Wize COMM 308 Final Exam Prep Booklet PDF

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The price (or fair value) of an asset is determined by its Financial Assets are just

on stream of cash flows

Law of One Price: if markets are competitive, then same goods must have same price In terms of financial assets, same good refers to same with same

How this works? People are motivated by self-interest (aka greedy) Hunt for mis-priced opportunities When price is under-priced, demand increases ( When price is over-priced, supply increases ( People ensure Law of One Price holds

) )

Principle of No Arbitrage: process described above Arbitrage: mis-pricing that leads to risk-less profit Arbitrageur: investor who profits from executing arbitrage strategy (ie. the greedy people)

“A dollar today is worth more than a dollar tomorrow.” 

This is because you can invest this dollar today (for instance in a bank account) and have one dollar plus earned interest tomorrow. This idea is called time value of money (TVM). How much more is a dollar today worth than a dollar tomorrow?

1. Future Value and Present Value How do you compare two different cash flows that occur at the same time period?

How about cash flows from different time periods?

Caution: it is wrong to compare cash flows occurring at different time periods by their magnitudes alone. Finance is fundamentally about manipulating cash flows and values from different time periods making them comparable. Once cash flows are in the same time period, you can do whatever you want to them: add, subtract, multiply, etc… Simple interest: interest is paid / earned only on the original principle amount. Simple interest is usually not applied in the business world.

F utureV alue = Pr inciple + Pr inciple × r ×  n where Principle = Original amount at time 0 r = Interest rate n = Number of periods

Example: You deposit $4,000 today at the bank. At the end of year 3 you deposit another $25,000. Your bank pays a simple interest rate of 8% a year. How much do you have in your bank account at the end of 5 years? 

Solution:

Compound interest: Interest is paid on the original principle amount and any earned interest. Compound interest is standard in the business world.

n

F utureV alue = Pr incople × (1 + r) where Principle = Original amount at time 0 r = Interest rate n = Number of periods

Example: You deposit $4,000 today at the bank. At the end of year 3 you deposit another $25,000. Your bank pays a compound interest rate of 8% a year (compounded annually). How much do you have in your bank account at the end of 5 years?

Solution:

By using the power of compounding, we can find the value of a current amount anytime in the future as long as we have some interest rate. This is called the future value (FV) of a cash flow or dollar amount. Conversely, if given a value that occurs in the future (FV), we can manipulate the equation to find out how much is needed to be invested / deposited today in order to grow to the future value when the future value occurs (period n). 

n

F utureV alue = Pr inciple × (1 + r) → F utureV alue = Pr inciple (1 + r)n

Let’s replace Principle in the above equation with Present Value. Voila, the most famous equation in finance.

Pr esentV alue = 

F utureV alue n (1 + r)

More generally, if there are multiple cash flows then the present value of all the cash flows is

PV =

CF1 1

(1 + r)

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+

CF2 2

(1 + r)

+ ... +

CFn (1 + r)n

By calculating the FV or PV of one of the cash flows, we are essentially “moving” that cash flow to the other’s time period, thus making both cash flows “comparable”. Once the cash flow amounts are in the same time period, you can do whatever you want to them (add, subtract, multiple, or divide).

Note: Present value (price) and discount rate are inversely-related.

General Steps to valuing cash flows: 1. Draw a timeline! 2. Determine the dollar amount of each cash flow and what time period they occur. 3. Determine if you’re looking for present value or future value. 4. Find the correct effective periodic interest rate to use to discount / future value the cash flows. 5. Put all together in the equation.

Concept Clarifier: Concept Clarifier: TVM At the end of Year 2, Rabi will receive a $60,000 cheque from his father. At the end of Year 5, Rabi will receive a $35,000 cheque from his mother. The effective annual interest rate is 3.25%. What is the total present value of Rabi’s cheques from his parents as at Year 0?

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To find the present value of future cash flows, we simply discount each future cash flow with an effective periodic interest rate.

There are times when the same dollar amount reoccur in the future over a fixed time interval. This is known as an annuity.

More precisely, annuities have the following characteristics: 1. Fixed dollar amount of payout over time; 2. Regular time interval of payout; and 3. End date.

