TVM

Default risk premium

Premium required by investor because of the risk of default An extra return that compensates investors for the possibility that the borrower will fail to make a promised payment at the contracted time and in the contracted amount.

r =

Real risk-free interest rate + Inflation premium + Default risk premium + Liquidity premium + Maturity premium

Two types of annuities

1. Ordinary annuity (cash flows occur at the end of each compounding period, most common) 2. Annuity due (payment occurs at the beginning of each period, first payment is today at t=0)

Future value factor

(1 + r)^N

Perpetuity (perpetual annuity)

A financial instrument that pays a fixed amount of money at set intervals over an infinite period of time = PMT / (i/y) E.g. preferred stock

N = 30 × 12 = 360; I/Y = 9 / 12 = 0.75; PV = -150,000(1 − 0.2) = -120,000; FV = 0; CPT → PMT = $965.55

An investor is looking at a $150,000 home. If 20% must be put down and the balance is financed at 9% over the next 30 years, what is the monthly mortgage payment?

Interest rate

A rate of return that reflects the relationship between differently dated cash flows; a discount rate / the sum of a real risk-free rate and premiums that compensate investors for bearing distinct types of risk denoted r If $9,500 today and $10,000 in one year are equivalent in value, then $10,000 − $9,500 = $500 is the required compensation for receiving $10,000 in one year rather than now. The interest rate—the required compensation stated as a rate of return—is $500/$9,500 = 0.0526 or 5.26 percent. Three ways to consider them: (1) Required rate of return (2) Discount rates (ex above is discount) (3) Opportunity cost In the example, if the party who supplied $9,500 had instead decided to spend it today, he would have forgone earning 5.26 percent on the money. So we can view 5.26 percent as the opportunity cost of current consumption

Maturity premium

An extra return that compensates investors for the increased sensitivity of the market value of debt to a change in market interest rates as maturity is extended. [Notes: The difference between the interest rate on longer-maturity, liquid Treasury debt and that on short-term Treasury debt reflects a positive maturity premium for the longer-term debt (and possibly different inflation premiums as well)]

Donald invested $3 million in an American bank that promises to pay 4% compounded daily. Which of the following is closest to the amount Donald receives at the end of the first year? Assume 365 days in a year. A. $3.003 million B. $3.122 million C. $3.562 million

B. $3.122 million ------------------- N = 365 I = (4/365) = 0.011 PV = 3,00,000 FV = 3,122,893.8

Jill Smith wishes to compute the required rate of return. Which of the following premiums is she least likely to include? A. Inflation premium B. Maturity premium C. Nominal premium

C. Nominal premium There is no such component in the list of premiums in CFA curriculum

Which is the following is least likely true? A. Discount rate is the rate needed to calculate present value B. Opportunity cost represents the value an investor foregoes C. Requires rate of return is the maximum rate of return an investment must receive to accept an investment

C. Requires rate of return is the maximum rate of return an investment must receive to accept an investment Minimum* not maximum

Inflation premium

Compensates investors for expected (annual) inflation in the upcoming premium and reflects the average inflation rate expected over the maturity of the debt Inflation reduces the purchasing power of a unit of currency—the amount of goods and services one can buy with it.

Cyndia Rojers deposits $5 million in her savings account. The account holders are entitled to 5% interest. If Cyndia withdraws cash after 2.5 years, how much cash would she most likely be able to withdraw?

FV = PV (1 + r)^n FV = 5 (1 + 0.05)^2.5 = $5.649 million

Alice would like to have $5,000 saved in an account at the end of three years. If the return on the account is 9% per year with monthly compounding, how much must Alice deposit today in order to reach her savings goal in three years?

The effective monthly rate is 9 / 12 = 0.75%, and we can calculate the present value of $5,000 three years (36 months) from now as 5,000 / (1.0075)36 = $3,820.74. Alternatively, since the EAR is 1.007512 − 1 = 0.093807, we can calculate the present value by discounting 5,000 at the EAR for three years. 5,000 / 1.0938073 = $3,820.74, which is the same result.

Liquidity premium

The premium an investor demands because of the lack of liquidity of an investment An extra return that compensates investors for the risk of loss relative to an investment's fair value if the investment needs to be converted to cash quickly. [Notes: US T-bills, for example, do not bear a liquidity premium because large amounts can be bought and sold without affecting their market price. Many bonds of small issuers, by contrast, trade infrequently after they are issued; the interest rate on such bonds includes a liquidity premium reflecting the relatively high costs (including the impact on price) of selling a position.]

EAR

The rate of interest that investors actually realize as a result of compounding is known as the effective annual rate (EAR) or effective annual yield (EAY) EAR represents the annual rate of return actually being earned after adjustments have been made for different compounding periods. (1 + periodic rate)^m − 1 periodic rate = stated annual rate/m m = the number of compounding periods per year Obviously, the EAR for a stated rate of 8% compounded annually is not the same as the EAR for 8% compounded semiannually, or quarterly. Indeed, whenever compound interest is being used, the stated rate and the actual (effective) rate of interest are equal only when interest is compounded annually. Otherwise, the greater the compounding frequency, the greater the EAR will be in comparison to the stated rate.

Real risk-free interest rate

The rate that you get on a security that has no risk and is extremely liquid. Make an assumption here that there is no inflation This is a theoretical rate of return but in practice nominal risk-free interest rate is what you see when there is no risk outside of inflation (eg. T-bills) the single-period interest rate for a completely risk-free security if no inflation were expected. In economic theory, the real risk-free rate reflects the time preferences of individuals for current versus future real consumption.

Nominal risk-free interest rate

The sum of the real risk-free interest rate and the inflation premium If real risk-free interest rate = 3% and the inflation premium = 2%, then the nominal risk-free interest rate = 5% If you hear the term risk-free rate, the assumption is we're talking about this [Notes: Many countries have governmental short-term debt whose interest rate can be considered to represent the nominal risk-free interest rate in that country. The interest rate on a 90-day US Treasury bill (T-bill), for example, represents the nominal risk-free interest rate over that time horizon. US T-bills can be bought and sold in large quantities with minimal transaction costs and are backed by the full faith and credit of the US government.]

John plans to invest $2,500 in an account that will earn 8% per year with quarterly compounding. How much will be in the account at the end of two years?

There are eight quarterly compounding periods in two years, and the effective quarterly rate is 8 / 4 = 2%. The account will grow to 2,500(1.02)8 = $2,929.15. Alternatively, since the EAR is 1.024 − 1 = 0.082432, we can grow the $2,500 at 8.2432% for two years to get 2,500(1.082432)2 = $2,929.15, which is the same result.

Discount

To reduce the value of a future payment in allowance for how far away it is in time; to calculate the present value of some future amount. Also, the amount by which an instrument is priced below its face (par) value.

Compute EAR if the stated annual rate is 12%, compounded quarterly

m = 4 periodic rate is 12/4=3% = 0.03 Thus, EAR = (1 + 0.03)^4 − 1 = 1.1255 − 1 = 0.1255 = 12.55%. This solution uses the [y^x] key on your financial calculator. The exact keystrokes on the TI for the above computation are 1.03 [y^x] 4 [=]. On the HP, the strokes are 1.03 [ENTER] 4 [y^x].