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FINANCIAL ENGINEERING CLUB
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AGENDA
 Interest rates and returns
 Bonds
 Bond risk
 Other fixed income instruments
INTEREST RATES
INTEREST RATES
 Compensation to the owner of an asset (generally cash) for loss of the asset’s use
 Ex) You deposit $1000 into your savings account. They pay you a small interest rate since
they may use your deposit for various reasons and you cannot readily use it without a
withdrawal.
For
𝑃
𝑟
𝑁
principal amount invested/borrowed
rate (per arbitrary period)
number of periods invested/borrowed
 Simple interest: 𝑃 𝑥 𝑟 𝑥 𝑁
 Ex) $100 invested now at 6% per month for 10 months: interest earned = $100 x .06 x
10 = $60.
INTEREST RATES
 Compound Interest: Interest earned each period is reinvested at the same rate




Compound interest earned = 𝑃 𝑥 [(1 + 𝑟)𝑁 −1]
Ex) $100 invested now at 6% per month , compounded monthly for 10 months:
Amount earned = $100 x .106^ 10 = $179.08
Interest earned = $179.08 – 100 = 79.08
 Interest at a rate r per period, compounded N1 times per period, but invested over
N2 compounding periods earns the interest:
 𝑃 𝑥 [(1 +
𝑟 𝑁
) 2 −1]
𝑁1
 When 𝑟 is per year this is known as the effective annual interest rate
INTEREST RATES
 Ex) What is the value of $100, invested at a rate of 5% annually for two years, compounded
monthly?
 100 x (1+.05/12)^24 = 110.4941
 Continuously compounded interest rate: lim 1 +
𝑁→∞
𝑟 𝑁
𝑁
= 𝑒𝑟
 What is the value of $100, invested at a rate of 5% annually for two years, continuously compounded?
 100 x (𝑒 .05 )2 = 100 x 𝑒 .1 = 110.5171
MORE SCENARIOS
$1000
$1000
$1000
$1000
$1000
$1000
$1000
 Ex) Suppose you invest $1000 at 5% per year today and in every subsequent year
until 2020 (7 investments). What is your investment worth in 2030?
????
7 investments of $1000.
• First one for 16 years
• Second for 15 years
• Etc
February 2030
February 2020
February 2019
February 2018
February 2017
February 2016
February 2015
February 2014
⋯
$1000 𝑥 ((1.05)16 + (1.05)15 + …
+ (1.05)10 ) = 1000 𝑥 13.26247
= $𝟏𝟑, 𝟐𝟔𝟐. 𝟒𝟕
MORE SCENARIOS
$1000
$1000
$1000
$1000
$1000
$1000
$1000
 Alternative approach—consider the value of the investment in 2020, once all
investments have been pooled, then accrue interest from 2020 to 2030.
????
Investment in February 2020
• This is known as an annuity
• Valueannuity = P x
February 2030
February 2020
February 2019
February 2018
February 2017
February 2016
February 2015
February 2014
⋯
• $1000 x
(1.05)7 −1
.05
(1+𝑟)𝑁 −1
𝑟
= 8142.01
Invest this for 10 years:
8142.01*(1.05)^10 = $13262.36
TIME VALUE OF MONEY
 Up until now, we have been considering how much money is worth in the future,
after being invested at different rates
 One can always invest their free cash at some interest rate
 Opportunity cost: the cost of a choice; the amount of economic value forgone by doing
A instead of B
 There is an opportunity cost to holding cash—it can always be invested. Ideally it would
earn interest in the future and be worth more.
 Money today is worth more in the future
 It can be invested
 Money in the future is worth less today
 You can invest current cash to grow into a future sum
TIME VALUE OF MONEY
 Future value: the value of an asset at a specific date in the future
 Effectively what we have been calculating
FV = Present Value * (1 + 𝑟)𝑡
 Present value: the value today of an asset in the future, if it exists
 The reverse of what we have been calculating
PV =
FV
(1+𝑟)𝑡
This method of reducing a future
cash flow to its value today is known
as discounting it back to today
PRESENT VALUE
 Ex) How much money would you need to invest at 5% per year to earn $1000 in 3
years?
 Alternatively, how much is $1000 in 3 years worth now at a rate of 5%?
FV
1000
PV = (1+𝑟)𝑡 = (1.05)3 = $863.84
 Ex) At what rate would you need to invest $100 annually to earn $250 in three
years?
