Mathematics of Finance

Simple interest Notations

p – interest rate per conversion period (in percent) t – time in units that correspond to the rate, moment K₀ – opening capital, original principal, present value K_t – amount, capital at the momentt

Zt – simple interest earned in the term t i – interest rate:i= ^p

100

T – number of days

• The most frequently period considered is the year, but a half-year, quarter, month etc. can be used either. The number of days per year or month differs from country to country. Usual usances are 30

360, act 360, act

act, where act means the actual number of days. In what follows, in all formulas containing the quantity T the underlying period is the year with 360 days, each month having 30 days (i. e., the first usance is used).

Basic interest formulas

T = 30·(m₂−m₁) +d₂−d₁ – number of days for which interest is paid;m₁, m₂– months;d₁, d₂– days Z_t=K₀· p

100·t=K₀·i·t – simple interest Z_T =K₀·i·T

360 = K₀·p·T

100·360 – simple interest on a day basis K₀= 100·Z_t

p·t = Z_t

i·t – capital (at t= 0) p=100·Z_t

K₀·t – interest rate (in percent) i= Z_t

K₀·t – interest rate

t= 100·Z_t K₀·p = Z_t

K₀·i – term, time

32 Mathematics of finance

Capital at the momentt K_t=K₀(1 +i·t) =K₀

1 +i· T 360

– amount, capital at the mo- mentt

K₀= K_t

1 +i·t = K_t

1 +i·₃₆₀^T – present value i= Kn−K₀

K₀·t = 360·Kn−K₀

K₀·T – interest rate t= K_n−K₀

K₀·i – term, time

T = 360· K_n−K₀

K₀·i – number of days for which

interest is paid Periodical payments

• Dividing the original period intomparts of length _m¹ and assuming periodi- cal payments of sizerat the beginning (annuity due) and at the end (ordinary annuity), respectively, of each payment interval, the following amount results:

R=r

m+m+ 1 2 ·i

– annuity due R=r

m+m−1 2 ·i

– ordinary annuity

Especiallym= 12 (monthly payments and annual interest):

R=r(12 + 6,5i) – annuity due; R=r(12 + 5,5i) – ordinary annuity Different methods for computing interest

Letti=DiMiYibe day, month and year of thei-th date (i= 1: begin,i= 2:

end); let t=t₂−t₁ describe the actual number of days between begin and end; letT_i denote the number of days in the ’broken’ year i; basisi = 365 oder 366.

method formula

30/360 t= [360·(Y₂−Y₁) + 30·(M₂−M₁) +D₂−D₁]/360

act/360 t= (t₂−t₁)/360

act/act t= T₁

basis 1+Y₂−Y₁−1 + T₂ basis 2

∗

∗If the term considered lies within one year, only the first summand remains.

Compound interest 33

Compound interest

When considering several conversion periods, one speaks about compound interest if interest earned is added to the capital (usually at the end of every conversion period) and yields again interest.

Notations

p – rate of interest (in percent) per period n – number of conversion periods

K₀ – opening capital, present value

K_n – capital afternperiods, amount, final value

i – (nominal) interest rate per conversion period:i= p 100 q – accumulation factor (for 1 period):q= 1 +i

qⁿ – accumulation factor (fornperiods) m – number of parts of the period d – discount factor

im,ˆim – interest rates belonging to each of themparts of the period δ – intensity of interest

Conversion table of basic quantities

p i q d δ

p p 100i 100(q−1) 100 d

1−d 100(e^δ−1)

i p

100 i q−1 d

1−d e^δ−1

q 1 + p

100 1 +i q 1

1−d e^δ

d p

100 +p

i 1 +i

q−1

q d 1−e^−δ

δ ln

1 + p 100

ln(1 +i) lnq ln

1 1−d

δ

34 Mathematics of finance

Basic formulas

K_n =K₀·(1 +i)ⁿ=K₀·qⁿ – compound amount formula K₀= K_n

(1 +i)ⁿ = K_n

qⁿ – present value at compound interest, amount at time t= 0

p= 100

n

K_n K₀ −1

– rate of interest (in percent)

n=logK_n−logK₀

logq – term, time

n≈69

p – approximate formula for the time

within which a capital doubles Kn =K₀·q₁·q₂·. . .·qn – final value under changing rates of in-

terestpj,j= 1, . . . , n (with qj= ₁₀₀^pj ) p_r= 100

1+i 1+r−1

≈100(i−r) – real rate of interest considering a rate of inflation r

Mixed, partly simple and partly compound interest K_t=K₀(1 +it₁)(1 +i)^N(1 +it₂) – capital after time t

• Here N denotes the integral number of conversion periods, andt₁, t₂ are the lengths of parts of the conversion period where simple interest is paid.

