Financial and investment Financial and investment mathematicsmathematics

RNDr. Petr Budinský, CSc.RNDr. Petr Budinský, CSc.

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FINANCIAL MATHEMATICSFINANCIAL MATHEMATICSFuture value – different types of compoundingFuture value – different types of compounding

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ExampleExample::Assume FV = 100.000 CZK and interest rate = 12 %. Assume FV = 100.000 CZK and interest rate = 12 %. Calculate future value in 3 years time. Calculate future value in 3 years time.

… …

… …

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Present value calculated from future valuePresent value calculated from future value

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ExampleExample::Assume cash flows given by following table and interest rateAssume cash flows given by following table and interest rate

r = 6 %, compoundedr = 6 %, compounded

a) a) Once yearlyOnce yearly

year 1 2 3 4

cash flows 0 100 200 300

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ExampleExample::Assume cash flows given by following table and interest rateAssume cash flows given by following table and interest rate

r = 6 %, compoundedr = 6 %, compounded

b) b) ContinouslyContinously

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Yield calculated in case of fixed cash flowsYield calculated in case of fixed cash flows

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Equalities and inequalities among yieldsEqualities and inequalities among yields

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ExampleExample::Assume an investment P = 10.000 Kč for 5 years, after 5 Assume an investment P = 10.000 Kč for 5 years, after 5 years you earn an amount FV = 21.000 CZK. Calculate the years you earn an amount FV = 21.000 CZK. Calculate the yield.yield.

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ExampleExample::Assume a loan 1.000.000 CZK for 10 years. This loan is paid Assume a loan 1.000.000 CZK for 10 years. This loan is paid by same installments C at the end of each year with the by same installments C at the end of each year with the yield yyield y(1)(1) = 8 % p.a. Calculate the installment = 8 % p.a. Calculate the installment C.C.

C C can be splitted to the interest rate payment - 80.000 can be splitted to the interest rate payment - 80.000 CZK and to the amount 69.029,49 CZK by which the CZK and to the amount 69.029,49 CZK by which the loan will be decreased to 930.970,51 CZKloan will be decreased to 930.970,51 CZK..

1.000.000 = C (1/(1+ y) + 1/(1+ y)1.000.000 = C (1/(1+ y) + 1/(1+ y)22 + ... +1/(1+ y)+ ... +1/(1+ y)1010))

1.000.000 = C[1-1/(1 + y)1.000.000 = C[1-1/(1 + y)1010]/y]/y

C = 1.000.000 ⋅ 0,08/[1 −1/1,08C = 1.000.000 ⋅ 0,08/[1 −1/1,0810 10 ] = 149.029,49 Kč] = 149.029,49 Kč

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Table of paymentsTable of payments

Installment Interest rate part Principal payment Remaining part

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BondsBonds

zero-coupon bond: annuity:

perpetuity:

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Closed formula for bond priceClosed formula for bond price

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Rules for bondsRules for bonds Rule 1Rule 1::

If the yield y is equal to the coupon rate c, then the bond price P is equal to face value FV, if yield y is higher, resp. less than the coupon rate c, then the bond price P is smaller, resp. greater than the face value FV.

Rule 2Rule 2::If the price of the bond increases, resp. decreases, this results in a decrease, resp. increase of the yield of the bond. Reverse: decrease, resp. rise in interest rates (yields) results in an increase, resp. decrease in bond prices.

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Rule 3Rule 3::If the bond comes closer to its maturity, then the bond price If the bond comes closer to its maturity, then the bond price comes closer to the face value of bond.comes closer to the face value of bond.

Rule 4Rule 4::

The closer is the bond to its maturity the higher is the The closer is the bond to its maturity the higher is the

velocity of approaching the face value by the price of thevelocity of approaching the face value by the price of the

bond.bond.

Rules for bondsRules for bonds

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Rule for bondsRule for bonds Rule 5Rule 5::

The decrease in a bond yield leads to an increase in bond price by an amount higher than is the amount corresponding to the decrease (in absolute value) in the price of the bond if the yield increases by same percentage as previously decreased.

