Time value of money chapte

1. Time Value:
Compounding and Discounting
Dr. Julio Huato
SFC - jhuato.sfc@gmail.com
Fall 2012

2. Questions
• Why is time value important in macro and finance?
• What are the present value (PV) and future value (FV) of an
asset?
• What are the PV and FV of a simple asset (one-shot cash
flow in the future)?
• What is the PV of a perpetuity?
• What are the PV and FV of an ordinary annuity and of a
mixed-stream asset?
1

3. • How much do we need to deposit periodically in order to
accumulate a fixed sum of money in the future?
• What are the different types of bonds or loans and how do we
determine their value?

4. Time value of money
Wealth generates benefits and entail costs over their lifetime. Net
benefits = total benefits - total costs. In a market economy,
benefits and costs can be measured as cash flows.
USD 1 on 10-5-2012 = USD 1 on 10-5-2022
Time line diagrams.
Financial tables/calculators/spreadsheets.
2

5. One-shot cash flows
Let V0 = $100 today (i.e. at t = 0).
Principal: Amount on which interest is paid. Compound interest:
Interest earned on previous interest that has increased the previous
principal.
Say the banks pay an annual interest rate i = 5%, which is then
added to the principal.
3

6. FV and compounding
Vt = V0 (1 + i)t
If V0 = $100, t = 10, and i = 5% = 0.05, then
V1 0 = $100 × (1 + 0.05)10 = $162.89
(1 + i) is called the gross interest rate. (1 + i)t is called the future
value interest factor. What’s the meaning of the FVIF in plain
terms?
In general, compound growth of any variable means that the value
of the variable increases each period by the factor (1 + g). When
money is invested at the compound interest rate i, the growth rate
is i.
4

7. FV and compounding
Vt = V0 (1 + i)t
Note that the greater V0 , the greater i, and the greater t, then Vt
will also be greater. And vice versa.
5

8. PV and discounting
V0 = Vt (1 + i)−t
If V10 = $162.89 and i = 5% = 0.05, then
V0 = $162.89 (1 + 0.05)−10 = $100.
Here, the interest rate i is called the discount rate and (1 + i)−t is
called the discount factor. What’s the meaning of the DF in plain
terms?
Note that the greater Vt , the smaller i, and the smaller t, the
greater will V0 be.
6

9. Finding i
Suppose a company needs to borrow. It issues bonds for $129
each promising to pay its holder $1,000 at the end of 25 years.
No coupons. One single payment at the end of 25 years. What
(fixed) annual interest rate is the bond paying?
We know the PV ($129), the FV ($1,000). We don’t know the
interest rate i. Let’s figure it out:
V0 = Vt (1 + i)−t
Vt
(1 + i)t =
V0
Vt 1/t
1+i=
V0
Vt 1/t
i= −1
V0
$1, 000 1/25
i= −1
$129
7

10. Frequent compounding
So far, we have implicitly assumed that the interest rate com-
pounds annually (once a year). What if it compounds more often?
Suppose you have $100 and earn an annual interest rate of 6%
to be compounded monthly. Each month the bank pays you 1/12
of the annual rate, 0.06/12 = 0.005 or a half percent (50bp).∗
Since the interest is compounded monthly, your $100 earn [1 +
(i/12)]12 − 1 = (1.005)12 − 1 = 1.0617 − 1 = 0.617. Your true
annual interest rate is not 6% but 6.17%!
We need to modify the FV formula when dealing with more fre-
quent compounding. Let m be the number of periodic payments
per year (e.g. 2 semi-annually, 4 quarterly, 12 monthly,etc.).
Then:
i mt
Vt = V0 1 +
m
∗ 1%=0.1=100bp.
8

11. WARNING!
Do NOT ever compare, add up, or subtract cash flows
that occur at different times without previously dis-
counting them (or compounding them) to a common
date. They are apples and oranges!
9

12. FV and PV of multiple cash flows
If an asset generates multiple cash flows, how do we find its FV
(or PV)? One way is to use computational brute force, i.e. we
calculate the FV (or PV) of each cash flow with the formulas we
know and then add up all those FVs (or PVs).
Example: You save $300 each year for 3 years to buy a computer
starting today (3 deposits of $300 each). You earn 2% annually
on your savings balance. How much money will you have in 3
years? (Answer: You will have $993.04. Show how that result
was obtained by applying the formulas learned so far.)
We better find a way to simplify the computation of asset that
generate multiple cash flows.
10