• To simplify calculations, in mathematics of finance usually the compound amount formula with noninteger exponent is used instead of the formula of mixed interest, i. e.K_t=K₀(1 +i)^t, wheret=t₁+N+t₂.

Anticipative interest (discount)

In this case the rate of interest is determined as part of thefinal value ( discount factor p. 33).

d=K₁−K₀

K₁ = K_t−K₀

K_t·t – (discount) rate of interest under antici- patice compounding

K_n= K₀

(1−d)ⁿ – final value

K₀=K_n(1−d)ⁿ – present value

Compound interest 35

More frequently compounding Kn,m=K₀·

1 + _mⁱn·m

– amount aftern years when interest rate is convertedmtimes per year

i_m=_mⁱ – relative rate per period

ˆi_m= ^m√

1 +i−1 – equivalent rate per period i_eff= (1 +i_m)^m−1 – effective rate of interest per year p_eff= 100

(1 + p

100m)^m−1

– effective rate of interest per year (in percent)

• Instead of one year one can take any other period as the original one.

• Compounding interest m times per year with the equivalent rate per period ˆim leads to the same final value as compounding once per year with the nominal ratei; compounding interestmtimes per year with the relative rate per period im leads to that (greater) final value, which results when compounding once per year with the effective ratei_eff.

Continuous compounding

K_n,∞=K₀·e^i·n – amount at continuous compounding

δ= ln(1 +i) – intensity of interest (equivalent to the interest ratei)

i= e^δ−1 – nominal rate (equivalent to the intensityδ) Average date of payment

Problem: At which date of pay- mentt_m equivalently the total debt K₁+K₂+. . .+K_k has to be payed?

K_k- K₂

K₁ t₁

0 t₂ . . . t_k

liabilities to pay simple interest:

t_m=K₁+K₂+. . .+K_k−K₀

i , where K₀= K₁

1 +it₁ +. . .+ K_k 1 +it_k compound interest:

t_m=ln(K₁+. . .+K_k)−lnK₀

lnq , where K₀=K₁

q^t1 +. . .+K_k q^tk continuous compounding:

t_m=ln(K₁+. . .+K_k)−lnK₀

δ , where K₀=K₁e^−δt1+. . .+K_ke^−δtk

36 Mathematics of finance

Annuities Notations

p – rate of interest

n – term, number of payment intervals or rent periods R – periodic rent, size of each payment

q – accumulation factor:q= 1 + ^p

100

Basic formulas

Basic assumption: Conversion and rent periods are equal to each other.

F_n^due=R·q· qⁿ−1

q−1 – amount of an annuity due, final value P_n^due= R

qⁿ⁻¹ · qⁿ−1

q−1 – present value of an annuity due F_n^ord=R·qⁿ−1

q−1 – amount of an ordinary annuity, final value P_n^ord= R

qⁿ · qⁿ−1

q−1 – present value of an ordinary annuity P_∞^due= Rq

q−1 – present value of a perpetuity, payments at the beginning of each period

P_∞^ord= R

q−1 – present value of a perpetuity, interest used as earned

n= 1 logq ·log

F_n^ord·q−1 R + 1

= 1

logq·log R

R−P_n^ord(q−1) – term Factors describing the amount of 1 per period

due ordinary

amount of 1

per period ¨s_n|=q·qⁿ−1

q−1 s_n|= qⁿ−1 q−1 present worth

of 1 per period ¨a_n|= qⁿ−1

qⁿ⁻¹(q−1) a_n|= qⁿ−1 qⁿ(q−1) Conversion period> rent period

Ifm periodic payments (rents) are made per conversion period, then in the above formulas the paymentsRare to be understood asR=r

m+^m+1

2 ·i (annuities due) and R = r

m+^m−1

2 ·i

(ordinary annuities), resp. These amounts R arise at the end of the conversion period, so with respect toR one always has to use formulas belonging to ordinary annuities.

