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Math MSc 1st Sem > IMF Theory Chapter 1 > Flashcards

IMF Theory Chapter 1 Flashcards

1
Q

3 compounding conventions

A

simple interest rule: A*(1+rt)

compounding interest:

  • A*(1+r)^n
  • A*(1+r/m)^k (k in {1,..,m})
  • effective interest rate r’: A(1+r’)=A(1+r/m)^m iff r’=…

continuous compounding:

  • Aexp(r)=lim A(1+r/m)^m as m–>inf
  • multiple years: Ae^{rt}
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2
Q

Future value of cash flows (x_0,…,x_n)?

assuming t_i=i years

A

FV_n:=Sum( x_i*(1+r)^{n-i} ; i=0,…,n )

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3
Q

Present value of cash flows (x_0,…,x_n)?

assuming t_i=i years

A

PV_0:=Sum( x_i / (1+r)^i ; i=0,…,n )

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4
Q

IRR?

A

Internal rate of return is the rate lambda at which:

PV_0:=Sum( x_i / (1+lambda)^i ; i=0,…,n )=0 holds.

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5
Q

Under which conditions does a unique IRR exist?

A
  • There exists k in {0,…,n-1} s.t.
  • for every i in {0,…,k}, the x_i have the same sign,
  • and for every i in {k+1,…,n} the x_i have the opposite sign.
  • If in addition at least one of the (x_i){i=0,…,k} and (x_i){i=k+1,…,n} is non-zero,
  • then the IRR exists and is unique.
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6
Q

Which of the following cash flow streams should we choose? Prevailing interest rates are 10%.

i) (-1,2)
ii) (-1,0,3)

A

PV criteria chooses (ii) but IRR chooses (i).

lec 1, page 6

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7
Q

Face value of a bond?

Coupon rate?

Emission price?
Par value?

Reimbursement price?

Maturity?

A

(aka nominal) this is the amount with which computations regarding the bond are made.

(yearly) coupon payment / face value

Price paid for a bond when it is emitted/created/first sold. Usually it is taken to be equal to the face value. In that case we say the bond is emitted at par value. If emission price is below face value, we say it is emitted below par value.

Reimbursement price is the amount paid back at maturity of a bond (excluding potential simultaneous coupon payment)

End date of the contract

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8
Q

Clean price of a bond?

Dirty price of a bond?

A

clean price := present value - accrued interest (%)
dirty price:=present value
where
accrued interest:=(#{days since last coupon} / #{days in current coupon period} ) * (coupon rate per period)

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9
Q

Purchase bond on may 8th, maturing august 15th several years from now. Coupon 9% and coupon payments made twice a year on Feb 15th and Aug 15th.
What is the accrued interest on May 8th?

Days in month:
Jan 31
Feb 28 (29 if leap year. assume leap year)
Mar 31
Apr 30 
Mai 31
Jun 30
Jul 31
Aug 31
Sept 30
Oct 31
Nov 30
Dec 31
A

Accrued interest = 83/(83+99) * 4.5% ~ 2.05%

lect. 2

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10
Q

YTM?

A

Yield to maturity of a bond is the interest rate that makes the present value of all associated future payments equal to the current value of the bond.
Note YTM is nothing but the IRR of the bond at its current price.

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11
Q

Derive the relationship between YTM and price for a bond with face value F, m coupon payments of c*F/m every year and n coupon periods remaining.

What properties does P(lambda) have?

A

P=Sum(cF/m(1+lambda/m)^{-k} ; k=1,…,n) + F(1+lambda/m)^{-n}
=cF/lambda(1-(1+lambda/m)^{-n})+F(1+lambda/m)^{-n}

This describes the price-yield curve.

i) P non-increasing
ii) If lambda=c, then P=F. Bond is “par bond”.
iii) P is convex
iv) |dP(0)/dlambda|=n(n+1)cF/(2m^2)+F*n/m, i.e. bonds with longer maturities are more sensitive to variations in yield

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12
Q

Duration of a fixed-income instrument?

Bounded?

Can maturity=duration?

A

Weighted average of the times at which the cash-flows of the instrument occur.
(t_i){i=0,…,n} with associated present values (PV{t_i}){i=0,…,n}, then the Duration is:
D:=Sum(PV{t_i}*t_i) / Sum(PV_{t_j})

Yes, above by t_0 and below by t_n since convex comb between t_0 and t_n.