ExampleExample::Assume Assume a 5-year bond with a face value FV = 1.000 CZK, FV = 1.000 CZK, coupon rate c = 10 % coupon rate c = 10 % and yield y = 14 %.= 14 %.

Yield 12 % 13 % 14 % 15 % 16 %

Price 927,90 894,48 862,68 832,39 803,54

Price change 65,22 31,80 0 -30,29 -59,14

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Example::

Rule for bondsRule for bonds

CZK

CZK

CZK

CZK

CZK

CZK

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Relationship Relationship of the bondof the bond price and time to price and time to maturity of the bondmaturity of the bond

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Bond pricing – general approachBond pricing – general approach

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A + B = 360A + B = 360

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ExampleExample::Assume a 5-year bond with a face value FV = 10.000 CZK issued at 6. 2. 1998 with maturity 6. 2. 2003 and with coupon rate c = 14.85%. The yield of this bond was y = 7% on 9. 11. 1999. Calculate the clean price PCL of the bond.

CZK

CZK

CZK

CZK

CZK

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The sensitivity of bond prices to changes inThe sensitivity of bond prices to changes in interest ratesinterest rates (yields)(yields)

Modified duration DModified duration Dmodmod is a positive number expressing the is a positive number expressing the increase (in %), resp. decrease (in %) of the bond price if the increase (in %), resp. decrease (in %) of the bond price if the yield decreases, resp. increases by 1%. yield decreases, resp. increases by 1%.

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Macaulay durationMacaulay duration

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Macaulay durationMacaulay duration

zero-coupon bond:

annuity:

perpetuity:

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ExampleExample::Bond parameters are as follows : FV = 1.000 CZK,: FV = 1.000 CZK,n = 5, c = 10 %, y = 14 %.n = 5, c = 10 %, y = 14 %.

CZKCZK

CZK CZK

CZK

CZK

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The dependence of duration on c and y and y

1.1.

2. 2.

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Dependence Dependence of duration D of duration D on time to maturityon time to maturity n n

nn

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EstimateEstimate of changes in bond prices of changes in bond prices

ExampleExample::

a)a)

b)b)

CZK

CZK

CZK

CZK

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Bond convexityBond convexity

Convexity is sometimes called the "curvature" of the bond.

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Calculation of convexity

CX = 2/yCX = 2/y22

Zero-coupon bond:

Perpetuity:

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INVESTMENT MATHEMATICSINVESTMENT MATHEMATICSRisks associated with the bond portfolios

When investing in bonds investor must take into account the two risks:

1. risk of capital loss (if yields increase )2. risk of loss from reinvestment (if yields decrease )

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ExampleExample::Assume 5 year zero-coupon bond with face value

FV = 1.000 CZK and yield y = 4%. This bond is an investmenta) a) for 3 yearsfor 3 years

CZK

CZK

CZK

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ExampleExampleb) b) for 7 yearsfor 7 years

Investment horizon

CZK

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Investment horizon X DurationInvestment horizon X Duration

When you invest in a particular bond and if our investment horizon is: Short - you suffer a loss in case yields increase(„capital loss"> „input of reinvestment“)

Long - you suffer a loss in case yields decrease(„loss of reinvestment "> „capital gain")

If the investment horizon is equal to (Macaulay) duration of the bond, then the "capital loss", resp. „loss of reinvestment" is fully covered by "reinvestment income" , resp. by „return on capital" , in case of both increase and decrease of yields.

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Example:Example:Assume 8-year bond, which has a face value Assume 8-year bond, which has a face value FV = 1.000 CZK with coupon rate c = 9,2 % and the yieldFV = 1.000 CZK with coupon rate c = 9,2 % and the yieldy = 9,2 %. Ty = 9,2 %. This bond is an investment for 1 year, 2 years, 1 year, 2 years, 3 years, …, 8 years - we assume 8 investment strategies. 3 years, …, 8 years - we assume 8 investment strategies.

Further assume 5 Further assume 5 scenarios of development of the yields:8,4 %, 8,8 %, 9,2 % (8,4 %, 8,8 %, 9,2 % (unchanged yield ), 9,6 % and 10 %. ), 9,6 % and 10 %.