Annuities 37

Basic quantities

a_n| – present value of 1 per period (ordinary annuity)

¨

a_n| – present value of 1 per period (annuity due)

s_n| – final value (amount) of 1 per period (ordinary annuity)

¨

s_n| – final value (amount) of 1 per period (annuity due)

a_∞| – present value of 1 per period (perpetuity, interest used as earned)

¨

a_∞| – present value of 1 per period (perpetuity, payments at the begin- ning of each period)

Factors for amount and present value a_n| = 1

q+ 1 q² + 1

q³ +. . .+ 1

qⁿ = qⁿ−1 qⁿ(q−1)

¨

a_n| = 1 +1 q+ 1

q² +. . .+ 1

qⁿ⁻¹ = qⁿ−1 qⁿ⁻¹(q−1) s_n| = 1 +q+q²+. . .+qⁿ⁻¹ = qⁿ−1

q−1

¨

s_n| = q+q²+q³+. . .+qⁿ = q·qⁿ−1 q−1 a_∞| = 1

q+ 1 q² + 1

q³ +. . . = 1

q−1

¨

a_∞| = 1 +1 q+ 1

q² +. . . = q

q−1 Conversion table

a_n| ¨a_n| s_n| s¨_n| qⁿ

a_n| a_n|

¨ a_n|

q

s_n| 1 +is_n|

¨ s_n| q(1 +d¨s_n|)

qⁿ−1 qⁿi

¨

a_n| qa_n| ¨a_n|

qs_n| 1 +is_n|

¨ s_n| 1 +d¨s_n|

qⁿ−1 qⁿd s_n|

a_n| 1−ia_n|

¨ a_n|

q(1−d¨a_n|) s_n|

¨ s_n|

q

qⁿ−1 i

¨ s_n|

qa_n| 1−ia_n|

¨ a_n|

1−d¨a_n| qs_n| s¨_n| qⁿ−1 d

qⁿ 1

1−ia_n|

1

1−d¨a_n| 1 +is_n| 1 +d¨s_n| qⁿ

38 Mathematics of finance

Dynamic annuities

Arithmetically increasing dynamic annuity

Cash flows (increases proportional to the rentR with factorδ):

- n−1 n . . .

1 0

R(1+(n−1)δ) R(1 +δ)

R -

n 1

0

R R(1+δ) R(1+(n−1)δ) 2 . . .

F_n^due = Rq q−1

qⁿ−1 +δ

qⁿ−1 q−1 −n

P_n^due = R

qⁿ⁻¹(q−1)

qⁿ−1 +δ

qⁿ−1 q−1 −n F_n^ord = R

q−1

qⁿ−1 +δ

qⁿ−1 q−1 −n P_n^ord = R

qⁿ(q−1)

qⁿ−1 +δ

qⁿ−1 q−1 −n P_∞^due= Rq

q−1

1 + δ q−1

, P_∞^ord= R

q−1

1 + δ q−1

Geometrically increasing dynamic annuity -

0 1 2 n−1 n

Rbⁿ⁻¹ R

. . . Rb Rb²

-

0 1 2 . . . n

Rbⁿ⁻¹ Rb

R

The constant quotient b = 1 + ^s

100 of succeeding terms is characterized by therate of increase s.

F_n^due = Rq·qⁿ−bⁿ

q−b , b=q; F_n^due = Rnqⁿ, b=q P_n^due = R

qⁿ⁻¹· qⁿ−bⁿ

q−b , b=q; P_n^due = Rn, b=q F_n^ord = R·qⁿ−bⁿ

q−b , b=q; F_n^ord = Rnqⁿ⁻¹, b=q P_n^ord = R

qⁿ ·qⁿ−bⁿ

q−b , b=q; P_n^ord = Rn

q , b=q

P_∞^due = Rq

q−b, b < q; P_∞^ord = R

q−b, b < q

Amortization calculus 39

Amortization calculus Notations

p – interest rate (in percent) n – number of repayment periods i – interest rate:i= ^p

100

q – accumulation factor:q= 1 +i S₀ – loan, original debt

Sk – outstanding principal after kperiods Tk – payoff in thek-th period

Z_k – interest in the k-th period A_k – total payment in thek-th period Kinds of amortization

• Constant payoffs: repayments constant:Tk =T = S₀

n, interest decreasing

• Constant annuities: total payments constant:Ak = A = const, interest decreasing, payoffs increasing

• Amortization of a debt due at the end of the payback period:A_k=S₀·i, k= 1, . . . , n−1; A_n=S₀·(1 +i)

• In anamortization schedule all relevant quantities (interest, payoffs, total payment, outstanding principal, etc.) are clearly arranged in a table.