For zero-coupon bonds: maturity=duration

Interpr: Weighted length of time one has to wait in order to receive the security’s payments.

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13
Q

Macaulay Duration?

Modified Duration?

Relationship of modified duration and yield

A

D_t=1/PV_t * Sum( (k-t)/mc_k(1+lambda/m^{k-t}) ; k>t ), where PV_t=Sum( c_k*(1+lambda/m)^{k-t} )

D^M:=D*(1+lambda/m)

D^M_t = -1/PV_t * dPV_t/dlambda

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14
Q

Consider bond with maturity =10y, coupon rate=6%, face value=100, YTM=6%, PV_0=100, what is the modified duration?

How does the value change when YTM changes to 6.1%?

Approximate the change to the first order

YTM changes to 8%?

Compare to approx

A

lect 2, page 6
D_0^M=1/(1.06)(Sum( 6i(1.06)^{-i} ; i=1,…,10) + 10010/1.06^{-10})~7.36

PV_0(0.061)=Sum( 6/1.061^{-i} ; i=1,..,10) + 100/(1.061)^{10}~99.267

dPV_t(lambda)=PV_t(lambda+dlambda)-PV_t(lambda)=dPV_t/dlambda(lambda)*dlambda+o(dlambda)=-D_t^M(lambda)*PV_t(lambda)*dlambda+o(dlambda)
then dPV_0(0.06)~0.736

PV_0(0.08)=Sum(6/(1.08)^i ; i=1,..,10)+100/(1.08)^{10}~86.58

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15
Q

Convexity of fixed-income?

Specify for m payments per year for n periods

A

C_t(YTM=lambda)=1/PV_t*PV_t’‘(lambda)

C_t=1/PV_tSum( (k-t)(k-t+1)c_k/[m^2(1+lambda/m)^{k-t+2}] ; k>t), where PV_t=Sum(c_k/(1+lambda/m)^{k-t} ; k>t)

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16
Q

2nd order approx of PV_t

A

dPV_t=-D_t^M(lambda)dlambda+0.5PV_t*C_t(dlambda)^2+o(dlambda^2)

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17
Q

P^00=current value of portfolio.
Given n fixed-income securities with values (P_i){i=1,..,n}, durations (D_i){i=1,..,n} and convexities (C_i){i=1,..,n}, how does one construct an immunized portfolio? n=3

A

One needs to find the quantities (q_i)_{i=1,..,n} s.t. the LSE:
-P^0=Sum(q_iP_i)
-D^0=1/p^0Sum(q_iP_iD_i)
-c^0=1/p^0Sum(q_iP_i*C_i)
Has a unique solution (which is the case as long as the associated determinant is not 0).

NOTE : He uses superscripts here to note that these are different assets and not the same one at different times.

18
Q

Immunize a 1M par value bond with YTM=8%, T=7.

P^1=Bond w. c=7%, T=6
P^2=Bond w. c=10%, T=6
P^3=Bond w. c=2%, T=9

A

lect 3, slide 2

19
Q

Define discount factor of money-market account.

A

For t>=0, we let B_t be the value at time t of a money-market account, i.e. unit of money at time 0 that provides us the quantity B_t at time t.
To this money-market account we can associate the discount factor:
d(t,T):=B_t/B_T.

When rates are deterministic, d(t,T) is nothing but the present value at time t, of one unit of money received at T>=t.

Note: B_t != B(0,t)

20
Q

Define zero-coupon bond.

How does it relate to the discount rate?

A

For some maturity T>=0. A zero-coupon bond with maturity T is a contract guaranteeing to its holder a payment of 1 at time T, and pays no intermediate coupons. The value of such a contract is denoted by B(t,T), for any t in [0,T]. Notice that B(T,T)=1.

When interest rates are deterministic B(t,T)=d(t,T). (later even when stochastic)

21
Q

State AoA (define a.o.)

A

There are no arbitrage opportunities in the market.

Def. An arbitrage opportunity over [t,T] is self-financing strategy (X_s)_s, s.t. X_t=0, P(X_T >= 0)=1 and P(X_T > 0)>0.

22
Q

Associate the present value of the cash flow (x1,..,xn) and zero-coupon bonds.

A

Given that AoA holds, PV_t=Sum( B(t,t_k)*x_{t_k} ; k=0,..,n )

23
Q

Define the continuously compounded spot rate at time t and maturity T.