3 250
250
250
100 = (1+𝑟)3 → (1 + 𝑟)3 = 100 → r = 100 − 1 = .357
WHICH INTEREST RATE?
 There are many different types of interest rates—which one do I use in my calculations?
BTMM <GO> In Bloomberg
TERM STRUCTURE OF INTEREST RATES
 Interest rates do not remain constant over time and over borrowing tenures
 Default risk—the chance that the borrower may default, or be unable to pay interest to
the lender. A longer borrowing time increases the default risk.
 Opportunity costs
 Interest rates reflect economic conditions
 The shape of an interest rate curve over time is known as the yield curve
TERM STRUCTURE OF INTEREST RATES
 The quantitative values of the interest rate at different term lengths is known as
the term-structure of interest rates
 This refers more to bond yields than regular interest rates
 More on this in Bonds
 Who determines how interest rates change?
 The Federal Reserve—the central banking system of the united states—uses monetary
policy to force the Fed Funds rate to a target set by them (the fed funds target rate)
 This helps determine several other rates
 For interbank lending, rates are based on LIBOR (London Interbank Offer rate), which is
determined by the British Banker’s Association
MORE ON INTEREST RATES
 Interest rates are pivotal in valuing future cash flows and therefore the majority of
financial products.
 Here we have calculated the present and future value of streams of cash flows
with certain interest rate environments
 There are sophisticated models for interest rates which are used heavily on
interest rate/fixed income derivatives such as floors, caps, floorlets, caplets, swaps,
swaptions, etc
 LIBOR Market Model (LMM):
 Hull-White
BONDS
BOND BASICS
 A debt instrument in which an investor loans money to the issuer (by buying the
bond) and the issuer agrees to repay the principal with interest over the life of the
bond until it matures.
 A bond has several key features
 PAR value (also known as face value) is the notional amount that is borrowed by the
issuer and hence the amount on which is paid interest
 You may by the bond for cheaper /more than (discount/premium to) PAR
 Maturity is the date at which the issuer has agreed to repay the principal
 Coupon the interest rate specifying the regular interest payments
 Usually a fixed amount at regular intervals over the life of a bond (like an annuity)
 In this regard bonds are referred to as fixed-income instruments
 Market Value—if the bond is traded in a secondary market, you may buy it after it is
issued at this price
BOND BASICS
 Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000
with a semiannual coupon of 5%. It matures in January 2019.
-$10,000
$500
$500
$500
$500
$500
$500
$500
$500
$500
$10,500
Jan 2014
July 2014
Jan 2015
July 2015
Jan 2016
July 2016
Jan 2017
July 2017
Jan 2018
July 2018
Jan 2019
 PAR = $10,000
 Coupon = 5%, semiannually
 Maturity is January 2019
WHERE DO BONDS COME FROM?
 Bonds are a major way institutions finance their operations
 Interest payments on bonds are tax-deductible, making them cheap financing option
 However, they are risky—too many short term obligations to creditors can cause default
 Governments also issue bonds
 (US) Treasury bills are short term (< 2 years until maturity) bonds that are generally zero-coupon
 (US) Treasury bonds are longer term instruments
 Bonds issued by governments are generally referred to as sovereign debt
WHERE DO BONDS COME FROM?
ZERO-COUPON BONDS
 A zero-coupon bond is a bond with no coupon.
 However it is bought at a steep discount to PAR
-$8,500
$10,000
 Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for
$8,500 that matures in 5 years.
What is the effective interest rate (yield) to the
borrower?
t=5
t=4
t=3
t=2
t=1
t=0
PV =
FV
(1+𝑟)𝑡
 8,500 =
10,000
(1+𝑟)5
 r = .033038
ZERO-COUPON BONDS
 Note that buying a zero-coupon bond is equivalent to lending money
 You lend the value at which you buy it to the issuer and you earn the yield
 Conversely, short-selling a zero-coupon bond is equivalent to borrowing money
 This concept is very important for future financial engineering applications
VALUING A BOND
 For valuation purposes a bond is simply a stream of future cash flows
 Must know your discounting rate r—the rate at which you can borrow cash
$10,500
$500
$500
$500
$500
$500
$500
$500
$500
$500
-$10,000
 Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a
semiannual coupon of 5%. It matures in January 2019. What is this bond’s value?
