Chapter 5 Time Value of Money

most important technique used in finance is Time Value of Money (TVM) also called discount cash flow (DCF)

-uses compound interest (interest is earned in prior periods' interest) rather than simple interest (interest is not earned on interest, it's interest rate x number of periods that interest is earned)

Annuity

A series of equal payments that occur on a regular basis

Annuity Due

BEG MODE an annuity whose payments occur at the beginning of each period

Ordinary Annuity

END MODE An annuity that pays at the end of each period.

Find the FV of these ordinary annuities. Compounding occurs once a year. a) $700 per year for 14 years at 0%

END MODE PV: 0 FV: ? PMT: -700 I/Y: 0 N: 14 Rework the problem assuming that they are annuities due. BEG MODE PV: 0 FV: ? PMT: -700 I/Y: 0 N: 14

Find the PV of these Ordinary annuities. Discounting occurs once a year. a) $400 per year for 10 years at 14%

END MODE PV: ? FV: 0 PMT: -400 I/Y: 14 N: 10 Rework as Annuity Due BEG MODE PV: ? FV: 0 PMT: -400 I/Y: 14 N: 10

You have $14,776.68 in a brokerage account, and you plan to deposit an additional $4,000 at the end of every future year until you account totals $230,000. You expect to earn 12% annually on the account. How many years will it take to reach your goal?

PV: -14776.68 FV: 230000 PMT: -4000 I/Y: 12 N: ?

Your parents will retire in 29 years. They currently have $210,000 saved, and they think they will need $1,550,000 at retirement. What annual interest rate must they earn to reach their goal, assuming they don't save any additional funds?

PV: -210000 FV: 1550000 PMT: 0 I/Y: ? N: 29

Bank 1 lends funds at a nominal rate of 8% with payments to be made semi-annually. Bank 2 requires payments to be made quarterly. If Bank 2 wants to charge the same EAR as Bank 1, what INOM will they charge their customers?

Step 1 find EAR of Bank 1 NOM: 8% C/Y: 2 (semi) EAR: ??? -> 8.16 Step 2 plug in EAR from Bank 1 to find INOM NOM: ??? -> 7.92 C/Y: 4 (quarterly) EAR: 8.16 They will charge INOM 7.92

effective annual rate (EAR)

The annual rate if interest actually being earned -If compounding periods are different then EAR must be used for comparison. -If compounding occurs more than once a year, EAR is higher than INOM.

You want to buy a car, and a local bank will lend you $15,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest rate will be 12% with interest paid monthly. a) what will be the monthly loan payment? b) what will be the loan's EAR?

a) FV: 0 PV: 15000 PMT: ? I/Y: 1 ( 12% / 12 = 1 ) N: 60 b) NOM: 12% C/Y: 12 EFF: ?

Find the interest rates earned on each of the following: a) you lend $750 and the borrower promises to pay you $780 at the end of 1 year. b) you borrow $51,000 and promise to pay back $111,815 at the end of 5 years. c) You borrow $9,000 and promise to make payments of $2,684.80 at the end of each year for 5 years.

a) FV: 780 PV: -750 PMT: 0 I/Y: ? N: 1 b) FV: -111815 PV: 51000 PMT: 0 I/Y: ? N: 5 c) FV: 0 PV: 9000 PMT: -2684.8 I/Y: ? N: 5

Find the amount to which $400 will grow: a) 7% compounded semiannually for 10 years b) 7% compounded daily for 10 years c) Why does the observed pattern of FVs occur?

a) FV: ? PV -400 PMT: 0 I/Y: 3.5 ( 7/2= 3.5 ) N: 20 ( 10*2= 20 ) b) FV: ? PV: -400 PMT: 0 I/Y: 0.0192 ( 7 /365 = 0.0192 ) N: 3650 ( 10*365= 3650 ) c) The FV increases bc as the compounding periods per year increase, interest is earned on interest more frequently.

Find the following values. Compounding/discounting occurs annually. a) an initial $800 compounded for 10 years at 7% b) the present value of $1,730 due in 10 years at 14% c) define PV d) how are PVs affected by interest rates?

a) FV: ? PV: -800 PMT: 0 I/Y: 7 N: 10 b) FV: 1730 PV: ? PMT: 0 I/Y: 7 N: 10 c) The PV is the value today of a sum of money to be received in the future and in general is less than the FV. d) The PV will decrease as the interest rate increases.

Sawyer Corporation's 2015 sales were $6 million. It's 2010 sales were $3 million. a. At what rate have sales been growing? b. suppose someone made this statement: "Sales doubled in 5 years. This relresents a growth if 100% in 5 years, so, dividing 100% by 5, we find the growth rate to be 20% per year." Is the statement correct?

a) FV: 6 PV: -3 PMT: 0 I/Y: ? N: 5 ( 2015-2010= 5 ) b) The statement is incorrect since the effect of compounding is not considered.

Find the PV of $700 due in the future under these conditions: a) 4% nominal rate, semiannual compounding, discounted back 7 years b) 4% nominal rate, quarterly compounding, discounted back 7 years

a) FV: 700 PV: ? PMT: 0 I/Y: 2 ( 4 / 2 = 2 ) N: 14 ( 7*2 = 14 ) b) FV: 700 PV: ? PMT: 0 I/Y: 1 ( 4 / 4 = 1 ) N: 28 ( 7*4 = 28 )

Perpetuity

an annuity in which the cash flows continue forever -payments of a perpetuity constitute an infinite series. ex. a preferred stock with no maturity is an example o a perpetuity

A dollar today is worth ___ than a dollar in the future because if you had it now you could invest that dollar and earn interest

more than

Future Value (FV)

or compounding, is the process of going from today's values to future amounts. -the greater the interest rate, the faster the growth rate

Present Value (PV)

or discounting, the value today of a future cash flow or series of cash flows -PV declines faster at higher interest rates -A graph of the discounting process shows how the PV of any sum to be received in the future decreases and approaches 0.

nominal interest rate (INOM)

the stated interest rate -also called the annual percentage rate (APR) which is periodic rate times number of periods per year. -if compounding periods for different securities are the same then use the APR for comparison -If loan/investment uses annual compounding then the INOM is also the EAR.