A

r(t,T) := -1/(T-t)log(B(t,T)) iff B(t,T)=exp(-(T-t)r(t,T))

spot rate = interest rate (for zero-coupons)

24
Q

Define the simply-compounded spot rate.

A

l(t,T) := (1-B(t,T)) / [(T-t)B(t,T)] iff B(t,T) := 1 / [1+(T-t)l(t,T)]

spot rate = interest rate (for zero-coupons)

25
Q

Define the k-times-per-year compounded spot rate y_k(t,T)

A 
y_k(t,T) := k*B(t,T)^{-1/(k*(T-t))}-k
iff B(t,T)=( 1 + y_k(t,T) / k)^{-k*(T-t)}

when k=1, y(t,T):=y_1(t,T)

spot rate = interest rate (for zero-coupons)

26
Q

Given:
Maturity | Coupon rate | Market price
0.5 | 0 | 99.05
0.75 | 0 | 98.45
1 | 0 | 97.85
2 | 3.5 | 98.50
3 | 4 | 98.30
4 | 3.5 | 99.59
Find the 1-times-per-year coupounded spot rates y(0,05), y(0,0.75), y(0,1) and y(0,2).

A

lect 3, slide 8

27
Q

FRA

A

A forward rate agreement (FRA) locks in an interest rate for an investment in the future.
Let t>=0 be current time, s>=t the expiry time and T>=s the maturity. The FRA is an agreement at time t, whereby the seller pays the holder an interest rate K on the nominal value N at time T for the period between times s and T in exchange for the floating payment based on the rate l(s,T).
The FRA value is therefore:
FRA_T(s,T,N,K)=N(T-s)(k-l(s,T))=N[(T-s)K+1-1/B(s,T)]
FRA_t(s,T,N,K)=N[ (1+(T-s)K)*B(t,T) - B(t,s) ].

BTW: contract is symmetric ie usually FRA_0=0

28
Q

Define simply-compounded forward interest rate.

Define k-times-compounded forward interest rate.

Define continuously-compounded forward interest rate.

A

Lecture 4, slide 4

29
Q

Define the Fisher-Weil quasi-modified duration (convexity).

A

D^{FW}_t := - 1/PV_tdPV_t(lambda=0)/dlambda, where we defined the shifted present value by PV_t(lambda)=Sum[ ck( 1+(y_m(t,k)+lambda)/m )^{m*(k-t)} ; k>t ].
Convexity D^{DW}_t as above with 2nd derivative.

30
Q

Nelson-Siegel model

A

r_{NS}(t,T):=b0+b1(1-exp(-(T-t)/tau))/(T-t)tau+b2*((1-exp((-T-t)/tau)/[(T-t)/tau]-exp((-T+t)/tau),
where
b0=shift parameter and long term interest rate
b1 and b2=reproduce approx. twist and curvature movements
tau=decaying factor

Sensitivities: S_t^{bi}:=1/PV_t*dPV_t/dbi

lect 4, slide 6

31
Q

Define forward contract.

A

A forward contract is an obligation for two parties to exchange an underlying asset (S_t)_{t>=0} at time T against a fixed value F_0(T;S_T) (the T-forward price).

We call the forward/contract price, the quantity F_t(T;S_T) and the forward/contract value, the quantity f_t(T,S_T).

Note that f_0(T,S_T)=0
f_T(T;S_T)=S_T-F_0(T;S_T)

One can show: F_t(T;S_T)=S_t/B(t,T) when there are no dividends/storage costs and AoA holds.
pg. 41 script

32
Q

What is the forward price (in absence of storage costs)?

E.g. a zero-coupon bond starting at forward maturity?

Forward value (absence of storage costs)?

A

F_t(T;S_T)=S_t/B(t,T)
Pf: Assuming strict inequalities yields arbitrage strategy. pg 41

F_t(t+j,B(t+j,t+M))=B(t,t+M)/B(t,t+j)

f_t(T;S_T)=S_t-F_0(T;S_T)*B(t,T)
Pf: “direct” AoA argument (bottom of 42)

BTW: in general f_t=(F_t-F_0)*B(t,T)

33
Q 
Prove FB_t( (T_i)_i, N, K)=N*B(t,T_0)
FB_t = floating-rate bond
A

lect 5, slide 4

34
Q

Define swap.

