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Rob's 2000 Notes

πŸ”Ž Algebra Review

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General principle:

  • If you’ve got something on one side, you can move it to the other side, but with the opposite operation.
  • Look for this principle in the key operations below. Then just remember the principle.

Key Operations:

Section titled β€œKey Operations:”

Add/Subtract x from both sides

Section titled β€œAdd/Subtract x from both sides”

You can add or subtract x from both sides to move a number over to the other side of an equation.

If you start with

iβˆ’1=5i - 1 = 5

You can add a 1 to both sides:

iβˆ’1+1=5+1i-1+1 = 5+1

You get

i=6i=6

You can divide both sides by the same number

Section titled β€œYou can divide both sides by the same number”

You can divide both sides by x to move a number over to the other side of an equation.

If you start with

4i=204i = 20

You can divide both sides by 4:

4iΓ·4=20Γ·44i\div4 = 20\div4

You get

i=5i=5

How to get rid of a pesky exponent

Section titled β€œHow to get rid of a pesky exponent”

To get rid of a pesky exponent, you can move it to the other side as a fraction.

For example,

(1+i)10=1.9(1+i)^{10} = 1.9

becomes

1+i=1.91101+i = 1.9^\frac{1}{10}

✏️ Suppose you have a 10 year zero coupon bond with a face value of $1000 that is currently selling for $500. What is its YTM?

βœ” When he says that something is β€œselling for” $500, he means that that is its price, so PB = $500, F=$1000, T=10.

We just write down the bond pricing formula:

PB=F(1+i)TP_B = \frac{F}{(1+i)^T} $500=$1000(1+i)10\$500 = \frac{\$1000}{(1+i)^{10}}

The first step that we do is the β€œswitcheroo:”

(1+i)10=$1000$500(1+i)^{10} = \frac{\$1000}{\$500}

Recopying, and simplifying we get:

(1+i)10=2(1+i)^{10} = 2

Now we move the β€œ10th power” exponent over to the other side as a β€œone 10th power:”

1+i=21101+i = 2^{\frac{1}{10}}

You can enter this in to a spreadsheet or Google as 2^(1/10). Do it exactly like that. don’t omit parentheses. πŸ™‹β€β™€οΈβ€PEMDAS is your friend! πŸ˜„β€

1+i=1.071771+i = 1.07177 i=1.07177βˆ’1i = 1.07177-1

Therefore, i=7.177%.

To understand the switch

What you are really doing

Section titled β€œWhat you are really doing”

You can take the same exponent of both sides to get rid of an exponent. For example if one side of an equation is to the tenth power, you can take the 110\frac{1}{10}th power of both sides. This will give you what you want.

If you start with

(1+i)10=1.9(1+i)^{10} = 1.9

You can take the β€œone tenth” power of both sides:

(1+i)10110=1.9110){(1+i)^{10}}^{\frac{1}{10}} = 1.9^{\frac{1}{10}})

You get

i=5i=5

Plug and Chug

Section titled β€œPlug and Chug”

If you stuck, you can often solve the problem using β€œplug and chug.” With some practice, you’ll find it easy!

  1. Identify the number you need and numbers you know.
    • Need to know:
    • Already know: Do you know an equation that connects the number you need to the numbers you know?
  2. Plug πŸ”Œ the numbers you know into the equation.
  3. Algebraically solve for the needed quantity (β€œchug” πŸš‚). If you like, you can check βœ” your algebra by plugging your answers back into the original equation.
  4. Think over 🧠 your result. Does it make sense based on what you learned in class?

I will often summarize these four steps with the following template:

Plug and chug:
1. Equation β†’
2. Plug πŸ”Œ β†’
3. Solve πŸš‚ β†’
4. 🧠 β†’

Notes:

  • Math fact: If you know all of the numbers in a formula except one, you can generally use algebra to find the last number.
  • This method is just how you solved basic word problems in Algebra class. We realize that it may have been years or even decades since you took algebra, so we’re trying to make it as β€œstep by step” as possible!

✏️ Suppose that a given company has an ROA of 1.5% and Assets of $1 billion. What is the firm’s profit after taxes?

βœ” Solving this is easy. You just find an equation, plug numbers into it and do very basic algebra.

Plug and chug:
1. Equation β†’ We’ll use the definition of ROA, ’ ROA = ProfitAfterTaxes/Asset
2. Plug πŸ”Œ β†’
1.5% = ProfitAfterTaxes/$1B
3. Solve πŸš‚ β†’
1.5%Γ—$1B = ProfitAfterTaxes ProfitAfterTaxes = 1.5%Γ—$1B = $15M
4. 🧠 β†’ Let’s check our work by plugging back into the formula. ROA = ProfitAfterTaxes/Asset ROA = $15M/$1B = .015 = 1.5%

✏️ Suppose that total deposits in the economy increase by $5B, but the money supply only increases by $4.5B. What happened to Cash in the hands of the public? βœ” Plug and chug:
1. Equation β†’
Ξ”MS=Ξ”Deposits+Ξ”Cash Held by Public
2. Plug πŸ”Œ β†’
$4.5B=$5B+Ξ”Cash Held by Public
3. Solve πŸš‚ β†’
$4.5B-$5B=Ξ”Cash Held by Public
Ξ”Cash Held by Public = $4.5B-$5B = -$.5B
4. 🧠 β†’ You can plug the numbers back in to check.

✏️ Suppose that rf = 5%, the market risk premium, Erm - rf, is 6%, and for a given stock its expected return, Ers, is 9%. What is Ξ²? βœ” Plug and chug:
1. Equation β†’
E(rS)=rF+Ξ²[E(rM)-rF]
2. Plug πŸ”Œ β†’
9%=5%+Ξ²[6%]
3. Solve πŸš‚ β†’
9%-5%=Ξ²[6%]
4%=Ξ²[6%]
4%\div6%=Ξ²
Ξ² = 4/6 = .66
4. 🧠 β†’

Math worksheet

Section titled β€œMath worksheet”

Math worksheet