FIN 3320 Exam 2

The present value of $600 that is paid at the end of year 2 discounted at 6% (Compounded Annually)

$588.2353

$500 per year for 8 years at 14% (Compounding Occurs Once a year)

$6,616.38

An initial $600 compounded for 1 year at 6%. What is the future value (Compounded annually)

$636

An initial $600 compounded for 2 years at 6%. What is the future value? (Compounded Annually)

$674.16

You want to buy a car, and a local bank will lend you $40,000. The loan will be fully amortized over 5 years (60 months), and the nominal interest rate will be 8% with interest paid monthly. What will be the monthly loan payment? What will be the loan's EAR?

$811.0558

$250 per year for 4 years at 7%(Compounding Occurs Once a year)

$1,109.958

What is the PV of a security that will pay $29,000 in 20 years if securities of equal risk pay 5% annually

$10,929.7950

The present value of $200 paid in 10 years at 4% (Compounded Annually)

$135.1128

$300 per year for 6 years at 4% (Discounting Occurs Once a Year)

$1572.6411

If you deposit $2,000 in a bank account that pays 6% interest annually, how much will be in your account after 5 years?

$2648.2657

$700 per year for 4 years at 0% (Compounding Occurs Once a year)

$2800

An initial $200 compounded for 10 years at 4%. Find the Future Value (Compounded Annually)

$296.0489

$500 per year for 6 years at 0%(Discounting Occurs Once a Year)

$3,000

What's the FV of a 5%, 5-year ordinary annuity that pays $800 each year?

$4,420.5050

An initial $200 compounded for 10 years at 8%. Find the Future Value (Compounded Annually)

$431.7850

$600 per year for 12 years at 8% (Discounting Occurs Once a Year)

$4521.6468

The present value of $600 that is paid at the end of year 1 at a discount rate of 6% (Compounded Annually)

$566.0377

Compound Interest

- Interest paid on prior period's ending balance - each period's interest is added to the principal - interest is paid on interest

Simple Interest

- Interest paid on initial principal only - each period's interest is NOT added to the original principal - interest is not paid on interest - FVn= $PV*(1+(i*n)) - when you put the interest rate into a formula, you do as a formula - when you put the interest rate into a calculator, you do as as an integer

Risk Intuition

- Investors prefer dollars today to dollars in the future (Time preferences) • Rather consume (or invest) now than later • Suggests discount rate should be positive • Investors don't like risk (risk aversion) • They require compensation for taking risks in the form of a higher expected return • Also suggests a positive discount rate

Realized Return

- Realized return or cash return measures the gain or loss on an investment in dollars • Example: You invested in 1 share of Apple (AAPL) for $95 and sold a year later for $200. The company did not pay any dividend during that period. What will be the cash return on this investment?

Annuity Interest Rate

- You can also solve for "interest rate" you must earn on your investment that will allow your savings to grow to a certain amount of money by a future date. - "What interest rate should I be targeting to help get me the money I need?

Annuity Number of Periods

- You may want to calculate the number of periods it will take for an annuity to reach a certain future value, given interest rate - "How long do I have to wait?"

Bottom Line

- a dollar today is worth more than a dollar tomorrow - the rate of interest quantifies the tradeoff between cash flows across time - when we get money matter

Effective Annual Rate (EAR)

- annual rate that accounts for compounding - How we compare rates with different - Interpretation: For a given APR and compounding period, EAR is the rate that will earn the same interest under annual compounding - Return over 1 year = (1+APR/m)^m = 1+EAR

Annual Percentage Rate (APR)

- periodic interest rate x periods per year - "Typical" interest rate quotes - All rates in past examples are APRs - We cannot compare two loans based on APR if they do not have the same compounding period

Perpetuity (Present Value)

- stream of level cash flows that never ends - cash flow is always in the amount CG - First Cash flow is in period 1 (time o cash flow is 0)

What is the future value of a 3-year, $100 ordinary annuity if the annual interest rate is 4%?

0 1 2 3 | | | | 100 100 100 104 108.16 $312.16 Go through the following discussion. One approach would be to treat each annuity flow as a lump sum. Here we have FVAN = $100(1) + $100(1.04) + $100(1.04)2 = $100[1 + (1.04) + (1.04)2] = $100(3.1216) = $312.16. Future values of annuities may be calculated in 3 ways: (1) Treat the payments as lump sums. (2) Use a financial calculator. (3) Use a spreadsheet.

The present value of $1,870 due in 10 years at 8% and at 4% (Compounded Annually)

8%: $866.1718 4%: $1263.3050

Draw time lines for (1) a $100 lump sum cash flow at the end of Year 2; (2) an ordinary annuity of $100 per year for 3 years; and (3) an uneven cash flow stream of -$50, $100, $75, and $50 at the end of Years 0 through 3.

A time line is a graphical representation that is used to show the timing of cash flows. The tick marks represent end of periods (often years), so Time 0 is today; Time 1 is the end of the first year, or 1 year from today; and so on. 0 1 2 Year | | | Lump sum 100 Cash flow 0 1 2 3 | | | | Annuity 100 100 100 0 1 2 3 | | | | Uneven cash flow stream -50 100 75 50 A lump sum is a single flow; for example, a $100 inflow in Year 2, as shown in the top time line. An annuity is a series of equal cash flows occurring over equal intervals, as illustrated in the middle time line. An uneven cash flow stream is an irregular series of cash flows that do not constitute an annuity, as in the lower time line. -50 represents a cash outflow rather than a receipt or inflow.