Basic formula for the total payment

A_k =T_k+Z_k – total payment (annuity) consisting of payoff and interest

Constant payoffs(conversion period = payment period) T_k= S₀

n – payoff in thek-th period

Zk=S₀·

1−k−1 n

i – interest in thek-th period A_k =S₀

n [1−(n−k+ 1)i] – total payment in thek-th period Sk =S₀·

1−k

n

– outstanding principal afterk periods P =S₀i

n

(n+ 1)a_n|− 1 qⁿ

q qⁿ−1

(q−1)² − n

q−1 – present value of all interest payments

40 Mathematics of finance

Constant annuities(conversion period = payment period) A=S₀·qⁿ(q−1)

qⁿ−1 – (total) payment, annuity

S₀= A(qⁿ−1)

qⁿ(q−1) – original debt

T_k=T₁q^k−1= (A−S₀·i)q^k−1 – payoff, repayment

S_k =S₀q^k−Aq^k−1

q−1 =S₀−T₁q^k−1

q−1 – outstanding principal Z_k=S₀i−T₁(q^k−1−1) =A−T₁q^k−1 – interest

n= 1 logq

logA−log(A−S₀i)

– length of payback period, period of full repayment

More frequently payments

In every periodm constant annuitiesA^(m)are payed.

A^(m)= A m+^m−1

2 i – payment at the end of each period A^(m)= A

m+^m+1

2 i – payment at the beginning of each period Especially: Monthly payments, conversion period = 1 year (m= 12)

A_mon= A

12 + 5,5i – payment at the end of every month

A_mon= A

12 + 6,5i – payment at the beginning of every month

Amortization with agio

In an amortization with additional agio (redemption premium) ofαpercent on the payoff the quantityT_k has to be replaced by ˆT_k =T_k·

1 + α 100

= Tk·fα. When considering amortization by constant annuities with included agio, the quantitiesS_α =S₀·f_α (fictitious debt),i_α= i

f_α (fictitious rate of interest) andqα= 1 +iα, resp., are to be used in the above formulas.

Price calculus 41

Price calculus Notations

P – price, quotation (in percent) K_nom – nominal capital or value K_real – real capital, market value

n – (remaining) term, payback period p,p_eff – nominal (effective, resp.) rate of interest

bn,nom; b_n,real – present worth of 1 per period (ordinary annuity) a=C−100 – agio for price above par

d= 100−C – disagio for price below par

R – return at the end of the payback period q_eff= 1 +^peff

100 – accumulation factor (effective interest rate) Price formulas

P = 100· K_real

K_nom – price as quotient of real and

nominal capital

P = 100· b_n,real

bn,nom = 100· n k=1

1 q^k_eff

n k=1

1 q^k

– price of a debt repayed by constant annuities

P =100 n

n· p

p_eff +b_n,real

1− p

p_eff – price of a debt repayed by constant payoffs

P = 1 qⁿ eff

·

p· q_effⁿ −1 q_eff−1 +R

– price of a debt due at the end of the payback period P = 100·p·(p_eff)⁻¹ – price of a perpetuity p_s= 100

C

p−a n

= 100 C

p+d

n

– simple yield-to-maturity;

= approximate effective interest rate of a debt due at the end of the payback period (price above and below par, resp.)

• Securities, shares (or stocks) are evaluated at the market by the price.

For given priceC in general the effective rate of interest (yield-to-maturity, redemption yield) can be obtained from the above equations by solving (ap- proximately) a polynomial equation of higher degree ( p. 44).

42 Mathematics of finance

Investment analysis

Multi-period capital budgeting techniques (discounted cash flow methods) are methods for estimating investment profitability. The most known are:capital (or net present) value method, method of internal rate of return, annuity method. Future income and expenses are prognostic values.

