COMPOUNDING and DISCOUNTING over TIME
In financial markets, the rate of interest i charged on borrowed
money reflects the supply of loanable funds relative to demand.
i is the equilibrium rental cost of money.
i accounts for three factors:
• the estimated risk of default by the borrower d (reflected in
a consumer's credit score or a company's bond rating)
• inflationary expectations e (the lender wants to offset any
decline in the loaned money's purchasing power), and
• an underlying rate of time preference or "discount rate" r,
reflecting the fact that spending money today is more fun
than waiting a year to spend it.
i = d + e + r.
Market interest rate = risk + inflation + discount rate
We will analyze…
1. how markets incorporate discounting under zero risk,
2. how markets respond to non-zero risk.
(We will generally assume zero inflation.)
A $100 investment earning a 10% compounded return:
VALUE
$100.00
$110.00
$121.00
$133.10
$146.41
$161.05
$177.16
$194.87
$214.36
$235.79
$259.37
$285.31
$313.84
$345.23
$379.75
$417.72
$459.50
$505.45
$555.99
$611.59
$672.75
Compound Growth over Time
$800
$700
$600
$500
value
YEAR
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
$400
$300
$200
$100
$0
0
5
10
15
year
20
25
$100 → 100x1.10 → 100x1.102 → 100x1.103 → ….
Compounding any Present Value (PV), at growth rate r, the the
Future Value (FV) in any year t will be:
FV = PV(1+r)t
For example $1,000 PV compounded at r = 0.12 (12%) annually
over 9 years equals $1,000(1.12)9 = $2,773.08
Discounting is the reverse of compounding:
the present value of some future payment or benefit FV
depends on the time t you have to wait for it and your rate of
time preference r:
PV = FV/(1+r)t
The basic discounting equation PV = FV/(1+r)t has
four variables: PV, FV, r and t.
Given any three of these you can solve for the fourth.
For example:
• If you invest PV = $100 today at r = 0.06 for t = 15 years, you
will have…
FV = 100(1.06)15 = $239.66.
• If you won a $1 million lottery prize paid out in 20 annual
installments of FV = $50,000, the present value of this stream of
payments under discount rate r = 0.08 (8%) would be…
PV = $50,000/1.080 + $50,000/1.081 + $50,000/1.082 + ...
+ $50,000/1.0819
= $50,000.00 + $46,511.63 + $43,266.63 + ... + $12,653.46
= $547,953.91
• The time t required to double your money (FV/PV = 2) at
r = 8% interest:
FV/PV = 1.08t = 2
Taking the logs of both sides:
t log(1.08) = log(2)
t = log(2)/log(1.08) = 9.0 years.
“Rule of 72”
Interest rate x years to double your money ≈ 72
r
t
rxt
2% 35.0 70
3% 23.4 70
4% 17.7 71
5% 14.2 71
6% 11.9 71
7% 10.2 72
8%
9.0
72
9%
8.0
72
10%
7.3
73
11%
6.6
73
12%
6.1
73
• The implicit rate of return r on a classic car you bought for
PV = $10,000 and sold t = 9 years later for FV = $17,000 is where
FV/PV = (1+r)9 = 1.7
1 + r = 1.7(1/9)
r = 1.7(1/9) - 1
= 0.061 or 6.1%.