Suppose we can borrow cash at
4% annually
Jan 2019
July 2018
Jan 2018
July 2017
Jan 2017
July 2016
Jan 2016
July 2015
Jan 2015
July 2014
Jan 2014
Price = PV(all the cashflows)
= 𝑃𝑉(𝐶𝐹1) + 𝑃𝑉(𝐶𝐹2) + … + 𝑃𝑉(𝐶𝐹10)
500
500
10,500
= (1.04).5 + (1.04)1 + … + (1.04)5 = 12,715.18
VALUING A BOND
 Note that the value of a bond is given by
PV(Bond) =
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖
𝑖
ti is the time at which cash flow i is realized
CFi is the ith cash flow at time ti , which may not be equal for all i
ri is the interest rate at time ti
YIELD TO MATURITY
 Note that the market price of the bond may not be equal to the present value of
the bond?
 What discount factor will equate the present value of the bond to the market value?
Market Value = PV(Bond) =
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖
𝑖
 This is known as the yield to maturity
 If you buy the bond and hold it, what is your equivalent yield or interest rate on the bond
YIELD TO MATURITY
 Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000
with a semiannual coupon of 5%. It matures in January 2019. You can buy this
bond in the secondary market for $11,000. What is its yield to maturity?
Recall that
Price = PV(all the cashflows) = PV(CF1) + PV(CF2) + … + PV(CF10)
500
500
10,500
= (1+𝑦).5 + (1+𝑦)1 + … + (1+𝑦)5 = 11,000
 y = .0774
How do I arrive at this?
Easy way: using
Excel’s solver
CALCULATING YIELD TO MATURITY
 More rigorous ways: Root-finding methods like Newton-Raphson, bisection
method
 Newton-Raphson: Given a function f(x) and an initial point 𝑥0 iterate via
𝑥𝑛+1 = 𝑥𝑛 −
𝑓(𝑥𝑛 )
𝑓′ (𝑥𝑛 )
CALCULATING YIELD TO MATURITY
 Bisection Method—Guaranteed to converge on an interval [𝑎, 𝑏] where 𝑓(𝑎) and
𝑓(𝑏) have opposite signs
 Binary search on the interval [𝑎, 𝑏] , evaluate 𝑓 at midpoint and update interval appropriately
to keep the signs of 𝑓(𝑎) and 𝑓(𝑏) opposite
 Terminate when within a reasonable range of 0 or a and b are very close
 Here f x =
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑥)𝑡𝑖
 𝑝𝑟𝑖𝑐𝑒 is the market price
− 𝑝𝑟𝑖𝑐𝑒
YIELD TO MATURITY
YIELD TO MATURITY
 Fundamental property of bond prices: they are inversely related to interest rates
BOND RISK
BOND RISK
 How can we measure the risk of the price of a bond?
 If you need to sell your bond today, you may have lost money
 Some risks of bonds (Qualitative)
 Default risk—probability that issuer will be unable to repay (default) principal and
interest rates
 Interest rate risk—implicit assumption of bond pricing/discounting that we will be able
to reinvest at the rate we discount at. This may not be true.
 What is the primary factor that directly affects a bonds price:
 Cash flows—these do not change after bond is issued
 Interest rates—subject to change
DURATION
 Formal definition: The average time until maturity of a bond, weighted by cash flow
 This version of duration is known as Macaulay Duration, named after Frederick Macaulay
-$8,500
$10,000
 Ex) Goldman Sachs sells a zero-coupon bond with a PAR-value of $10,000 for $8,500
that matures in 5 years.
Time
5
Cashflow
10,000
10,000
t=5
t=4
t=3
t=2
t=1
t=0
Duration = 10,000 ∗ 5 = 5
For a zero-coupon bond, the Macaulay duration is
equal to its maturity
DURATION
 In general Macaulay Duration =
1
𝑃𝑟𝑖𝑐𝑒
∗
𝑀 𝑡𝑖 ∗𝐶𝐹𝑖
𝑖=1 1+𝑟 𝑡𝑖
+
𝑀∗𝑡𝑀
(1+𝑟)𝑡𝑖
$10,500
Jan 2019
$500
July 2018
$500
Jan 2018
$500
July 2017
$500
Jan 2017
$500
July 2016
$500
Jan 2016
$500
July 2015
$500
Jan 2015
$500
July 2014
Jan 2014
-$10,000
 Ex) In January 2014, Goldman Sachs issues a bond with a PAR value of $10,000 with a
semiannual coupon of 5%. It matures in January 2019. What is this bond’s Macaulay Duration?
