Annuities

AN annuity is a financial instrument that pays $$CF$$ dollars for $$T$$ years.

$! PV = \sum_{t = 1}^{T}\frac{CF}{(1 + r)^t} $!

Make sure you know when the first cash flow begins: It starts tomorrow $$t = 1$$, not today!

Also it pays for T years, meaning, income at the end will be $$CF/(1 + r)^T$$.

The above formula can also be expressed as:

$! \begin{aligned} PV &= \frac{CF}{r} \Big( 1 - \frac{1}{(1 + r)^T} \Big) \\ &= \frac{CF}{r} - \frac{1}{(1 + r)^T} \frac{CF}{r} \\ &= PV_0 - PV_T \end{aligned} $!

Where $$PV_0$$ is value of the perpetuity today, and $$PV_T$$ is value of perpetuity at time $$T$$, but evaluated at today.

Annuity Example: Mortgage Loan

A 30 year mortgage is an annuity with 360 monthly payments, starting one from today. therefore monthly rate on a mortgage is computed as the quoted rate divided by 12. $! r_{monthly} = 0.09/12 = 0.0075 \text{ per month} $!

Q1: To buy a house you need to take out a $ 1,200,000 fixed rate mortgage with 30 years to maturity, and a quoted interest rate of 9 %, What will be your monthly payment?

$! \$ 1.2 m = \frac{C}{1 + r} + \frac{C}{(1+r)^2} + \ldots + \frac{C}{(1 + r)^{360}} \\ where\\ r = 9\% / 12 = 0.75 \% $!

$! PV = \frac{C}{r} \Big( 1 - \frac{1}{(1 + r)^{360}} \Big) $!

That gives $$ C =\$ 9655 $$, for 360 months.

Principle and Interest Decompositions

Q2 : Of the first month’s payment, how much was payed as interest and how much was paid as principal?

The monthly interest rate is 0.75 $, giving the first month’s interest 0.0075 * $1,200,000 = $9,000.

Therefore we have $9000 as interest, and $655 as principal. $$\rarr$$ **The remaining balance becomes 1,200,000 - 655.

Level Coupon Bonds

recap: Bonds are financial assets that has Fixed income.

Level coupon bond is the bond that has fixed level (same amount payments through the interval).

Most common bond has $$x$$ % semi -annual level coupon bound.

Here the $$x$$ is not the interest rate. It is the percentage of the principal to be paid every interval.

example 1 : 8% semi - annual level coupon bond pays $40 every six months on $1000 in principal. At maturity, it pays $1040, as principal + 40.

example 2 : WHat are the payments to a 5% semi - annual level coupon bond, $100 million, due in 2.5 years?

it has to pay you 2.5% * principal every 6 months, therfore 2.5 mil in 6 month, for 2.5 years, + 100 mil This gives 2.5 * 2.5 + 100 mil.

Typically, coupon price is same as face value. (the last payment you get). Of course this wouldn’t be the case for zero coupon bond.

Q3: Is coupon rate of a bond equal to the interest rate? No:

Q4: What is the interest rate on an IBM 3.5 % semi - annual coupon bond?

we actually don’t know, as the value you receive is 3.5%/2 * FV, and the price today (PV) is unknown. This explains the Q3 as well.

Yield to Maturity (YTM)

Suppose that a one year risk free, zero coupon bond with a $1000 has an initial price of $966.18

$! \begin{aligned} P_{0,T} &= \frac{F}{(1 + r_{0,T})^T}\\ 966.18 &= \frac{1000}{1 + r_{0,1}} \\ r_{0,1} &= \frac{1000}{966.18} - 1 = 0.035 \end{aligned} $!

Yield to Maturity is the rate of return earned on a bond held to maturity. also called “promised yield”. Also called IRR (internal rate of return) Also called annualized rate or return.

Q5: An insurance company offers a retirement annuity that pays $100,000 per year for 15 years and sells for $806,070, What is the implied interest rate that this insurance company is offering you?

$! \begin{aligned} P_0 &= 806,070\\ PV &= \frac{100,000}{r} \Big(1 - \frac{1}{(1+r)^{15}}\Big)\\ r &= 9 \% \end{aligned} $!

If they were give with inflation compensator 3%, that is, that amount increases by 3% every year, the inflation rate would be:

$! \begin{aligned} P_0 &= 806,070\\ PV &= \frac{100,000}{r-3} \Big(1 - \frac{(1+3\%)^{15}}{(1+r)^{15}}\Big)\\ r &= 11.76 \% \end{aligned} $!

Q6: In 2016 google had P/E ratio of 27. If the cost of capital was 8%, what was the market’s expected growth rate?

$! \begin{aligned} P &= \frac{E}{r - g} \\ 27 &= \frac{1}{8 - g} \\ g &= 4.2\% \end{aligned} $!

Q7: If the dealer said that you can have 12 month car loans with 9%, and they asked you to pay them $10900/12 = $908.33 per month, would they be correct?

No! If you are paying cash earlier than 1 year (that is, every month), you are actually paying them more than you have to!

To be clear, if you were to pay “at the interest rate of 9% per year”, you would have to pay C such that:

$! \$ 10,000 = \frac{C}{1 + r_m} + \frac{C}{(1+r_m)^2} + \ldots + \frac{C}{(1+r_m)^{12}} $!

where $$r_m$$ would be 0.721%, as $$ (1 + r_m)^{12} = 1 + 9\% $$

But if you were to pay “at the interest rate of 9% per year”,

What is the actual interest rate on the car dealer’s proposed deal loan?

$! \$ 10,000 = \frac{908}{1 + r_m} + \frac{908}{(1+r_m)^2} + \ldots + \frac{908}{(1+r_m)^{12}} \\ \\ r = 17.5 \% $!