13. PV of a consol or perpetuity
An asset that promises to pay a fixed annual payment forever
starting at the end of the present year (or beginning of the second).
The principal is not repaid, you just receive these annual payments
forever. Say the annual payment is C. What is the annual interest
rate on this perpetuity? Assume the market decides that the right
C
value of this security is P V P0 .∗ Then obviously: i = P V P0 .
Therefore, P V P0 = C/P V P0 . Usually, we know i of assets of
similar risk and C. We can then find the PV of the perpetuity as:
P V P0 = C/i. This is the PV of the perpetuity at zero or, to be
more precise, P V P0 .
∗ How does the market do it?
11

14. PV of a perpetuity
What is its PV at t some year in the future (P V Pt )? This perpe-
tuity would begin generating a cash flow C starting in year t + 1.
The value of the perpetuity at n will be the same as today’s!
P V Pt = P M T .
i
Note that P V Pt is a FV (PV at t), not really a PV! An actual
present value is P V P0 . So, to find the PV of that perpetuity
that generates an eternal cash flow beginning in year t + 1, we
need to discount P V Pt to the present to find its P V P0 . That is,
P V Pt
P V P0 = (1+i)t = C (1 + i)−t .
i
See how this lego toy works?
12

15. PV of a consol or perpetuity
Example: Find the P V P at the beginning of 2012 of a consol
that yields $100 annually starting at the end of 2012. The mar-
ket interest rate of equally safe investments is 5%. (Show that
P V P0 = $2, 000.
Example: Find the P V P at the beginning of 2012 of a perpetuity
that yields $100 annually starting at the end of 2016. The mar-
ket interest rate of equally safe investments is 5%. (Show that
P V P4 = $1, 645.40.)
13

16. Annuities (FV)
What will be the balance of a savings account in 10 years if one
deposits $1,000 at the end of each year and i is 5%? What will
the balance be if the deposits are made at the beginning of each
year?
These are annuities. The payments are the same each year. The
former (payments at the end of each year) is known as an ordinary
annuity. The latter (payments at the beginning of each year) is
known as an annuity due.
An annuity is just a finite number of periodic constant flows. Ap-
plying the FV formula we know, we construct a formula for the
FV of annuities.
14

17. FV of an ordinary annuity
Let C be the annual deposit (or payment), i the interest rate, and
t the term of the annuity. The FVA will be the sum of the FV’s
of each annual deposit (or payment).
F V At = C(1 + i)t−1 + C(1 + i)t−2 +
. . . + C(1 + i)2 + C(1 + i) + C (1)
t−1
F V At = C (1 + i)n (2)
n=0
t−1
The interest factor is F V IF Ai,t = t=0 (1 + i)n .
15

18. PV of an ordinary annuity
But there’s an easier way: The PV of an ordinary annuity can be
viewed as the PV of a perpetuity that begins to produce cash flows
at the end of year 1 minus the PV of a perpetuity that produces
its first cash flow at the end of year (t + 1). Hence:
C
P V At = 1 − (1 + i)−t
i
16

19. Miscellaneous
How do we figure out the FV and PV of mixed-stream assets
(different cash flows each period)? Mix and match the previous
formulas as may be required. If no fancy formula can be applied,
use the basic formulas for FV and PV of single cash flows. That
always works!
Compounding and discounting in continuous time: F V = P V ein
and P V = F V e−in
17

20. And?
The principles learned about FV and PV, compounding and dis-
counting, applies to all types of bonds. Not only to regular loans
or bonds with a fixed interest rate (fixed return), but to variable-
return securities as well (e.g. stocks, variable interest-rate credit
instruments, etc.)
With variable-return securities, the return is uncertain. But even
the most certain (‘fixed’) return is in fact uncertain. The future
is essentially unknown and no human institution is eternal.
When we compare FV’s and PV’s of different investments, we can
only make a meaningful comparison when the degree of risk in-
volved in the investments we compare are is similar. If the degrees
of risk are different, then we are comparing apples and oranges.
We need to find an analytical way to translate risk into return.
Then we will be able to convert uncertain cash flows into ‘certain’
cash flows (i.e. adjusted for risk) and thus use what we’ve learned
to compare these different instruments on an apples-to-apples ba-
sis.
18

21. What did we learn?
• Why is time value a big deal in finance?
• What’s the PV and FV of a security?
• What’s the PV and FV of a simple bond (one-shot cash flow
in the future)?
• What’s the PV of a perpetuity?
• What’s the PV and FV of an annuity (ordinary- and -due) and
those of a mixed-stream asset?
19

22. • What’s frequent compounding and how does that affect asset
valuation?
• How much do we need to deposit periodically in order to
accumulate a fixed sum of money in the future?
• How are loans amortized – i.e., how do we determine equal
periodic payments to repay a loan principal plus a stipulated
interest?
• How are interest or growth rates found? How do we find the
number of periods it takes for an initial deposit to grow to a
certain future amount, given the interest rate?