The present value of a perpetual annuity paying $X per year,
discounted at annual rate r, is:
$PV = $X + $X/(1+r) + $X/(1+r)2 + ... = $X/r
A mortgage leverages the growth of your home equity
purchase price
20% down
mortgage
rate
term (years)
monthly payment:
$200,000
$40,000
$160,000
4.00%
30
$763.86
Amortizing a mortgage (first 12 months)
Month Payment Loan Balance
0
$763.86 $160,000.00
1
$763.86 $159,769.47
2
$763.86 $159,538.17
3
$763.86 $159,306.10
4
$763.86 $159,073.25
5
$763.86 $158,839.63
6
$763.86 $158,605.24
7
$763.86 $158,370.05
8
$763.86 $158,134.09
9
$763.86 $157,897.34
10
$763.86 $157,659.80
11
$763.86 $157,421.47
12
$763.86 $157,182.34
Interest Paydown
$533.33 $230.53
$532.56 $231.30
$531.79 $232.07
$531.02 $232.84
$530.24 $233.62
$529.47 $234.40
$528.68 $235.18
$527.90 $235.96
$527.11 $236.75
$526.32 $237.54
$525.53 $238.33
$524.74 $239.13
$523.94 $239.92
Value
$200,000
$200,333
$200,667
$201,002
$201,337
$201,672
$202,008
$202,345
$202,682
$203,020
$203,358
$203,697
$204,037
Equity
$40,000.00
$40,563.86
$41,129.05
$41,695.57
$42,263.42
$42,832.60
$43,403.12
$43,974.98
$44,548.18
$45,122.74
$45,698.65
$46,275.91
$46,854.53
Your equity (value of house minus loan balance) increased 17.1%
After 10 years, your equity has grown from $40,000 to $118,050
Year Payment Loan Balance Interest Paydown
0
$9,253
$160,000 $6,400
$2,853
1
$9,253
$157,147 $6,286
$2,967
2
$9,253
$154,180 $6,167
$3,086
3
$9,253
$151,095 $6,044
$3,209
4
$9,253
$147,886 $5,915
$3,337
5
$9,253
$144,548 $5,782
$3,471
6
$9,253
$141,077 $5,643
$3,610
7
$9,253
$137,468 $5,499
$3,754
8
$9,253
$133,714 $5,349
$3,904
9
$9,253
$129,809 $5,192
$4,060
10
$9,253
$125,749 $5,030
$4,223
Value
$200,000
$204,000
$208,080
$212,242
$216,486
$220,816
$225,232
$229,737
$234,332
$239,019
$243,799
Equity
Equity Growth
$40,000
$46,853
17.1%
$53,900
15.0%
$61,147
13.4%
$68,601
12.2%
$76,268
11.2%
$84,155
10.3%
$92,270
9.6%
$100,618
9.0%
$109,209
8.5%
$118,050
8.1%
Consider a mining company with two turbidium mines, A and B.
The price of turbidium is rising 2% a year.
The marginal costs of mining turbidium from A and B are constant at
$33 and $38 respectively.
Marginal rent = price – marginal cost:
this is the company’s profit/unit.
In what year should the company extract its turbidium to maximize
its profits?
When the implicit rate of return falls to the company’s discount rate!
Benefit-cost analyses are sensitive to the choice of discount rate.
For example, suppose there are three options for using a vacant piece land
with the
a. The preserve option (recreation and wildlife habitat), yields an infinite
stream of small annual net benefits.
b. The landfill option yields a 4-year net benefit stream in years 2-5.
c. The condos option yields an immediate one-time net benefit (no
discounting required.
Net benefits from land use options
year
preserve landfill condos
0
$5
$0
$115
1
$5
$0
$0
2
$5
$20
$0
3
$5
$50
$0
4
$5
$50
$0
5
$5
$20
$0
6 to ∞
$5
$0
$0
Present values of alternative land use
options under various discount rates:
r
preserve landfill condos
3.0%
$166.67 $126.29 $115.00
3.5%
$142.86 $124.18 $115.00
4.0%
$125.00 $122.12 $115.00
4.5%
$111.11 $120.11 $115.00
5.0%
$100.00 $118.14 $115.00
5.5%
$90.91 $116.21 $115.00
6.0%
$83.33 $114.33 $115.00
6.5%
$76.92 $112.49 $115.00
7.0%
$71.43 $110.69 $115.00
MANAGING RISK
The conventional wisdom: "buy low and sell high," i.e.,
take a "long" position in some market and hope the price goes
up.
But the reverse strategy can also be profitable: sell high, then buy
it back low, taking a "short" position and hoping the price goes
down.
(How can you sell something you don't own? Just borrow it!)
Speculators actually help stabilize market prices:
They buy up market surpluses when prices are low,
and keep prices from falling lower.
They sell during market shortages when prices are high,
and keep prices from going higher.
Most investors in the stock market are bullish "longs" who buy
stocks, betting that their prices will rise.
If you buy 100 shares of Megabux Inc. (MBUX) at $50/share,
there is no limit on the upside: it could possibly go to
$115,000/share like Berkshire-Hathaway (Warren Buffet's
investment company) and your $5,000 initial investment would
now be worth $11.5 million.
And your downside risk is limited: the worst that can happen is
MBUX goes to zero and you lose your $5,000.
In contrast, a "short" is bearish on the stock, and bets that its price
will fall.
If you thought MBUX was going to tank, you could short 100 shares
at $50/share through your brokerage.
The broker borrows some other investor's shares, sells them for
you, and holds the proceeds for you.
The upside of this position is limited: the best outcome for you
would be if MBUX goes to zero, and the $5,000 from the short sale
is all yours; the unknown long whose shares you borrowed is out
$5,000.
But your downside risk is theoretically unlimited: if MBUX suddenly
soared to $80,000/share, you would be on the hook for the
additional $7,995,000 needed to buy back those shares!
(In practice, your broker would close out your position before your
losses exceeded the money in your brokerage margin account.)