Time Cashflow PV(time*CF)
0.5
500 245.1451689
1
500 480.7692308
1.5
500 707.1495257
2
500
924.556213
2.5
500 1133.252445
3
500 1333.494538
3.5
500 1525.532138
4
500 1709.608382
4.5
500 1885.96006
5
10500
43151.1731
Sum = 53,096.64081
r = .04
Price = 12,715.18
Macaulay Duration =
53,096.64081
= 4.175847
12,715.18
DURATION
 Why would average time until maturity be related to how sensitive a bond’s price
is?
 Re-examine the pricing formula: PV(Bond) =
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖
𝑖
 Duration is the sensitivity of a bond’s price to interest rates.
 A more useful metric—Modified duration:
−𝟏 𝝏𝑷
Modified Duration(r0) =
| , where P is the present value of the bond, r is the
𝑷 𝝏𝒓 𝒓𝟎
variable interest rate, and r0 is a numerical rate
DURATION
 Note that
−𝟏 𝝏𝑷
|
𝑷 𝝏𝒓 𝒓𝟎
=
−𝟏 𝝏𝑷
𝑷 𝝏𝒓
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖
𝑖
+
𝑀∗𝑡𝑀
(1+𝑟𝑖 )𝑡𝑖
Macaulay Duration =
=
−𝟏
𝑷𝒓𝒊𝒄𝒆
1
𝑃𝑟𝑖𝑐𝑒
∗
𝑡𝑖 ∗𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖 −1
𝑖
𝑀 𝑡𝑖 ∗𝐶𝐹𝑖
𝑖=1 1+𝑟 𝑡𝑖
+
+
𝑀∗𝑡𝑀
(1+𝑟𝑖 )𝑡𝑖
𝑀∗𝑡𝑀
1+𝑟 𝑡𝑖
 However, if rates are continuously compounded
−𝟏 𝝏𝑷
|
𝑷 𝝏𝒓 𝒓𝟎
=
−𝟏 𝝏𝑷
𝑷 𝝏𝒓
𝑀 𝐶𝐹𝑖
𝑖=0 𝑒 𝑟𝑖 𝑡𝑖
+
𝑀∗𝑡𝑀
𝑒 𝑟𝑖 𝑡𝑖
=
−𝟏
𝑷𝒓𝒊𝒄𝒆
𝑀 −𝑡𝑖 ∗𝐶𝐹𝑖
𝑖=0 𝑒 𝑟𝑖 𝑡𝑖
+
𝑀∗𝑡𝑀
𝑒 𝑟𝑖 𝑡𝑖
1
Macaulay Duration =
𝑃𝑟𝑖𝑐𝑒
∗
=
𝟏
𝑷𝒓𝒊𝒄𝒆
𝑀 𝑡𝑖 ∗𝐶𝐹𝑖
𝑖=1 𝑒 𝑟𝑖 𝑡𝑖
𝑀 𝑡𝑖 ∗𝐶𝐹𝑖
𝑖=0 𝑒 𝑟𝑖 𝑡𝑖
+
𝑀∗𝑡𝑀
𝑒 𝑟𝑖 𝑡𝑖
+
𝑀∗𝑡𝑀
𝑒 𝑟𝑖 𝑡𝑖
DURATION
 Example) Suppose your bond has a (modified) duration of 5. If the interest rate
rises by 1%, how does the price of your bond change?




Increases by 1%
Decreases by 1%
Increases by 5%
Decreases by 5%
DURATION
 Example) Suppose your bond has a (modified) duration of 5. If the interest rate
rises by 1%, how does the price of your bond change?




Increases by 1%
Decreases by 1%
Increases by 5%
Decreases by 5%
CONVEXITY
 Note that changes in bonds’ prices with respect to interest rates are not linear
 How many continuous derivatives would you say PV(Bond) =
𝐶𝐹𝑖
𝑀
𝑖=0 (1+𝑟 )𝑡𝑖
has?
𝑖
 Convexity is the second derivative of a bonds price wrt interest rates, normalized
by price:
−𝟏 𝝏𝟐 𝑷
Convexity(r0) =
|
𝑷(𝒓𝟎 ) 𝝏𝒓𝟐 𝒓𝟎
CONVEXITY
 Example) Suppose your bond has a (modified) duration of 5 and a convexity of 15.
