Economics 434 Financial Markets Professor Burton University of Virginia Fall 2015 October 8, 13, 2015

Fixed Income Does not have “unlimited upside” (which equity has) Takes over in case of bankruptcy Risks: – Duration – Credit October 8, 13, 2015

Default Free Securities

What is a default free security? Honors its promises In the real world – Govt securities – Securities backed by government All others may, under certain circumstances, default

Why do we care? About Default Free Securities? Simplifies the analysis – One less thing to worry about – Does apply to government securities markets There is risk – Why? – Time!

What do we mean by Rate of interest? Yield? Rate of return? Etc.?

Imagine! You loaned me $ 100 At 10 percent interest What would this mean? What would happen?

Very likely, the following would occur: At the end of (one year), you would pay me $ return of principal $ 100 times (r) where r =.10 Plus $ interest at 10% rate $ total payoff after one year

Which gives a useful formula that works both ways Future value of $ 100 when r = 10% is (1 + r) times $100 Present value of $ 110 when r = 10% is $ 110 divided by (1 + r) Present Value = Dollar Amount (1 + r)

Suppose r = 10 % for all future times: then {$ 100 (1 + r)} times (1 + r) = Future value of $ 100} Two years from now

If r is always 10%, what is the value today of $ 100 two years from now? PV (1 + r) (1 + r) = $ 100 PV is the answer!!! PV = $ 100 (1 + r) 2

Generally, If r 1, r 2, r 3,……r n …. are the future one year rates Then Present Value of $ 100 received n periods from now is $ 100 (1 + r 1 ) (1 + r 2 ) (1 + r 3 ) ……(1 +r n )

Present Value (and “Future Value” Present value means “today’s value of a future stream of income” Future value means “the value on a specific future today of a specific amount of money today”

Present Value (money one year from today) Suppose the default free interest rate for one year is R Then M dollars received one year from today is currently worth: Present Value =

Present Value (money two years from today) Suppose the default free interest rate for two years is R per year Then M dollars received two years from today is currently worth: Present Value =

Present Value (money two years from today) Suppose the default free interest rate for three years is R per year Then M dollars received three years from today is currently worth: Present Value =

Present Value (money six months from today) Suppose the default free interest rate for one year is R per year Then M dollars received six months from today is currently worth: Present Value =

Present Value (one day from today – i.e., tomorrow..assume 365 days in a year) Suppose the default free interest rate for one year is R per year Then M dollars received tomorrow is currently worth: Present Value =

Critical assumption – that R is constant….not true, obviously R never changes and is the same for all periods of time This is, of course, not true The future R’s are not known exactly, though their markets already exist

We will simplify matters: R T (R with a subscript) will mean the one year default free rate starting at the beginning of year T from now) For example: – R 2 will mean the one year rate starting one year from today. – R 1 will mean the one year rate today. – The future value of $ 100 two years from today will be …… FV = $ 100(1+R 1 )(1+R 2 )

So, assume we know R 1, R 2, ….R N.. Then Future Value of $ 100 is: $ 100 times (1 + R 1 ) times (1 + R 2 ) times…..(1 + R N ) Therefore, Present Value of $ 100 three years from now is PV =

Useful Fact = 1 r

What does this formula tell us? $ 1 every year forever starting one year from now is worth: – $1 – Divided by r – If r is 5 %, then $ 20 – If r is 10% then $ 10 Excellent shortcut

Present Value is the most Crucial Concept in Finance Value of future stream of payments As valued today Emphasis on “discounting” future revenue streams Common practice to use higher rates to reflect higher uncertainty of receipt of future payments

October 8, 13, 2015