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How to discount annuities? Approach 1: do the same as before! o ie, discount each annuity cash flow individually and then add them up.

Approach 2 (Preferred Approach): use the present and future values of annuities equations

1 − (1 + r) P V Annuity = C ( r

−n

)

n

((1 + r) − 1) F V Annuity = C r

where C = Dollar amount of annuity r = Effective periodic interest rate n = Number of payment periods in the annuity

Concept Clarifier: Concept Clarifier: Annuity 1 Joey is calculating the present value of his future federal GST rebates. He estimates that he will receive $300 in each quarter of a calendar year. Joey’s bank account offers an APR of 1.25% that compounds monthly. What is the present value of Joey’s total future federal GST rebates over a seven-year period?

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• Ordinary Annuity: cash flows occur at the end of each period (previous section).

• Annuity Due: cash flows occur at the beginning of each period instead of at the end

• Calculate PV of an annuity due using the shortcut equation:

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1 − (1 + r) P V AnnuitiesDue = C ( r

−n

) (1 + r)

OR

where

C= Dollar amount of annuity due r = Effective periodic interest rate n = Number of payment periods in the annuity

Calculate FV of an annuity due using the shortcut equation: n

F V = C ( (1+rr)

−1

) (1 + r )

where C = Dollar amount of annuity due r = Effective periodic interest rate n = Number of payment periods in the annuity

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Concept Clarifier: Concept Clarifier: Annuity Due Carmen and Carrie are sisters. Carmen donates $30 to World Vision Canada at the beginning of each month. Carrie donates $30 to World Vision Canada at the end of each month. Their bank accounts both offer an effective annual interest rate of 1.25%. Calculate, separately, the present value of each sister’s donations over the next ten years.

Concept Clarifier: Concept Clarifier: Annuity Due (Future Value) Joe and Cindy are siblings. Joe saves $1,000 at the beginning of each calendar quarter. Cindy saves $1,000 at the end of each calendar quarter. Their bank accounts offer an effective annual interest rate of 1.25%. Calculate, separately, the future value of each sibling’s savings over the next five years.

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Growing Annuity: is an annuity where the amounts grow at a constant rate (g) each period until the end date. Because the annuity amounts are growing each period, the dollar value of the annuities are not equal. Growing annuities have the following characteristics: 1. Constant growth rate (g) over the life of the growing annuity; 2. Regular time interval of payout; and 3. End date.

• Calculate PV of a growing annuity using the shortcut equation: n

1+g C1 P V GrowingAnnuity = ) ) (1 − ( 1+r r−g where C1 = Dollar amount of first period annuity payment r = Effective periodic interest rate g = Constant growth rate of annuity payment n = Number of payment periods in the growing annuity

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• Like ordinary annuities, there’s a shortcut to calculate FV of growing annuity:

F V GrowingAnnuity =

C1 ((1 + r )n − (1 + g)n ) r−g

where C1 = Dollar amount of first period annuity payment r = Effective periodic interest rate g = Constant growth rate of annuity payment n = Number of payment periods in the growing annuity

Concept Clarifier: Concept Clarifier: Growing Annuity Donald purchased a car four years ago with a five year financing plan at 12% APR compounded monthly, with monthly payments. There was a typo in the contract and Donald is now bound by it. The typo states that the last year of his financing plan the monthly payments grow at a rate of 0.5% per month. He just paid $450, which is his last monthly payment before this typo clause is effective. Donald doesn’t like to calculate each month how much he needs to pay and would like to deposit a lumpsum in his account, which pays him 1.25% a month, to cover the remaining payments. How much does he need to deposit?

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• When an annuity doesn’t have an end date, it is known as a perpetuity. o ie, the annuity cash flows continue at each period forever.

• There are three characteristics of a perpetuity: 1. Fixed dollar amount; 2. Fixed amount occurs at regular time interval; 3. No end date. Continues forever. • Must calculate PV of perpetuity using the shortcut equation (there’s no end date!):

P V P erpetuity = where C = Dollar amount of perpetuity r = Effective periodic interest rate Concept Clarifier: Concept Clarifier: Perpetuity 

C r

David has a farm. On average, his farm yields $120,000 at the end of each year. He intends to sell his farm to Joe. What is the value of his farm? We assume an effective daily interest rate of 0.01%.