Forward contracting:
Parties contract to exchange a specified quantity for a
specified price at a specified future date.
Requires direct negotiation
Non-performance risk
Hedging with Commodities Futures:
Example: a farmer sells corn futures at planting time, buys
them back when she sells her corn:
May (now)
September
gain/loss
cash
position:
plant at
$4.25/bu
sell at
$3.72/bu
-$0.53/bu
futures
position:
sell (short) at
$4.15/bu
buy back at
$3.60/bu
+$0.55/bu
net result:
$4.27/bu
In this example, the hedge strategy paid off handsomely; without
it, the farmer would have received $3.72/bu instead of $4.27.
On a farm with 500 acres of cropland yielding 160 bushels/acre,
or 80,000 bushels total, that's $44,000 of revenue achieved by
shorting 16 contracts.
Taking offsetting ‘long” and “short” positions in markets with
parallel price movements minimizes price risk.
On the other hand, suppose local corn prices rose to, say,
$4.75/bu between May and September, while the futures price
rises in parallel from $4.15/bu to $4.65/bu.
Closing out the short futures position costs her the extra revenue
she gained in the cash market, e.g.:
May (now)
September
gain/loss
cash
position:
plant at
$4.25/bu
sell at
$4.75/bu
+$0.50/bu
futures
position:
short at
$4.15/bu
buy back
$4.65/bu
-$0.50/bu
net result:
$4.25/bu
So the futures position effectively locks in her price either way.
She got $4.25, which is what she wanted at the outset, but she
may envy neighboring farmers who didn't hedge and got 50
cents more per bushel.
Instead of shorting corn futures, she could buy a put" option on
corn futures:
A futures contract is a "derivative" of the actual commodity market.
Commodity options are derivatives of the commodity futures market.
There are two types of options--puts and calls--and two parties in any
options market--the writer (seller) of the option, and the buyer of the
option.
The buyer of a "call" option acquires the right, but not the obligation,
to buy the underlying asset from the option writer (seller) at a specified
"strike price" on or before a specified expiration date.
The buyer of a "put" option acquires the right, but not the obligation,
to sell the underlying asset to the option writer (seller) at a specified
"strike price" on or before the specified expiration date of the option.
Our farmer could buy put options on September corn futures
(one option per futures contract) at a $4.20/bu strike price.
She would pay the writer of these options a "premium" to
compensate him for assuming her price risk.
Here's how the farmer's option strategy performs under the
price decrease scenario, where she exercises the option:
Price falls:
cash
position:
put
options:
net result:
May (now)
plant at
$4.25/bu
September
sell at
$3.72/bu
buy puts at $4.20 exercise puts:
strike price for
short futures at
$0.10/bu premium $4.20; close out
position at $3.60
gain/loss
-$0.53/bu
+$0.50/bu
(net of
premium)
$4.22/bu
And here's how the farmer's option strategy performs under a
price increase scenario, letting the option expire:
Price rises:
May (now)
September
gain/loss
cash
position:
plant at
$4.25/bu
sell at
$4.75/bu
+$0.50/bu
put
options:
buy puts at $4.20 let puts
strike price for
expire
$0.10/bu premium
net result:
-$0.10/bu
(the option
premium)
$4.65/bu!
Other futures & options markets
• Treasury bonds & notes (bets on interest rates)
• S&P 500, Dow, Nikkei indexes (bets on overall stock market
performance)
• Foreign currencies: Euro, yen, pound, … (bets on exchange
rates)
• Agricultural commodities: corn, soybeans, wheat, cattle, hogs, …
• Energy: crude oil, natural gas, heating oil, gasoline, ….
• Minerals: gold, silver, copper, platinum, …
ENERGY
Here’s the 3-month chart for December 2013 Light Sweet
Crude Oil (WTI) futures--1,000 barrels/contract, priced
$/barrel--as of Monday 11/4/2013
(1 barrel = 42 gallons)
Active CME crude oil (WTI) futures contracts
“out of the money”
“in the money”
CME call options on Dec13 light sweet crude (WTI) as of
11/4/2013
“in the money”
“out of the money”
CME put options on Dec13 light sweet crude (WTI) as of
11/4/2013
Reminder: “Reserves” are dynamic!
As resources get scarcer, their prices increase.
Demand-side effects:
• Conservation
• Improved efficiencies in resource use
• Substitution
Supply-side effects:
• Sub-economic reserves become economical
• New discoveries
• More efficient extraction technologies
• Development of substitute resources
Resource markets anticipate future prices.
Current Reserves ÷ Current Annual Consumption
does NOT predict “Years until we run out!”
Rising price signals scarcity, slows consumption, motivates
supply responses