Cash Flow in the Calculator

CF; 2 ; CE/C Input cash flows in the calculator's "CFLO" register: CF 0 = 0 (enter; down arrow) CF 1 = 100 (enter; down arrow; down arrow) CF 2 = 300 (enter; down arrow; down arrow) CF 3 = 300 (enter; down arrow; down arrow) CF 4 = -50 (enter; down arrow; down arrow) Enter I/YR = 4, press NPV button NPV = $597.48. (Here NPV = PV.

Discount Rate

interest rate used in the denominator of present value calculations

A 5-year $100 ordinary annuity has an annual interest rate of 4%. (1) What is its present value?

Input N = 5, I/YR = 4, PMT = 100, FV = 0, and press the PV button.

What's the future value of $100 after 3 years if it earns 4%, annual compounding?

Input the following values: N = 3, I/YR = 4, PV = 100, PMT = 0, and solve for FV = $112.49.

What's the present value of $100 to be received in 3 years if the interest rate is 4%, annual compounding?

N = 3, I/YR = 4, PMT = 0, FV = 100, and then solve for PV = $88.90.

What annual interest rate would cause $100 to grow to $119.10 in 3 years?

N = 3, PV = -100, PMT = 0, FV = 119.10, then press the I/YR button to find I/YR = 6%.

Meet Your Calculator

N: number of periods that interest is paid I/Y: interest rate (10% = 10) PV: present value PMT: payment (can be 0) FV: future value

Annuity Future Value

Q: What if you wanted to find the future value of an annuity? - people who regularly set aside the same amount every month

What is risk?

Risk loosely refers to dispersion in possible outcomes (some better than others) No dispersion = no risk (receive the same percentage return no matter what) Greater dispersion = greater risk

Annuity

level stream of cash flows for a limited time - same, even spacing - cash flow is always in the amount CF - first cash flow is in period 1 (time o cash flow is 0) - last cash flow is in period T

A 5-year $100 ordinary annuity has an annual interest rate of 4%. 4) What would the present value be if this was a perpetuity?

The present value of the $100 perpetuity is $2,500. The PV is solved by dividing the annual payment by the interest rate: $100/0.04 = $2,500.

A 5-year $100 ordinary annuity has an annual interest rate of 4%. (2) What would the present value be if it was a 10-year annuity?

The present value of the 10-year annuity is $811.09. To solve with a financial calculator, input N = 10, I/YR = 4, PMT = 100, FV = 0, and press

A 5-year $100 ordinary annuity has an annual interest rate of 4%. (3) What would the present value be if it was a 25-year annuity?

The present value of the 25-year annuity is $1,562.21. To solve with a financial calculator, input N = 25, I/YR = 4, PMT = 100, FV = 0, and press the PV button.

Define (a) the stated (or quoted or nominal) rate, (b) the periodic rate, and (c) the effective annual rate (EAR or EFF%).

The quoted, or nominal, rate is merely the quoted percentage rate of return, the periodic rate is the rate charged by a lender or paid by a borrower each period (periodic rate = INOM/M), and the effective annual rate (EAR) is the rate of interest that would provide an identical future dollar value under annual compounding.

What's the difference between an ordinary annuity and an annuity due? What type of annuity is shown here? How would you change it to the other type of annuity? 0 1 2 3 | | | | 0 100 100 100

This is an ordinary annuity—it has its payments at the end of each period; that is, the first payment is made 1 period from today. Conversely, an annuity due has its first payment today. In other words, an ordinary annuity has end-of-period payments, while an annuity due has beginning-of-period payments. The annuity shown above is an ordinary annuity. To convert it to an annuity due, shift each payment to the left, so you end up with a payment at Year 0 but no payment at Year 3 as illustrated below. 0 1 2 3 | | | | 100 100 100

You are planning to invest $6,000 at the end of each year in an account that pays 5%. How long will it take before the account is worth $50,000?

Using a Financial Calculator 1/y = 5.0 PV = 0 PMT = -6,000 FV = 50,000 N = 7.14

In 20 years, you are hoping to have saved $100,000 towards your child's college education. If you are able to save $2,500 at the end of each year for the next 20 years, what rate of return must you earn on your investments in order to achieve your goal?

Using a Financial Calculator N = 20 PMT = -$2,500 FV = $100,000 PV = $0 i = 6.77

Measuring Risk

Variance = average squared deviation from the mean • Represents the dispersion of a given distribution • Variance is a natural measure of risk • Standard deviation = square root of variance Higher variance (or standard deviation) represents greater dispersion and, hence, greater risk

Discount Factor

factor by which a cash flow is multiplied to calculate a present value

Future Value

what a cash flow will be worth in the future (words like future, compounding)

Present Value

what a cash flow would be worth to you today (words like today, discounting) - when we want to figure out what something is worth today, we discount (take everything that is happening in the future and we smash it all by discounting it back into what it is worth today)

Expected Return

• Expected return is the rate of return that the investor expects to earn from an investment in the future. • It is the weighted average of the possible returns, where the weights are determined by the probability that it occurs.

Why is the T-bill return independent of the economy? Do T-bills promise a completely risk-free return?

• T-bills will return the promised 3.0%, regardless of the economy. • No, T-bills do not provide a completely risk-free return, as they are still exposed to inflation. Although, very little unexpected inflation is likely to occur over such a short period of time. • T-bills are also risky in terms of reinvestment risk. • T-bills are risk-free in the default sense of the word. -T-bill s have reinvestment rate risk and inflation risk