• Perpetuity Due: perpetuity cash flows occur at the beginning of each period instead of at the end.

P V P erpetuity = C +

C r

where C = Dollar amount of perpetuity r = Effective periodic interest rate

Concept Clarifier: Concept Clarifier: Perpetuity Due Stephanie has a business. On average, her business generates $50,000 at the beginning of each year. She intends to sell her business to attain funds to buy David’s farm. What is the value of her business if she is days away from receiving her next annual inflow from her business? We assume a quarterly APR of 2%. 

Growing Perpetuity: is a perpetuity where the perpetuity payments grow at a constant rate (g) each period in perpetuity. Again, because the payments grow each period, the dollar value of the payments are not equal. Growing annuities have the following characteristics: 1. Constant growth rate (g) each period in perpetuity; 2. Regular time interval of payout; and 3. No end date. Continues forever. • Calculate PV of a growing perpetuity using the shortcut equation:

P V GrowingP erprtuity =

where

C1 r−g

C1 = Dollar amount of first period perpetuity payment r = Effective periodic interest rate g= Constant growth rate of annuity payment

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Concept Clarifier: Concept Clarifier: Growing Perpetuity Loan shark Vinnie took Intro Finance back in the day so he understands he shouldn’t put all his eggs in the loan shark business. He’s diversified into other lines of businesses. His newest “business” is reigning in money and brought in $2,000 for him last month. This amount has been growing at an acceptable rate of 3% each month and Vinnie thinks both this growth rate and the business will continue forever. How much is this business worth if the discount rate for a business of this risk is 20% per month?

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Annual Percentage Rate (APR)

APR is a quoting convention where interest rates (monthly, quarterly, semiannually, etc…) are multiplied by the number of periods in a year → “annual” rate.

Not particularly useful because APR does not take into account effects of compounding! Thus comparing a loan quoted in APR rate with another loan’s effective annual interes rate (EAR) requires some work. Simply comparing the APR with EAR is like comparing a temperature in Fahrenheit with a temperature in Celsius – they are on different scales so one needs to be converted into the other for comparison to be possible. An interest rate is “APR” when: A quotation is provided; Compounding frequency is indicated; or Periodicity of the interest rate is not annual and is not specified to be “effective” (e.g. “monthly” interest rate).

APR is quoted as [APR, periodicity of compounding], ex [0.10, 4] is read 10% APR compounded quarterly. APR must be converted to the appropriate effective periodic rate before cash flow valuation is performed. 

k

AP R ) −1 EAR = (1 + k

1

rperiodic = (1 + EAR) m − 1

where EAR = Effective annual interest rate(akarannual ) k = Compounding frequency of APR rperiodic = effective periodic interest rate m= Periodicity of the desired effective periodic interest rate

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Common number of compounding periods within a year.

Example: For the APR [10%, 2], what is the effective annual rate? Solution:

EAR = (1 +

AP R k ) k

−1

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Effective Periodic Interest Rate

Often, cash flow (or payment) periodicity ≠ periodicity of compounding of the APR. To correctly value cash flows, the effective periodic interest rate must be the same periodicity of the cash flows

For instance, if cash flows are paid semi-annually in the future, then the effective semi -annual rate must be used to discount these cash flows. Likewise, an effective monthly interest rate is required to discount cash flows that are paid monthly.

Rule: cash flow payment frequency dictates the effective periodic interest rate to use.

ref f ectiveannual=EAR= 1+AP R )k −1 ( k

1

rperiodic = (1 + EAR) m − 1

Note: if compounding is continuous, the equation becomes

r periodic = eAP R×t − 1 where

e = Exponential function t = Number of years in the desired period you want to calculate 

General Steps to finding the effective periodic interest rate: 1. Determine the compounding frequency of the APR. 2. Determine the periodicity / frequency of the cash flows. 3. Calculate the EAR. 4. Calculate the effective periodic interest rate using the periodicity of the cash flows.

