Finance 1: Background 101

Evaluating Cash Flows How would you value the promise of $1000 to be paid in future? -from a friend? -from a bank? -from the US treasury? We need to evaluate based on two criteria: 1. The risk that you will not be paid 2. The time of the payment

Discounting Cash Flows In order to find the value of cash flows we need to: 1. Find the cash flows and when they will occur (may be unknown, and then you need to use expected cash flows and probability) 2. Use appropriate discount rate (higher if higher risk and higher expected inflation) -- a general rule is to find the interest rate paid on an investment of comparable duration and risk in the market. 3. Discount each cash flow to the present value of the cash flow.

Example The $1000 will be paid in exactly one year by the US government. 1. Let’s say the one year rate on government bonds is 4%. So if we invest $100 today (PV), then the value in one year (FV) is FV=($100)(1+.04)=PV(1+r) So PV=FV/(1+r) --- 1/(1+r) is called the discount factor and r is the discount rate 2. We find the PV of the future cash flows: PV = $1000/1.04 = $ So we would expect to pay $ for a 1 year gov’t bond that paid $1000 in one year (with no coupon).

Bonds Bonds are a means by which an entity can borrow money on the market. The government can issue bonds, corporations can issue bonds, etc. Face value: the amount that you will receive when the bond reaches maturity (when bond is due to be paid off). Coupon rate: most bonds pay a “coupon” every 6 months that is a percentage of the face value of the bond. Zero coupon bond: Only pays a lump sum at the end of the term.

Bond pricing Bonds are sold to the market with a set face value (usually $1000) and a set coupon. The market determines the price based the risk of default, inflation rate, and duration. Together, these will determine a discount rate on the bond. Look at See if you can find the implied discount rate of 30 yr treas. (find each cash flow and discount to PV) Yield to maturity or internal rate of return (r):

Finding YTM or IRR On calculator. On excel: Yield(settlement,maturity,rate,price,redemption,frequency) sett- date of sale (serial) mat - date of maturity (serial) rate - coupon rate redemption - % of face returned at maturity frequency - # pmts per year Or IRR(values)

Term structure of rates The YTM calculation implies that each cash flow has the same discount rate. This is not exactly true. Payments at later dates may have a higher discount rate since the further you go into the future, the less certain you can be of no default, and thus the higher the risk. (See bloomberg-yield curve)

Net Present Value Net present value is like the premium that you earn on capital. NPV = PV of all cash flows (including the investment) Ex. Invest $350,000 today to make $400,000 in 1 year at a 7% discount rate. Find NPV. NPV = / = $23,832 NPV rule: Invest to maximize NPV of the investment Rate of return rule: invest up to point where marginal rate of return = rate of return of equivalent investment in capital market.

Example You have an opportunity to buy a plot of land for $100,000 and build over the next three years and then sell the building in three years. T=0t=1t=2t= Set up the NPV calculation (discuss constant vs. changing r)

Perpetuity A set amount of money will be invested and then paid back in constant fixed amounts for perpetuity. Ex. Want to endow a position at a university to pay $ per year forever,5% rate:

Growing Perpetuity What if you want the salary to keep pace with inflation - 3% growth each year?

Annuity An annuity pays a fixed sum each year for a specified number of years. (ex. Home mortgage) One way to handle annuity calculations: Annuity (t year) = perpetuity (starting now) - perp (starting in yr t+1) Rhs (w/o C) is called annuity factor

Another way to see it

Example What is the present value of a mortgage if there are 20 years remaining with an 8% rate and monthly payments of $1000?

Example This annuity formula is also useful for making calculations of the value of a series of level annual (or monthly, etc) investments. Ex. $5000 per year for 40 years at 8%. We can find the present value of the payments making the annuity calculation, and then find the future value by applying 40 years of interest. (Try it) Could also do as a geometric series.

Bonds revisited We now have another way of looking at bond valuation: PV(bond) = PV(coupon payments)+PV(final amount) = (coupon x annuity factor) + (fin)(discount factor) Find PV of 30 year bond with $1000 face with 6.5% coupon given a 5% discount rate

Accounting for inflation Let’s say we are looking forward to retirement and we foresee an annual income of $75000 in 40 years. Problem: $75000 in 40 years is not likely to buy what it does today because of inflation. We therefore need to discount to today’s dollars: We should also consider discounting our interest rates to account for inflation. If we invest $100 today for 1 year at 5% interest, if there is 3% inflation, how much will we have in real dollars in 1 year?

Stock Pricing We can look at the price of a stock today as: