Valuing an Interest Rate Cash Flow Series with Unequal Payments

Summary

The document works through a time-value-of-money question: find the level payment C that makes a series of inflows equivalent to a schedule of outflows at a stated annual interest rate. The answer lays out the cash flows by period, applies a discount factor to each inflow, sums their present values, and compares that total with the present value of the initial and recurring payments.

Using the discount factors and cash flow timing shown in the answer, the inflows have a reported present value of 4,501.58, while the payment series has a reported factor of 5.816013485 multiplied by C. Equating the values gives C of 774.00. This is a worked valuation example, not a trading strategy. The prompt specifies annual compounding, but the answer assumes a 30/360-style day count and uses factors based on 0.88 per period; readers should check that the chosen discount convention and timing match the intended problem before relying on the result.

Key ideas

• Value each dated inflow and outflow at a common valuation date using the chosen discount convention.
• The answer sums discounted inflows and expresses the payment series as a discount factor multiplied by C.
• Equating the two present values gives a reported payment amount of 774.00.
• The result depends on the timing assumptions and discount factors used in the worked schedule.
• The answer’s discount convention should be checked against the prompt’s annual compounding assumption.

Tags

# Cash flow diagram, interest rate inflow series


# Cash flow diagram, interest rate inflow series












I have a econ midterm coming up soon and stumbled upon this question. My approach is:

2C=800/(1.12^2)+1200/(1.12^6)=125.71 or C=1245.71/2=622.85

But I have a gut feeling this is wrong. I believe the answer is somewhere around $781.

Consider the following cash flow diagram. What value of C makes the inflow series equivalent to the outflow series at an interest rate of 12% compounded annually?

## Answer by Phil H (score 1)

https://quant.stackexchange.com/a/16627

Interest rate is 12%, we'll assume some kind of simple day count scheme like 30/360.

Cash flows and discount factors for C payer

```
t   disc.fact.   rcv.cf  pay.cf  rcv.pv
0   1            0       -2C     0
1   .88          800     0       704
2   .7744        800     -C      619.52
3   .681472      800     -C      545.18
4   .59969536    800     -C      479.76
5   .527731916   1200    -C      633.28
6   .464404086   1200    -C      557.28
7   .408675596   1200    -C      490.41
8   .359634524   1200    -C      431.56
```

Receive total PV = 4,501.58

Pay total PV:

$$ \text{PayPV} = -2C -C.(1-0.12)^2 - C.(1-0.12)^3 .. \\ = -C \left(2 + \sum_{n=2}^{8} f^n \right) \\ = -C (5.816013485)$$

where $f=(1-0.12)$. Equate for Par:

$$ \text{RcvPV} + \text{PayPV} = 0\\ 4501.58 - 5.816013485 C = 0 \\ C = 774.00$$

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.