Concept Clarifier: Concept Clarifier: Interest Rates 1 What is the effective quarterly rate for an APR of 12%, compounded monthly?

Concept Clarifier: Concept Clarifier: Interest Rates 2 Convert the effective daily interest rate of 0.056% to an effective quarterly interest rate.

Concept Clarifier: Concept Clarifier: Interest Rates 3 Loan shark Vinnie offers his clients loans at an annual interest rate of 50% per year, compounded continuously. What is the effective monthly interest rate if his clients pay him back in one month? How about in one year?

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In the most general case, firms are funded by two groups: stockholders and bondholders. Bond: a financial security issued by a corporation or government that wishes to borrow money from the public. o Bond issuers are the borrowers o Bond purchasers are the lenders Bonds are often referred to as “Fixed Income Securities” because of their predetermined fixed cash flows. Bond covenant specifies the terms and conditions for the bond (i.e. what the borrower can and cannot do). Traded “Over-the-Counter” (OTC) in public debt markets Types of bonds: o Coupon bonds (aka interest-only bonds): regular coupon (interest) payments on payment dates and face value payment at bond maturity.

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o Zero coupon bonds (aka strip bonds, pure discount bonds): no coupon payments. Only face value payment at bond maturity.

Face Value (F): principal amount that the issuer promises to pay on maturity date. Normally $1,000, unless otherwise stated. Maturity Date: end date of the bond – when the issuer pays the bondholder the face value and fina coupon payment. Payment Frequency (k): how often coupon payments are paid in a year. o  = 1 if annual o  = 2 if semi-annual o  = 4 if quarterly o  = 12 if monthly Coupon Rate (CR): interest rate the issuer pays for borrowing. Quoted in APR. Coupon Payment (C): dollar amount of interest payments each period. Calculated using coupon rate. The dollar amount of each coupon payment is the same for all coupon payments. Coupons

C =F ×( where

CR ) k

C = Coupon payment F = Face value of bond CR = Coupon rate of bond k = Number of coupon payments per year. Payment frequency of coupon

bond. Price of Bond Formula 

1 − (1 + r) PV = C × ( r

−n

where

)+

F (1 + r)n

C = Coupon payment F = Face value of bond CR = Coupon rate of bond n = Number of total periods

Note: the correct r (effective periodic rate) to use in the bond pricing equation is based on the bond’s yield to maturity (YTM) and NOT the coupon rate!

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Yield to Maturity (YTM): overall rate of return of a bond if held to maturity and all coupons payments are reinvested at the YTM rate. Quoted in APR. Yield to Maturity (YTM)

Y TM = r × k where

r = Effective periodic discount rate k = Number of coupon payments per year. Payment frequency of coupon

bond.

General Steps to value a bond: 1. Draw a timeline! 2. Find the number of coupon payments in a year and calculate the coupon payment. 3. Find the correct effective periodic rate to use in the bond pricing equation. 4. Put all together in bond pricing formula.

Sometimes, you’ll be given bond price and all bond essentials and need to find YTM. −n

1 − (1 + r) PV = C × ( r



)+

F (1 + r)n

Try as hard as you may, you won’t be able to solve for (ie, isolate) r from the bond pricing equation. o Must use “trial and error” to find YTM!

Remember: price and discount rate are inversely related. YTM = Coupon Rate is where Bond Price = Face Value. As YTM (and thus r) increases, price decreases and vice versa.

The “current yield” gives you a good estimate for the YTM.

CurrentY ield =

AnuualCoupon ($) CurrentBond Pr ice

where Annual Coupon ($) = Dollar amount of all coupons in the year

Realized Rate of Return (ROR): Annual rate of return an investor actually earned during the holding period (ie. when the investor actually owned the instrument). 1

Psale + F V RC n ) −1 ROR = ( Ppurch 

where

 = Selling price of bond FVRC = FV of reinvested coupons ℎ = Purchase price of bond n = Number of years investor held the bond

Bond pricing is affected by market interest rate changes. o All things equal, bonds of longer terms are more sensitive to market interest rate changes.

o All things equal, bonds of lower coupon rates are more sensiti...