Keith Diaz
Time value of money
The dollar one year from today is worth less than the dollar today, because of
1. Risk: a dollar today is certain, whereas a dollar later may not materialize due to reduction
2. Real interest: interest before we consider inflation
3. Inflation: it is worth more today because inflation reduces buying power
Concept important to leases, mortgages, bonds, retirement contributions, stock valuation, etc.
Compounding interest means you calculate the interest not just on the dollar amount (base) but
also on any other interest accumulated
Single Amount – Future Value (FV)
 r = interest or discount rate
 PMT = amount of payment
 N = number of periods in years
Example: what will you have in 3 years if you deposit $100 into an account that earns 4% interest
compounded annually?
 r = 0.04
 PMT = $100
 N = 1
Example: how much money will you have in 5 years if you deposit $200 into an account that earns 3%
compounded annually?
Single Amount – Present Value (PV)
Example: how much do you have to deposit today to have $100 in a year (assume 4% interest
compounded annually?) What’s the PV of $100 to be received in 1 year (assume 4% discount 1rate)
= $96.15
Example: how much money do you have to deposit today to have $100 after 3 years (assuming 4%
interest compounded annually)?
Example: how much do you have to deposit today to have $3000 four years from now (assuming a 5%
discount rate)?
= $2468.11 Keith Diaz
Multiple Amounts – Future Value (FV)
Annuity: multiple payments of the same value, equally apart from each other in time
[ ] used when you deposit starting at the end of this year
[ ] used when you deposit starting today, time zero
Example: What will you have after 3 years if you deposit $100 each year for 3 years (beginning at the
end of this year) into an account that earns 4% interest compounded annually?
[ ]
= $312.16
Example: what will you have after 3 years if you deposit $100 each year for 3 years (starting today)
into an account today that earns 4% interest compounded annually?
[ ]
= $324.65
Example: how much must we put into an account each year earning 4% if we want to have $20 000 at
the end of 10 years?
[ ] [ ]
[ ]
If question doesn’t mention if the deposit is today, assume it is in the future and usordinaryannuity
Multiple Amounts – Present Value (PV)
[ ] use if starting at end of year
[ ] use if starting today
One less period if you use annuity due than if you use ordinary annuity, during which your money
accumulates interest before starting your withdrawals
Example: how big must your trust fund be today if you want to receive a payment of $500 each year
for the next 3 years? Assume an interest/discount rate of 4%.
1 1
PV ordinaryannuity00 3
0.04 0.04(10.04)
= $1387.55 Keith Diaz
Example: you borrowed $20 000 to fund your education. How big will you educational loan payment
be if you want to have the loan paid off in 4 years, you make the first payment at the end of this year,
and the discount rate is 3%?
[ ]
[ ]
PMT = 5380.54
Example: how big must your trust fund be today if you want to receive a payment of $500 each year
for the next 3 years starting today? Assume an interest/discount rate of 4%.
[ ]
= $1443.05
Combinations example
Example: how much would you pay for an investment that will give you $1000 after 4 years and a
payment of $50 a year as well? Assume 3% interest compounded annually.
[ ] +
= [ ]
= 1074.34
Perpetuity
Annuity that goes on forever (ex: dividend on a preferred share)
Example: what is the value of an investment in a 5% preferred with a par value of $12 if interest rates
are 3%?
Payment and compounding periods
Payment and interest periods are not always the same
Always multiply “n” (number of years) by number of payments per year
 n must always represent the number of payments you give/receive over the time of the
investment period
Adjust compounding rate to match payment frequency (r must match n) this is your new “r”
 If payment and interest periods are the same, but more than once per year (or single payment
but compounding more than once a year), divide r by the number of payments per year Keith Diaz
Example: what is the present value of 4 years of $50 payments received every 6 months and
compounded semiannually at 3%?
 Payment = $50

More
Less