It is valued at $100. If the interest rate (currently at 5%) rises by 1%,
How much does duration change?
by 1*15 = 15%
how does the price of your bond change?
DURATION AND CONVEXITY
Calculate how bond prices change given changes in yield:
P2 =P1 + -duration * (Δr) + .5*convexity* (Δr)2
(Taylor’s expansion)
P1
In the previous example:
• P1 = 100
• Δr = .01*5 = .05
• duration = 15
• convexity = 5
P
If interest rates change by Δr =.01, the bonds price changes to P2 =
ΔP
2
P2-duration * (Δr)
Convexity correction
r2
r1
Δr
100 + 5*(.05) + .5*15*(.05)2
= 100.0313
IMMUNIZATION
 Suppose you, as a borrower, had several (floating-rate) interest expenses.
 Since interest rates may change, you wish to hedge your risk here.
 What is your risk?
 Hedging—Buying and selling of assets so as to use some of their features to
‘cancel’ out risks of another.
 If I can match the duration of my liability (the owed interest rate expense) with
that of an asset (specifically a bond), I can buy the bond and have a net duration of
zero.
IMMUNIZATION
 Ex) You owe $1000 in 2 years. You would like to invest in bonds now to meet that obligation in the
future. Your borrowing rate is 9%/year and you can invest in the following two bonds:
Coupon Maturity Yield Price Duration
Bond 1
0
1 year
0.09 91.74
Bond 2
0
5 years
0.09 64.99 5 years
First note that PV(1000) =
1000
1.092
1 year
How much do you invest in each?
= $841.68
You want to invest 𝑥 in Bond 1 and 𝑦 in Bond 2 to meet this obligation: 91.74𝑥 + 64.99𝑦 = 841.68
Want to match the duration of your obligation (2 years): 91.74𝑥 ∗ 1 + 64.99𝑦 ∗ 5 = 2 ∗ 841.68
Thus x = 6.88 ‘units’ of bond 1 and y = 3.24 ‘units’ of bond 2
IMMUNIZATION
 What was the point of immunization?
 Hedging higher-order derivatives is simple
 It follows the same process
 Hedging n derivatives will lead to a system of n linear equations—therefore we need n
bonds
𝑥1 𝑃1 + 𝑥2 𝑃2 + ⋯ + 𝑥𝑛 𝑃𝑛 = 𝑃
𝑥1 𝑃1 𝐷1 + 𝑥2 𝑃2 𝐷2 + ⋯ + 𝑥𝑛 𝑃𝑛 𝐷𝑛 = 𝑃 ∗ 𝐷
⋮
𝑥1 𝑃1 𝐷1,𝑛 + 𝑥2 𝑃2 𝐷2,𝑛 + ⋯ + 𝑥𝑛 𝑃𝑛 𝐷𝑛,𝑛 = 𝑃 ∗ 𝐷 𝑛
𝑃𝑖 is the present value of bond i
𝑥𝑖 is the amount of ‘units’ of bond i
𝑥𝑖 is the duration of bond i
𝑥𝑖,𝑗 is the jth price-normalized derivative of bond i
𝑃, 𝐷, 𝐷𝑛 refer to the obligation
IMMUNIZATION
This is summarized conveniently by
𝑃1 … 𝑃𝑛
𝑑𝑃1
𝑑𝑟
⋮
𝑑 𝑛 𝑃1
𝑑𝑟 𝑛
⋯
⋱
⋯
𝑑𝑃𝑛
𝑑𝑟 𝑥1
⋮ =𝑃∗
⋮
𝑑 𝑛 𝑃𝑛 𝑥𝑛
𝑑𝑟 𝑛
𝑑𝑃
𝑑𝑟
⋮
𝑑𝑛 𝑃
𝑑𝑟 𝑛
QUESTIONS?
NEXT LECTURE (2/26/2014)
 The world of trading
 Trading ecosystem
 Market microstructure
 Roles of brokers, traders, exchanges
UPCOMING EXTERNAL EVENTS
 IMC Financial Markets Tech Talk on Hadoop – Thursday, Feb 13th 6pm – 2240 DCL
 UIUC MSFE Information Session – Wednesday, Feb 28th 4pm – 106B Engineering Hall
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Committee
applications due
at midnight!
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Wed. 19th Feb. 6-7pm
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