Notations

I_k – income at momentk

E_k – expenses, investments at momentk

C_k – net income, cash flow at momentk:C_k=I_k−E_k K_I – present value of income

K_E – present value of expenses

C – net present value, capital value of the investment n – number of periods

p – conventional (or minimum acceptable) rate of interest q – accumulation factor:q= 1 +₁₀₀^p

Capital value method K_I=

n k=0

I_k

q^k – present value of income; sum of all present values of future income

K_E= n k=0

E_k

q^k – present value of expenses; sum of all present values of future expenses

C= n k=0

Ck

q^k =KI−KE – capital value of net income, net present value

• For C = 0 the investment corresponds to the given conventional rate of interest p, for C > 0 its maturity yield is higher. If several possibilities of investments are for selection, then that with the highest net present value is preferred.

Method of internal rate of return

Theinternal rate of return (yield-to-maturity)is that quantity for which the net present value of the investment is equal to zero. If several investments are possible, then that with the highest internal rate of return is chosen.

Annuity method F_A= qⁿ·(q−1)

qⁿ−1 – annuity (or capital recovery) factor A_I =K_I·F_A – income annuity

A=K_E·F_A – expenses annuity

A_P =A_I−A – net income (profit) annuity

• ForA_I =Athe maturity yield of the investment is equal top, forA_I > A the maturity yield is higher than the conventional rate of interestp.

Depreciations 43

Depreciations

Depreciations describe the reduction in value of capital goods or items of equipment. The difference between original value (cost price, production costs) and depreciation yields thebook-value.

n – term of utilization (in years) A – original value

w_k – depreciation (write-down) in thek-th year

R_k – book-value afterk years (R_n – remainder, final value) Linear (straight-line) depreciation

w_k=w= A−R_n

n – annual depreciation

R_k =A−k·w – book-value afterkyears

Arithmetically degressive depreciation(reduction by d each year) w_k=w₁−(k−1)·d – depreciation in thek-th year d= 2· nw₁−(A−R_n)

n(n−1) – amount of reduction Sum-of-the-years digits method (as a special case): w_n=d wk= (n−k+ 1)·d – depreciation in thek-th year d=2·(A−R_n)

n(n+ 1) – amount of reduction

Geometrically degressive (double-declining balance) depreciation (reduction byspercent of the last year’s book value in each year)

R_k =A· 1− s

100 k

– book-value afterk years s= 100·

! 1− ⁿ

R_n A

"

– rate of depreciation wk=A· s

100· 1− s

100 k−1

– depreciation in thek-th year Transition from degressive to linear depreciation

Under the assumption R_n = 0 it makes sense to write down geometrically degressive until the yeark with k =n+ 1− ¹⁰⁰_s and after that to write down linearly.

44 Mathematics of finance

Numerical methods for the determination of zeros

Task: Find a zerox^∗of the continuous functionf(x); letεbe the accuracy bound for stopping the iteration process.

Table of values

For chosen values xfind the corresponding function values f(x). Then one obtains a rough survey on the graph of the function and the location of zeros.

Interval bisection

Givenx_L withf(x_L)<0 andx_R withf(x_R)>0.

1. Calculatex_M =¹

2(x_L+x_R) andf(x_M).

2. If|f(xM)|< ε, then stop and takexM as an approximation ofx^∗. 3. If f(x_M)<0, then setx_L :=x_M (x_R unchanged), if f(x_M)> 0, then setx_R:=x_M (x_L unchanged), go to 1.

Method of false position (linear interpolation, regula falsi) Givenx_L withf(x_L)<0 andx_R withf(x_R)>0.

1. Calculatex_S =x_L− x_R−x_L

f(x_R)−f(x_L)f(x_L) andf(x_S).

2. If|f(x_S)|< ε, then stop and takex_S as an approximation ofx^∗. 3. Iff(xS)<0, then setxL :=xS (xR unchanged), iff(xM)>0, then set xR:=xS (xL unchanged), go to 1.

•Forf(xL)>0, f(xR)<0 the methods can be adapted in an obvious way.

Newton’s method

Givenx₀∈U(x^∗); let the functionf be differentiable.

1. Calculatex_k+1=x_k− f(x_k) f(x_k).

2. If|f(x_k+1)|< ε, then stop and takex_k+1 as an approximation ofx^∗. 3. Setk:=k+ 1, go to 1.

• If f(xk) = 0 for somek, then restart the iteration process with another starting pointx₀.

•Other stopping rule:|x_L−x_R|< ε or |x_k+1−x_k|< ε.

Descartes’ rule of signs.The number of positive zeros of the polynomial n

k=0

a_kx^k is equal toworw−2,w−4, . . . , wherewis the number of changes in sign of the coefficientsa_k (not considering zeros).