Combination of the chosen investment strategy with a particular yield scenario will provide 40 different options. For each of these options is calculated the realized yield. . All results are summarized in the table. The price of a bond

P = 1.000 CZK.P = 1.000 CZK.

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Investment strategies

„„1. line“ – 1. line“ – income from coupons

nC;nC; „„2. line“ – 2. line“ – income from reinvestment of coupons after deduction of

the coupons

„„3. line“ – 3. line“ – capital gain (the difference between the sale and repurchase price of the bond)

„„4. line“- the total return (in CZK) the sum of 1., 2. and 3. line4. line“- the total return (in CZK) the sum of 1., 2. and 3. line „„5. line“ –the total return y5. line“ –the total return ynn in % (p.a.): in % (p.a.):

soso

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Bond portfolio durationBond portfolio duration

The duration of a coupon bond is a weighted average of durations (time to maturities) of the individual cash flows represented by coupons and face value, the weights correspond to the share of individual discounted cash flow as a proportion of the total price of the bond.

The duration of a coupon bond is mean lifetime of the bond.

The duration of a portfolio consisting of bonds is the weighted average of durations of individual bonds, where the weights correspond to investments in individual bonds as proportions of the total investment in the bond portfolio.

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Example:Example:

Assume an investment 1.000.000 CZK for 4 years, we have zero-coupon bonds with maturities of 1 year, 2 years, ..., 7 years with uniform yields y = 8% (assuming a flat yield curve). Create portfolios A, B, C, D as follows (n is the time to maturity of each bond)

CZK

CZK

CZK

CZK

CZK

CZK

CZK

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Change Change in in the value Vthe value V00 in case of in case of

changchange of the yielde of the yield

Realized amounts V0 (CZK)

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Bond portfolio convexityBond portfolio convexity

Bond portfolio convexity is the weighted average of convexities of individual bonds, where the weights correspond to investments in individual bonds as proportions of the total investment in the bond portfolio..

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The effect of convexity on the behavior of The effect of convexity on the behavior of bond portfoliosbond portfolios

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V4+ (D) > V4

+ (C) > V4+ (B) > V4 (A)

V4- (D) > V4

- (C) > V4- (B) > V4 (A)

A … FVA = 1 000 000 x 1,084 = 1 360 489 Benchmark V4(A) = 1 000 000 x 1,084 = 1 360 489 B … V4(B) = 500 000 x 1,083 x 1,081 + 500 000 x 1, 085 / 1,08 = 1 360 489 V4

+(B) = 500 000 x 1,083 x 1,091 + 500 000 x 1, 085 / 1,09 = 1 360 547 V4

-(B) = 500 000 x 1,083 x 1,071 + 500 000 x 1, 085 / 1,07 = 1 360 548 C … V4(C) = 500 000 x 1,082 x 1,082 + 500 000 x 1, 086 / 1,08 2 = 1 360 489 V4

+(C) = 500 000 x 1,082 x 1,092 + 500 000 x 1, 086 / 1,09 2 = 1 360 720 V4

-(C) = 500 000 x 1,082 x 1,072 + 500 000 x 1, 086 / 1,07 2 = 1 360 724 D … V4(D) = 500 000 x 1,081 x 1,083 + 500 000 x 1, 087 / 1,08 3 = 1 360 489 V4

+(D)= 500 000 x 1,081 x 1,093 + 500 000 x 1, 087 / 1,09 3 = 1 361 009 V4

-(D) = 500 000 x 1,081 x 1,073 + 500 000 x 1, 087 / 1,07 3 = 1 361 019

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REALIZOVANÉ VÝNOSY SC

ÉNÁŘ

E

A B C D 5% 1 360 489 1 361 029 1 362 649 1 365 350 6% 1 360 489 1 360 727 1 361 440 1 362 629 7% 1 360 489 1 360 548 1 360 724 1 361 019 8% 1 360 489 1 360 489 1 360 489 1 360 489 9% 1 360 489 1 360 547 1 360 720 1 361 009 10% 1 360 489 1 360 718 1 361 405 1 362 532 11% 1 360 489 1 361 000 1 362 532 1 365 088