A

A swap is a comb. of contracts between a bank and parties A and B resp.. The bank pays x% to B in exchange for Euribor (floating rate), and the bank pays A Euribor (floating rate) in exchange for y%. Obviously: profit:=y-x>=0.
The (floating rate - fixed rate)-leg is called the payer IRS or payer forward-start (PFS).
The (fixed rate - floating rate)-leg is called the receiver IRS or receiver forward-start (RFS).

The holder (of RFS) receives the amount N*(T_i-T_{i-1})*K and pays N*(T_i-T_{i-1})*l(T_{i-1}-T_i), therefore:
RFS_t( (T_i)_i, N,K)=Sum( FRA_t(T_{i-1},T_i,N,K) ; i=1,..,n )
=...=N*[ Sum( dT_i * K * B(t,T_i) ) + B(t,T_n) - B(t,T_0) ] ~ bond with coupon payments - floating-rate bond.
(Floating-rate bond = bond paying l(T_{i-1},T_i)=:FB_t( (T_i)_i, N, K), so
RFS_t( (T_i)_i, N,K)=C*B_t( (T_i)_i, N,K)-FB_t( (T_i)_i, N, K)
35
Q

Define European call and put options.

Their value at time T.

Difference American call and put options?

A

A buyer buys the right (not obilgation) to buy an underlying (S_t)_t at time T at the price K.
C_T(T,K;S)=(S_T-K)^+

A buyer buys the right to sell an underlying (S_t)_t at time T for the strike K. (Seller has obligation to sell at T at K).
P_T(T,K;S)=(K-S_T)^+

American options can be exercises at any time between 0 and T.
C^A_t(T,K;S)=(S_t-K)^+, P^A_t(T,K;S)=(K-S_t)^+

36
Q

Define the no-dominance principle.

When does this hold?

A

ND states that if the values of two self-financing portfolios (i.e. X_0=Y_0=0) are a.s. ordered at time T, then they must be ordered similarly, at any time t in [0,T].
I.e. if X_0=Y_0=0 and P(X_T>=Y_T)=1, then P(X_t>=Y_t)=1 for all t in [0,T].
Similarly, if P(X_T=Y_T)=1, then P(X_t=Y_t)=1 for all t in [0,T].

37
Q 
Elaborate the following:
i) Bounds of C_t(T,K;S)?
ii) Properties of K onto C_t(T,K;S)
iii) C_t(T_1,K*B(T_1,T_2);S) vs. C_t((T_2,K;S)?
Properties of T onto C_t(T,K;S)?
iv) C_t(T,K;S) and Lipschitz continuity?
v) Asymptotic property?
A

Let AoA hold. European call option prices with underlying S satisfy P-a.s., the following properties:
i) for all (K,T,t) in [0,inf)x[0,inf)x[0,T]:
(S_t-KB(t,T))^+ =< C_t(T,K;S) =< S_t
ii) for all (T,t) in [0,inf)x[0,T], the map K onto C_t(t,K,S) is non-increasing and convex
iii) for all T_1, T_2, t, K in [0,inf)x[T_1,inf)x[0,T]x[0,inf)
C_t(T_1,KB(T_1,T_2);S)=

38
Q

Under what conditions do American and European call option values coincide?
i.e. C^A_t(T,K;S)=C_t(T,K;S)

A

When AoA holds, interest rates are positive and (K,T,t) in [0,inf)^2x[0,T].

39
Q

State the Call-Put parity result.

A

When AoA holds:

i) C_t(T,K;S)-P_t(T,K;S)=S_t-KB(t,T)
ii) S_t-K =< C^A_t(T,K;S)-P^A_t(T,K;S)=< S_t- KB(t,T)

40
Q 
Define A(R^n).
(script A)
A

A(R^n):={R^n-valued, F-adapted processes}

41
Q

Let AoA hold. Put option prices written on the underlying asset S satistfy:
i) (K,T,t) in [0,inf)^2x[0,T]:
(KB(t,T)-S_t)^+ =< P_t(T,K;S) =< KB(t,T)
ii) (T,K) in [0,inf)x[0,T], K onto P_t(T,K;S) is non-decr. and convex.
iii) (T,t,K_1,K_2) in [0,inf)x[0,T]x[0,inf)^2, we have:
|P_t(T,t,K_1;S)-P_t(T,K_2;S)|=<b></b>
Remember to add a card for forward contract with storage costs
Proof of no-dominance

A

..

Math MSc 1st Sem (5 decks)

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