Topic 2: Time value
of money
ECMI Discrete Financial Models
Rafał Weron
Capitalization vs. discounting
discounting
today
future
€100
€110
capitalization
2
Simple interest
• Calculated on the original principal only
– Accumulated interest from prior periods is not used in
calculations
• where
– PV – present value (price); FV – future value;
– r – (annual) interest rate; T – maturity (in years)
3
Compound interest
• Calculated each period on the original principal and
all interest accumulated during past periods
• where
– PV – present value (price); FV – future value;
– r – (annual) interest rate; T – maturity (in years)
4
Semiannual, quarterly compounding
• Compounding m-times per year
• where
– PV – present value (price); FV – future value;
– r – (annual) interest rate; T – maturity (in years);
– m – compounding frequency (per year)
5
Continuous compounding
• Continuous compounding, i.e. for m
• where
– PV – present value (price); FV – future value;
– r – (annual) interest rate; T – maturity (in years)
6
Compounding frequency vs. FV
• Let PV= €1000, r=10%, T=1 year. Then:
Compounding
m
Future value
annual
1
FV = 1000 (1+0.1) = €1100.00
semiannual
2
FV = 1000 (1+0.1/2)2 = €1102.50
quarterly
4
FV = 1000 (1+0.1/4)4 = €1103.81
monthly
12
FV = 1000 (1+0.1/12)12 = €1104.71
daily
365
FV = 1000 (1+0.1/365)365 = €1105.16
continuous

FV = 1000 e0.1 = €1105.17
7
Effective rate
•
•
•
•
•
•
For annual compounding re = 10%
For semiannual compounding re = 10.25%
For quarterly compounding re  10.38%
For monthly compounding re  10.47%
For daily compounding re  10.52%
For continuous compounding re  10.52%
8
Current value of a future cash flow
discounting
today
future
€?
€110
9
Current value of a future cash flow
• Simple interest
• Compound interest
• Continuous compounding
10
Discount factors
• Simple interest
• Compound interest
• Continuous compounding
11
Future value of a series of cash
flows
€10
€10
€?
12
Future value of a deferred annuity
• where
1
2
3
C
C
C
...
n
C
C – coupon; n – #coupon payments (n = mT); r – interest rate (r = rp.a./m)
• Actuarial name and notation for FV (with C=1):
annuity-immediate, s n
13
Future value of an annuity due
0
1
2
C
C
C
...
n-1
n
C
• where
C – coupon; n – #coupon payments (n = mT); r – interest rate (r = rp.a./m)
• Actuarial name and notation for FV (with C=1):
annuity-due, sn
14
Current value of a series of future
cash flows
€?
€10
€110
15
Current value of a deferred annuity
C
C
C
1
2
3
C
...
n
• where
C – coupon; n – #coupon payments (n = mT); r – interest rate (r = rp.a./m)
• Actuarial name and notation for PV (with C=1):
annuity-immediate, a n
16
Current value of an annuity due
C
C
C
1
2
C
...
n-1
n
• where
C – coupon; n – #coupon payments (n = mT); r – interest rate (r = rp.a./m)
• Actuarial name and notation for PV (with C=1):
annuity-due, an
17
Current value of a perpetuity
C
C
C
...
1
2
3
...
• where
C – coupon; r – interest rate (r = rp.a./m)
• Actuarial name and notation for PV (with C=1):
perpetuity-immediate, a 
18
Future value of a series of
irregular cash flows
• where
1
2
3
C1
C2
C3
...
n
Cn
– Ct – t-period coupon; n – #coupon payments (n = mT);
– r – interest rate (r = rp.a./m)
19
Current value of a series of
irregular cash flows
C1
C2
C3
1
2
3
Cn
...
n
• where
– Ct – t-period coupon; n – #coupon payments (n = mT);
– r – interest rate (r = rp.a./m)
20
Current value … in the case of
floating interest rates
C1
r1
C2
r2
C3
r3
Cn
rn
• where
– Ct – t-period coupon; rt – t-period interest rate
21
Net Present Value (NPV)
• NPV is the difference between
– current value of future cash flows generated by
the investment
– and the capital outlay at the beginning of the
investment (I0)
22
Internal Rate of Return (IRR)
• Mathematically the IRR is defined as a discount rate
that results in a NPV of zero of a series of cash flows
– Numerical methods have to be used to compute IRR
– IRR ‘assumes’ that the coupons are reinvested at the same
rate
23