ChangeChange in in the value V the value V44 in case of in case of changchange in e in

the yieldthe yield

Realized income

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DE = 0,7 x 1 + 0,3 x 11 = 4

CXE (1,082) = 0,7 x 1 x 2 + 0,3 x 11 x 12 = 41

IH = 4

•

1 11

3 7

7/10 3/10

w1 D1 + w2 D2 = DP = IH w1 + w2 = 1 w1 + 11 w2 = 4 w1 + w2 = 1 w1 = 0,7 ; w2 = 0,3

Portfolio E: 700 000 Kč (n=1) a 300 000 Kč (n=11)

n = 1 … FV = 756 000 Kč … D1 = 1 n = 11 … FV = 699 000 Kč … D2 = 11

CZK CZK

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BENCHMARK … V4 (A) = 1 360 489

V4 (E) = 700 000 x 1,081 x 1,083 + 300 000 x 1,0811 / 1,087 = 1 360 489

V4+

(E) = 700 000 x 1,081 x 1,093 + 300 000 x 1,0811 / 1,097 = 1 361 688

V4- (E) = 700 000 x 1,081 x 1,073 + 300 000 x 1,0811 / 1,077 = 1 361 741

DA = DB = Dc = DD = DE = 4

CXA < CXB < CXc < CXD < CXE

V4+ (E) > V4

+ (D) > V4+ (C) > V4

+ (B) > V4+ (A)

V4- (E) > V4

- (D) > V4- (C) > V4

- (B) > V4- (A)

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Example:Example:

Assume an investment of 2.800.000 CZK for 5 years, and we have available two zero-coupon bonds A, B:

Create a portfolio hedged against interest rate risk and calculate yield to the investment horizon, provided that day after the purchased of the portfolio yield increased, resp. decreased by 1%.

A … n = 3, y = 4% B … n = 10, y = 4%

3wA + 10 wB = 5 wA + wB = 1 7wB = 2 wA = 5/7, wB =

2/7

A … 2 000 000 Kč B … 800 000 Kč V5 = 2 000 000 x 1,043 x 1,042 + 800 000 x 1,0410 / 1,045 = 3 406 628 Kč V5

+=2 000 000 x 1,043 x 1,052+ 800 000 x 1,0410/ 1,055 = 3 408 173 Kč V5

-= 2 000 000 x 1,043 x 1,032 + 800 000 x 1,0410/ 1,035 = 3 408 234 Kč

CZK

CZK

CZK

CZK

CZK

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Derivative contractsDerivative contracts

1. forward contracts2. option contracts (options)

Forward contract is an obligation, option contract is the right, to buy or sell-agreed number of shares-at agreed price-on the agreed date

Call option is right to buy.Put option is right to sell.

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Forward contractForward contract

profitShort position

Long position

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Option contractOption contract

profit

Call option Put option

Call option will be exercised only if ST ˃ X and profit of this option is equal to max {ST – X, 0}.

Put option will be exercised only if X ˃ ST and profit of this option is equal to max {X - ST, 0}.

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Graphs Graphs of profits of profits and losses and losses when usingwhen using optionsoptions

Call option, long position Call option, short position

Put option, long position Put option, short position

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Portfolio consisting of optionsPortfolio consisting of options

A. Combination of only one option type, eg. call options - portfolio is called "spread". All call options included in the portfolio have the same underlying stock and the same time to expiration, they differ in an exercise price – it is called "vertical spread". „Horizontal spread“ arises if, on the contrary same exercise price at different time expirations.

B. Combination of both types of options - portfolio is made up of call options and put options. Portfolio is called "combination".

C. Combination of call options or put options with underlying stocks. We speak about "hedging„.

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Bullish SpreadBullish Spread

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Bearish SpreadBearish Spread

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Butterfly SpreadButterfly Spread

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Condor SpreadCondor Spread

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Bottom StraddleBottom Straddle

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Bottom StrangleBottom Strangle

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HedgingHedging

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Example:Example:

Number PriceExcercise

pricePosition

Share

Call option

Put option

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