regular intervals, that end after a specified number of periods; what does the last cash flow reflect? What is the equation?

Annuity; car loans, mortgages, some bonds; PV (annuity) = C/r [ 1 - (1/[1+r]^n)]; FV (annuity) = C/r ([1+r]^n-1) (if we want to know how a savings account will grow over time if the investor deposits the same amount every period); tells you how much you need to start with so that earning x% and pulling out x amount each time, will give you nothing left after N-periods

a stream of equal cash flows arriving at a regular interval and ending after a specified time period; What are examples of these? What is the formula for PV of an annuity? What about FV of an annuity (what is the formal useful for)? What does an annuity tell you?

perpetuity; it arrives at the end of the first period (referred to as payment in arrears); PV (C in perpetuity) = C/r

a stream of equal cash flows that occurs at regular intervals and lasts forever (a scholarship); When does the first cash flow occur? What is the equation for PV of a perpetuity?

annual stated percentage rate (APR)

the return on an investment that is expressed as a per-year percentage, and that does not account for compounding that occurs throughout the year